# The Orbital Stability of Solitary Wave Solutions for the Generalized Gardner Equation and the Influence Caused by the Interactions between Nonlinear Terms

Authors: Wei-Guo Zhang; Xing-Qian Ling; Xiang Li; Shao-Wei Li
Journal: Complexity
 (2019)

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## Abstract

In this paper, the orbital stability of solitary wave solutions for the generalized Gardner equation is investigated. Firstly, according to the theory of orbital stability of Grillakis-Shatah-Strauss, a general conclusion is given to determine the orbital stability of solitary wave solutions. Furthermore, on the basis of the two bell-shaped solitary wave solutions of the equation, the explicit expressions of the orbital stability discriminants are deduced to give the orbitally stable and instable intervals for the two solitary waves as the wave velocity changing. Moreover, the influence caused by the interaction between two nonlinear terms is also discussed. From the conclusion, it can be seen that the influences caused by this interaction are apparently when0<p<4, which shows the complexity of this system with two nonlinear terms. Finally, by deriving the orbital stability discriminant d′′(c) in the form of Gaussian hypergeometric function, the numerical simulations of several main conclusions are given in this paper.


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## Body

## 1. Introduction

The Gardner equation originates from the study of the conservation law of KdV equation by Miura, Gardner, and Kruskal [1]. Because the Gardner equation possesses both quadratic and cubic term, many researchers recall the mixed KdV-MKdV equation,(1)ut+βuux+γu2ux+uxxx=0,as Gardner equation [2–5]. Equation (1) has important applications in many fields, such as nonlinear lattice, plasma physics, hydrodynamics, and solid state physics [6–9]. References [2–8] used different methods to solve the solitary wave solutions and some explicit solutions, such as the inverse scattering method and Hirota bilinear method [6, 7], the method of undetermined coefficients, and the extended tanh-function method [2, 3]. Recently, [10, 11] studied the generalized Gardner equation,(2)ut+α+βup+γu2pux+uxxx=0,p>0,and used sub-ODE method and G′/G expansion method to solve the solitary wave solutions and other solutions of (2).Equation (2) is the extension of the Gardner equation in [1]. When p=1,α=0, (2) turns into (1). When γ=0,α=0,β=1, (2) turns into the generalized KdV equation,(3)ut+upux+uxxx=0,p>0.This paper mainly studies the orbital stability of solitary wave solutions of generalized Gardner equation (2). The theory of the orbital stability of solitary wave solutions can be referred to in [12, 13]. We found that the stability of solitary wave solutions studied by the mostly above literatures has only one nonlinear term in the equation. For example, for the stability of the solitary wave solutions for the generalized KdV equation (3) (it can be seen as the special case of the generalized Gardner equation (2)), many literatures have studied on it. By using variational method, [14–19] studied the stability of solitary wave solutions of the generalized KdV equation (3) and got the following conclusion: if 0<p<4, the solitary wave solution of (3) is stable, and if p>4, it is instable. Meanwhile, [20] studied the orbital stability and instability of the solitary wave solutions of the nonlinear evolution equation which is much more complicated than (3) and obtained the following conclusion: if 0<p<4, the solitary wave solution of (3) is orbitally stable, and if p>4, it is orbitally instable. Besides, [21, 22] obtained some sufficient conditions when the solutions of the equation,(4)ut+upux+uxxx+γuxxxxx=0,p=const,p>0, are stable, by using variational method and Lyapunov method, respectively. The reason why previous literature focused on the stability of solitary wave solutions for the evolution equations with only one nonlinear term may be that the analysis to the existence and other relative problems for the solitary wave solutions will become more complex and difficult when there are two or more than two nonlinear terms in the equation.In this paper, based on our acknowledge on the solving the exact solitary wave solutions for nonlinear evolution equations, we give the exact solitary wave solutions of (2) with two nonlinear terms to directly state the existence. This reduces the high assumptions for the nonlinear parts of the equation when proving the existence of solitary wave solutions by analysis. Thus, it makes us apply the theory of orbital stability of Grillakis-Shatah-Strauss to the study of the orbital stability of solitary wave solutions for (2) possible.The generalized Gardner equation (2) has another nonlinear term γu2pux, comparing with the generalized KdV equation. So people want to know how about the orbital stability of the solitary wave solution of (2) naturally and the differences between the orbital stability of (2) and (3). In addition, there is only one bell-shaped solitary wave solution(5)ux-ct=cp+1p+22sech2pc2ξ1/p in (3), but there are two solutions u1(x-ct) and u2(x-ct) in (2) (as shown in Lemma 1 below). Naturally, people want to know whether the orbital stability of the solutions u1(x-ct) and u2(x-ct) is the same. In this paper, firstly by the application of the orbital stability theory of Grillakis-Shatah-Strauss, we will get a general conclusion to judge the orbital stability of the generalized Gardner equation. Further, on the basis of the two exact bell-shaped solitary wave solutions of (2), we can deduce the explicit expressions of the orbital stability discriminants of two solutions and then give the intervals which make two solutions stable as the wave velocity changing. And we will also discuss the influences on the stable intervals caused by the two nonlinear terms in (2). The results in this paper show that, when the whole order of the nonlinear terms in (2) is high, just as when p≥4, the solitary wave solutions of (2) with two nonlinear terms and (3) with a single nonlinear term possess the same stability, both of which are instable. But when 0<p<4, the orbital stability of (2) and (3) has apparent differences, which reflects the influences caused by two nonlinear terms of (2) on the orbital stability. Especially, in the case of 0<p<2, β<0, we prove the following result: there exists p0(1<p0<2); if p∈(0,p0), for any c∈(α,+∞), the solitary wave solution u1(x-ct) is orbitally stable; if p∈(p0,2), although the higher nonlinear term of (2) is less than 4, there is still a part of wave velocity interval in (α,+∞) making u1(x-ct) orbitally instable. The same conclusions of solitary wave solution u2(x-ct) can also be deduced in the situation of 0<p<2, β>0. All the above conclusions indicate the complexity caused by the interaction between two nonlinear terms of (2).To give explicit explanations of our conclusions, we use the transformation method to establish the orbital stability discriminantd′′(c) of the solitary wave solution in the form of Gauss hypergeometric function. Then the numerical simulating illustrations of several main conclusions can be got.It is worth pointing out the discussion of the influence of the orbital stability of solitary wave solutions of the equation caused by the interactions between two nonlinear terms; the expression of orbital stability discriminantd′′(c) in the form of Gaussian hypergeometric function and giving the numerical simulations of the wave velocity intervals of the stability are original in this paper.The method in this paper is an extension of Grillakis-Shatah-Strauss orbital stability theory. If an equation satisfies the conditions of Grillakis-Shatah-Strauss orbital stability theory and the solitary wave solutions of this equation can be solved, then the method in this paper can be referred to study the stability and the influence caused by the nonlinear terms.Here we give some related results for the generalized Gardner equation as follows.Lemma 1.
Assumec>α.
(1) Ifγ>0 or γ≥0 and β>0, then (2) has a bell-shaped solitary wave solution(6)u1x-ct=ψ1ξ1/p=A1sech2τ1/2x-ct2+B1sech2τ1/2x-ct1/p,among which(7)τ1=pc-α,A1=c-αp+1p+22p+1β22p+1+γc-αp+1p+22,B1=-1+βA1c-αp+1p+2.
(2) Letp make q1/p (q is negative) meaningful in R (such as p is odd), if γ>0, or γ≥0 and β<0, (2) also has bell-shaped solitary wave solution(8)u2x-ct=ψ2ξ1/p=A2sech2τ2/2x-ct2+B2sech2τ2/2x-ct1/p,among which(9)τ2=τ1,A2=-A1,B2=-2-B1.Remark 2.
The above results can be deduced by the conclusion in [10, 23].Remark 3.
It is easy to examine that the functionψ1(ξ) in (6) is always positive under the condition of Lemma 1(1), and the function ψ2(ξ) in (8) is always negative under the condition of Lemma 1(2).


## 2. General Conclusions of the Orbital Stability of the Solitary Wave Solution for the Generalized Gardner Equation (2)

The generalized Gardner equation (2) can be rewritten as the following Hamilton system:(10)dudt=JE′u,u∈X,where X=H2(R) with dual space X∗=H-2(R) and(11)J=∂∂x:DJ⊂X∗→X,(12)Eu=∫R12ux2-α2u2-βup+2p+1p+2-γu2p+22p+12p+2dx,(13)E′u=-αu-βup+1p+1-γu2p+12p+1-uxx.The inner product ofX is(14)f,g=∫Rf·g+f′·g′+f′′·g′′dx,∀f,g∈X. And there is a natural isomorphism between X and X∗, I:X→X∗, satisfying If,g=(f,g), where f,g=∫Rf·gdx, I=∂4/∂x4-∂2/∂x2+1.LetT be a unitary operator group with a single parameter on X, satisfying(15)Tsu·=u·-s,∀s∈R,u·∈X;obviously, T′(0)=-∂/∂x.We can deduceB=-1 by JB=T′(0), so it can be defined that(16)Qu=12Bu,u=-12∫Ru2dx,Q′u=Bu=-u,Q′′u=-1.From (15), we know solitary wave solutions (6), (8) of (2) in Lemma 1 can be rewritten as T(ct)ui(x)(i=1,2). Next, we consider the orbital stability of T(ct)ui(x). To avoid duplication, we denote ϕc(x) as one of u1(x) and u2(x).Definition 4 (see [21]).
Theϕc-orbit {T(ct)ϕc} is stable if for all ε<0 there exists δ>0 with the following property. If u0-ϕcX<δ and u(t) is a solution of (2) in some interval [0,t0) with u(0)=u0, then u(t) can be continued to a solution in [0,+∞) and(17)sup0<t<+∞⁡infs∈R⁡ut-TsϕcX<ε.Otherwise, the ϕc-orbit is called unstable.To prove the orbital stability of the solitary wave solutionT(ct)ϕc, we need to verify whether (2) and the solution T(ct)ϕc satisfy the three assumptions in [12].Firstly, from Theorems 1 and 2 in [24], we can deduce the existence of solutions for the initial value problem of system (10).Lemma 5 (see [19]).
Lets≥2, for any fixed u0∈Hs(R), there exists a unique solution u∈C([0,∞);Hs(R)) in (2), which satisfies u(0)=u0.It is easy to show thatE(u),Q(u) which are defined in (12) and (16) satisfy(18)Eut=Eu0=Eu0,Qut=Qu0=Qu0.Lemma 6.
ϕ c is the bounded solution of (2) and satisfies E′(ϕc)-cQ′(ϕc)=0.Proof.
By substitutingϕc into (2), we can obtain(19)αϕcx+βϕcpϕcx+γϕc2pϕcx+ϕcxxx-cϕcx=0.Integrating on both sides of (19), we can get(20)-βϕcp+1p+1-γϕc2p+12p+1-ϕcxx+c-αϕc=C,C=const.And when x→∞, ϕc,ϕcx,ϕcxx→0, then C=0; that is,(21)βϕcp+1p+1-γϕc2p+12p+1-ϕcxx+c-αϕc=0.So(22)E′ϕc-cQ′ϕc=-βϕcp+1p+1-γϕc2p+12p+1-ϕcxx+c-αϕc=0.Next we define operatorHc:X→X∗, Hc=E′′(ϕc)-cQ′′(ϕc), where(23)E′′u=-α-βup-γu2p-∂xx,Q′′u=-1. After calculating, we can get(24)Hc=-βϕcp-γϕc2p-∂xx+c-α,c>α.It is easy to know that Hc is a self-conjugate operator, Hc=Hc∗, which means I-1Hc is a bounded self-conjugate operator in X and the eigenvalues of Hc consist of the real number λ such that Hc-λI irreversible.From (19), we can know λ=0 is one eigenvalue of Hc,(25)HcT′0ϕcx=-Hcϕcx=0.Let Z′=kϕcx∣k∈R, which contains the core Z=u∈X∣Hcu=0 of Hc.Now we are ready to prove the following lemma.Lemma 7.
For eachc∈(c-α,+∞), Hc has only one negative single eigenvalue. Its kernel space is spanned by T′(0)ϕc, and its other eigenvalues are positive, bounded, and far away from 0.Proof.
Sincex=0 is the only zero of ϕcx, we know 0 is the second eigenvalue of Hc from Sturm-Liouville Theorem [25]. So Hc only has one negative eigenvalue -σ2, and its corresponding eigenfunction is denoted as χ; then Hχ=-σ2χ, χ,χ=1.
ForHc, when x→∞, ϕc→0, so -βϕcp-γϕc2p→0. From Wely essential spectrum Theorem [26], the essential spectrum of Hc satisfies essHc=[c-α,+∞), (c>α), so the desired result holds.From the above discussion, we can make a spectral decomposition toHc. Let (26)Z=k1ϕcx∣k1∈R,N=k2χ∣k2∈R,P=p∈X∣p,χ=p,ϕcx=0.For anyn(≠0)∈N, Hck2χ,k2χ=-k22σ2χ,χ=-k22σ2<0, so Hcn,n<0.For anyz(≠0)∈Z, Hck1ϕcx,k1ϕcx=0, so Hcz,z=0.For anyp(≠0)∈P, according to Lemma 3.2 in [12], for any real function p∈H1(R) which satisfies (p,χ)=(p,ϕcx)=0, there exists a positive β>0 making Hcp,p≥βpX2, where β is irrelevant to p; then Hcp,p>0.Therefore, spaceX can be decomposed into direct sum X=N⊕Z⊕P, where Z is the kernel space of Hc, N is a finite-dimension space, and P is a closed subspace.We defined(c):R→R as d(c)=E(ϕc)-cQ(ϕc). As a result of Lemmas 5–7, according to [12], we can get the following general conclusion of the solitary wave solution of (2).Theorem 8.
Letu1(x-ct), u2(x-ct) be the solitary wave solutions of the generalized Gardner equation (2) given in Lemma 1, among which coefficients p,α,β,γ and wave velocity satisfy the hypothesis in Lemma 1, respectively. Denote ϕc(c) as one of u1(x) and u2(x); then if d′′(c)>0, T(ct)ϕc(x) is orbitally stable; else if d′′(c)<0, T(ct)ϕc(x) is orbitally instable.Remark 9.
Since the skew-symmetric operatorJ=∂/∂x:D(J)(⊂X∗)→X is not onto, by directly using the conclusion in [20] or making similarly deduction, we can obtain the conclusion that if d′′(c)<0, T(ct)ϕc(x) is orbitally instable in Theorem 8.It can be inferred from Theorem8 that we can examine the symbol of d′′(c) to study the orbital stability of the solitary wave solution of the generalized Gardner equation (2). Here(27)d′c=E′ϕc,ϕc′-cQ′ϕc,ϕc′-Qϕc=-Qϕc=12∫Rϕc2dx.For convenience, we give the explicit expressions of the orbital stability discriminants of the solitary wave solution (6) and the solitary wave solution (8) of the generalized Gardner equation (2), respectively.Firstly, we consider the orbital stability discriminantd′′(c) of the solitary wave solution (6) for (2). The solitary wave solution (6) can be rewritten as(28)u1x-ct=A1sech2τ1/2x-ct2+B1sech2τ1/2x-ct1/p=1d1+d2cosh2d3x-ct1/p, among which(29)d1=B1A1=-β22p+1+γc-αp+1p+22+β2p+1c-αp+1p+22p+1<0,(30)d2=4A1=2β22p+1+γc-αp+1p+22c-αp+1p+22p+1>0,(31)d3=τ12=pc-α2>0.In (27), we take ϕc(x)=u1(x). Noting that cosh2(d3x) is an even function, we have(32)d′c=12∫R1d1+d2cosh2d3x2/pdx=∫0+∞1d1+d2cosh2d3x2/pdx.Due to(33)ddc1d1+d2cosh2d3x2/p=-2pd1′+d2′cosh2d3x+2d2d3′xcosh⁡d3xsinh⁡d3xd1+d2cosh2⁡d3x2/p+1=-2pd1′-d1d2′/d2+d2′/d2d1+d2cosh2d3x+2d2d3′xcosh⁡d3xsinh⁡d3xd1+d2cosh2⁡d3x2/p+1,and when p>0, for any fixed c>α, each term on the right hand of formula (33) converges in [0,+∞),(34)∫0+∞xcosh⁡d3xsinh⁡d3xd1+d2cosh2d3x2/p+1dx=12d2d3∫0+∞xdd1+d2cosh2d3xd1+d2cosh2⁡d3x2/p+1=p4d2d3∫0+∞1d1+d2cosh2⁡d3x2/pdx.So when p>0, for any fixed c>α, ∫0+∞d/dc1/d1+d2cosh2d3x2/pdx converges.For any real numbera,b>α, if wave velocity c satisfies α<a<c<b, pa-α/2<d3<pb-α/2, and cosh⁡pa-α/2x<cosh⁡(d3x) in (0,+∞), then there exists a constant K, independent of c, satisfying(35)0<1d1+d2cosh2d3x<1d1+d2cosh2pa-α/2x=4epa-αxe2pa-αx1+22d1/d2+1e-pa-αx+e-2pa-αx<4Kepa-αx.Ifc>α, ∫0+∞d/dc(1/d1+d2cosh2(d3x))2/pdx not only converges, but also uniformly converges in any closed subintervals in (α,+∞). Hence, according to (27), (33), and the derivation rule of the integral with parameter, we can deduce the orbital stability discriminant of the solitary wave solution (6) of (2),(36)d′′c=∫0+∞ddc1d1+d2cosh2d3x2/pdx=2p∫0+∞d1′-d1d2′/d2+d2′/d2d1+d2cosh2d3x+2d2d3′xcosh⁡d3xsinh⁡d3xd1+d2cosh2⁡d3x2/p+1dx=-2pd2′d2+d3′p2d3∫0+∞1d1+d2cosh2d3x2/pdx-2pd1′-d1d2′d2∫0+∞1d1+d2cosh2d3x2/p+1dx.To avoid confusion, we denote the stability discriminant of the solitary wave solution (8) for the generalized Gardner equation (2) as d¯(c). By taking ϕc(x)=u2(x) in (27), we have(37)d¯′c=12∫RA2sech2τ2/2x2+B2sech2τ2/2x2/pdx=∫0+∞1d¯1+d¯2cosh2d¯3x2/pdx,where d¯1=B2/A2, d¯2=2/A2, d¯3=τ2/2. According to (8) and (9), we have(38)d¯1=d1+d2,d¯2=-d2,d¯3=d3.Similar to the discussion of the solitary wave solution (6), since ∫0+∞d/dc(1/d¯1+d¯2cosh2(d¯3x))2/pdx not only converges, but also uniformly converges in any closed subintervals in (α,+∞) when c>α, then according to (37) and the derivation rule of the integral with parameter, we have the orbital stability discriminant of the solitary wave solution (8),(39)d¯′′c=∫0+∞ddc1d¯1+d¯2cosh2d¯3x2/pdx=-2pd¯2′d¯2+d¯3′p2d¯3∫0+∞1d¯1+d¯2cosh2d¯3x2/pdx-2pd¯1′-d¯1d¯2′d¯2∫0+∞1d¯1+d¯2cosh2d¯3x2/p+1dx.According to (38),(40)d¯1′-d¯1d¯2′d¯2=d1′-d1d2′d2,d¯2′d¯2=d2′d2,d¯3′d¯3=d3′d3.By substituting (40) into (39), we can get(41)d¯′′c=∫0+∞ddc1d¯1+d¯2cosh2d¯3x2/pdx=-2pd2′d2+d3′p2d3∫0+∞1d¯1+d¯2cosh2d¯3x2/pdx-2pd1′-d1d2′d2∫0+∞1d¯1+d¯2cosh2d¯3x2/p+1dx.


## 3. The Orbital Stability of the Bell-Shaped Solitary Wave Solution of the Generalized Gardner Equation (2) When p>2

### 3.1. The Orbital Stability of the Solitary Wave Solution (6) When p>2

#### 3.1.1. In the Case ofp>2, γ>0, β>0

Note that the orbital stability discriminant of the solitary wave solution (6) of (2) is (36); we can get (42)d2′=-2β22p+1+γc-αp+1p+22c-α2p+1p+22p+1β22p+1+γc-αp+1p+22,d1′=-12d2′+βc-αp+1p+2′=1c-α2p+1p+22β22p+1+γc-αp+1p+222p+1β22p+1+γc-αp+1p+22-β,(43)-2pd1′-d1d2′d2=βγp+2pc-αβ22p+1+γc-αp+1p+22From (43), we have that when β>0, γ>0, -2/p(d1′-d1d2′/d2)>0, and 0<d1+d2/d1+d2cosh2(d3x)<1(x≠0), then(44)∫0+∞1d1+d2cosh2d3x2/p+1dx<1d1+d2∫0+∞1d1+d2cosh2d3x2/pdx.Substituting (43) and (44) into (36), we can get when β>0, γ>0,(45)d′′c<-d3′d3-2pd1′+d2′d1+d2∫0+∞1d1+d2cosh2d3x2/pdx.In (45),(46)d1′+d2′d1+d2=-β2p+1+β22p+1+γc-αp+1p+222c-αβ22p+1+γc-αp+1p+22,d3′d3=12c-α>0,so the coefficients in the right hand of (45) satisfy(47)-d3′d3-d1′+d2′d1+d2=12c-α2pβ2p+1+β22p+1+γc-αp+1p+22β22p+1+γc-αp+1p+22-1=12pc-α-β2pp-42p+1-γc-αp+1p+22p-222β2p+1+p-2Fα,β,γ,c,pFα,β,γ,c,p, where F(α,β,γ,c,p) = β2(2p+1)+γ(c-α)(p+1)(p+2)2).We denote the numerator in the right hand of (47) as G1, which means(48)G1=-β2pp-42p+1-γc-αp+1p+22p-22.In (48), it can be shown that if γ>0, c>α, p≥4, then G1≤0, and d′′(c)<0 follows from (45) and (47). When 2<p<4, G1≤0 is equivalent to(49)c≥α+β2p4-p2p+1γp+1p+22p-22.So in the case of γ>0, β>0, c>α, if 2<p<4 and c≥α+β2p(4-p)(2p+1)/γ(p+1)p+22p-22, then G1≤0, and d′′(c)<0 follows from (45) and (47).We can get following Proposition10 from the above discussions.Proposition 10.
In the case ofγ>0, β>0, c>α, if p≥4, or 2<p<4 and c≥α+β2p(4-p)(2p+1)/γ(p+1)(p+2)2(p-2)2, the stability discriminant of the solitary wave solution (6) satisfies d′′(c)<0.


#### 3.1.2. In the Case ofγ>0, β<0, c>α

Sinceβ<0, from (43), we have -2/p(d1′-d1d2′/d2)<0, which implies that(50)-2pd1′-d1d2′d2∫0+∞1d1+d2cosh2d3x2/p+1dx<0.Therefore, according to (36), we can get when β<0,(51)d′′c<-2pd2′d2+d3′p2d3∫0+∞1d1+d2cosh2d3x2/pdx<0.After calculating, we can get that the coefficients in the right hand of (51) satisfy(52)-2pd2′d2+d3′p2d3=β22p+14-p+γc-αp+1p+222-p2pc-αβ22p+1+γc-αp+1p+22.We useG2 to denote the numerator of (52), which means(53)G2=β22p+14-p+γc-αp+1p+222-p.From (53), we can obtain the following conclusion: if γ>0, c>α, p≥4, then G2<0; if 2<p<4 and c≥α+β2(2p+1)(4-p)/γ(p+1)(p+2)2(p-2), then G2≤0. So we can get the following Proposition 11 from (51).Proposition 11.
In the case ofγ>0, β<0, c>α, if p≥4, or 2<p<4 and c≥α+β2(2p+1)(4-p)/γ(p+1)(p+2)2(p-2), the stability discriminant of the solitary wave solution (6) satisfies d′′(c)<0.


#### 3.1.3. The Orbital Stability of the Solitary Wave Solution (6) in the Case of 2<p<4, γ>0, β>0

From (43), we can get that -2/p(d1′-d1d2′/d2)>0 if β>0. So we only need to consider which value c takes; the first coefficient of (36) will be not less than 0; then we will have d′′(c)>0. From (52) and (53), the first coefficient of (36) can be rewritten as(54)-2pd2′d2+d3′p2d3=G22pc-αβ22p+1+γc-αp+1p+22.From (53) and (54), we can obtain that, if c satisfies(55)c≤α+β22p+14-pγp+1p+22p-2,then G2≥0. So we can deduce d′′(c)>0 from (36). And under this condition, we have Proposition 12.Proposition 12.
In the case of2<p<4, γ>0, β>0, if c satisfies(56)α<c≤α+β22p+14-pγp+1p+22p-2, then the stability discriminant of the solitary wave solution (6) satisfies d′′(c)>0.Due to(57)β2p2p+14-pγp+1p+22p-22>β22p+14-pγp+1p+22p-2,we can get Theorem 13 from Propositions 10–12 and Theorem 8.Theorem 13.
Assumep>2, γ>0, c>α.
(1) Ifp≥4, or 2<p<4 and c≥α+β2p(2p+1)(4-p)/γ(p+1)(p+2)2(p-2)2, the solitary wave solution (6) is orbitally instable no matter β>0 or β<0.
(2) In the case ofβ<0, if 2<p<4 and c≥α+β2(2p+1)(4-p)/γ(p+1)(p+2)2(p-2), the solitary wave solution (6) is orbitally instable.
(3) In the case of2<p<4,β>0, if α<c≤α+β2(2p+1)(4-p)/γ(p+1)(p+2)2(p-2), the solitary wave solution (6) is orbitally stable.


### 3.2. The Orbital Stability of the Solitary Wave Solution (8) When p>2

#### 3.2.1. In the Case ofp>2, γ>0, β<0

Note thatd¯2=-d2<0, and cosh⁡x≥1 in [0,+∞); combining this with (38), we can deduce the following conclusion: no matter β>0 or β<0, d¯1+d¯2cosh2(d¯3x)≤d¯1+d¯2=d1<0, which means 0<d1/d¯1+d¯2cosh2(d¯3x)<1(x≠0),(58)∫0+∞1d¯1+d¯2cosh2d¯3x2/pdx>∫0+∞1d¯1+d¯2cosh2d¯3x2/pd1d¯1+d¯2cosh2d¯3xdx.And from (43), we can deduce that -2/pd1(d1′-d1d2′/d2)>0 when γ>0, β<0. So from (58), we have(59)-2pd1d1′-d1d2′d2∫0+∞1d¯1+d¯2cosh2⁡d¯3x2/pdx>-2pd1′-d1d2′d2∫0+∞1d¯1+d¯2cosh2⁡d¯3x2/p+1dxThen we substitute (59) into (41) and get(60)d¯′′c<-2pd1′d1-d3′d3∫0+∞1d¯1+d¯2cosh2d¯3x2/pdx.Since(61)d1′d1=β2p+1-β22p+1+γc-αp+1p+222c-αβ22p+1+γc-αp+1p+22,then the coefficient in the right hand of (60) satisfies(62)-2pd1′d1-d3′d3=12c-α2pβ22p+1+γc-αp+1p+22-β2p+1β22p+1+γc-αp+1p+22-1=-12pc-αβ2p2p+1p-4+γc-αp+1p+22p-22p-2Fα,β,γ,c,p-2β2p+1Fα,β,γ,c,p,where F(α,β,γ,c,p) = β2(2p+1)+γ(c-α)(p+1)(p+2)2).Here we set the numerator in the right hand of (62) (63)G3=β2p2p+1p-4+γc-αp+1p+22p-22.Sinceβ<0, similar to (48) and (49), from (63), we can deduce that if γ>0,c>α,p≥4, or 2<p<4 and c≥α+β2p(2p+1)(4-p)/γ(p+1)(p+2)2(p-2)2, then G3≥0. And we can get Proposition 14 from (60) and (62).Proposition 14.
In the case ofγ>0, c>α, β<0, if p≥4, or 2<p<4 and c≥α+β2p(2p+1)(4-p)/γ(p+1)p+22(p-2)2, the stability discriminant of the solitary wave solution (8) satisfies d¯′′(c)<0.


#### 3.2.2. In the Case ofp>2, γ>0, β>0

From (43), we can deduce that -2/p(d1′-d1d2′/d2)>0 when β>0, ∫0+∞(1/d¯1+d¯2cosh2(d¯3x))2/p+1dx<0. So from (41), we have(64)d¯′′c<-2pd2′d2+d3′p2d3∫0+∞1d¯1+d¯2cosh2d¯3x2/pdx.Then we can deduce the following proposition from (52) and (53).Proposition 15.
In the case ofγ>0, c>α, β>0, if p≥4, or 2<p<4 and c≥α+β2(2p+1)(4-p)/γ(p+1)p+22(p-2), the stability discriminant of the solitary wave solution (8) satisfies d¯′′(c)<0.


#### 3.2.3. The Orbital Stability of the Solitary Wave Solution (8) When 2<p<4, γ>0, β<0

In the case ofγ>0, β<0, we have -2/p(d1′-d1d2′/d2)<0, d¯1+d¯2cosh2(d¯3x)<0, so the second term of (41) is greater than 0. And d¯′′(c)>0 only if the coefficient of the first term in (41) is greater than 0. That is, -2/pd2′/d2+d3′p/2d3≥0. And from the discussion in Section 3.1.3, we can deduce that if α<c≤α+β2(2p+1)(4-p)/γ(p+1)p+22(p-2), -2/p(d2′/d2+d3′p/2d3)≥0, so d¯′′(c)>0, which means the solitary wave solution (8) is orbitally stable.Combining with the above discussion and Propositions14 and 15, we can get the following theorem.Theorem 16.
Assumep>2,γ>0, and set p make q1/p (q is negative) meaningful in R.
(1) Ifp>4, or 2<p<4 and c≥α+β2p(2p+1)(4-p)/γ(p+1)p+22(p-2)2, the solitary wave solution (8) is orbitally instable no matter β>0 or β<0.
(2) In the case ofβ>0, if 2<p<4 and c≥α+β2(2p+1)(4-p)/γ(p+1)p+22(p-2), the solitary wave solution (8) is orbitally instable.
(3) In the case of2<p<4,β<0, if α<c≤α+β2(2p+1)(4-p)/γ(p+1)p+22(p-2), the solitary wave solution (8) is orbitally stable.


## 3.1. The Orbital Stability of the Solitary Wave Solution (6) When p>2

### 3.1.1. In the Case ofp>2, γ>0, β>0

Note that the orbital stability discriminant of the solitary wave solution (6) of (2) is (36); we can get (42)d2′=-2β22p+1+γc-αp+1p+22c-α2p+1p+22p+1β22p+1+γc-αp+1p+22,d1′=-12d2′+βc-αp+1p+2′=1c-α2p+1p+22β22p+1+γc-αp+1p+222p+1β22p+1+γc-αp+1p+22-β,(43)-2pd1′-d1d2′d2=βγp+2pc-αβ22p+1+γc-αp+1p+22From (43), we have that when β>0, γ>0, -2/p(d1′-d1d2′/d2)>0, and 0<d1+d2/d1+d2cosh2(d3x)<1(x≠0), then(44)∫0+∞1d1+d2cosh2d3x2/p+1dx<1d1+d2∫0+∞1d1+d2cosh2d3x2/pdx.Substituting (43) and (44) into (36), we can get when β>0, γ>0,(45)d′′c<-d3′d3-2pd1′+d2′d1+d2∫0+∞1d1+d2cosh2d3x2/pdx.In (45),(46)d1′+d2′d1+d2=-β2p+1+β22p+1+γc-αp+1p+222c-αβ22p+1+γc-αp+1p+22,d3′d3=12c-α>0,so the coefficients in the right hand of (45) satisfy(47)-d3′d3-d1′+d2′d1+d2=12c-α2pβ2p+1+β22p+1+γc-αp+1p+22β22p+1+γc-αp+1p+22-1=12pc-α-β2pp-42p+1-γc-αp+1p+22p-222β2p+1+p-2Fα,β,γ,c,pFα,β,γ,c,p, where F(α,β,γ,c,p) = β2(2p+1)+γ(c-α)(p+1)(p+2)2).We denote the numerator in the right hand of (47) as G1, which means(48)G1=-β2pp-42p+1-γc-αp+1p+22p-22.In (48), it can be shown that if γ>0, c>α, p≥4, then G1≤0, and d′′(c)<0 follows from (45) and (47). When 2<p<4, G1≤0 is equivalent to(49)c≥α+β2p4-p2p+1γp+1p+22p-22.So in the case of γ>0, β>0, c>α, if 2<p<4 and c≥α+β2p(4-p)(2p+1)/γ(p+1)p+22p-22, then G1≤0, and d′′(c)<0 follows from (45) and (47).We can get following Proposition10 from the above discussions.Proposition 10.
In the case ofγ>0, β>0, c>α, if p≥4, or 2<p<4 and c≥α+β2p(4-p)(2p+1)/γ(p+1)(p+2)2(p-2)2, the stability discriminant of the solitary wave solution (6) satisfies d′′(c)<0.


### 3.1.2. In the Case ofγ>0, β<0, c>α

Sinceβ<0, from (43), we have -2/p(d1′-d1d2′/d2)<0, which implies that(50)-2pd1′-d1d2′d2∫0+∞1d1+d2cosh2d3x2/p+1dx<0.Therefore, according to (36), we can get when β<0,(51)d′′c<-2pd2′d2+d3′p2d3∫0+∞1d1+d2cosh2d3x2/pdx<0.After calculating, we can get that the coefficients in the right hand of (51) satisfy(52)-2pd2′d2+d3′p2d3=β22p+14-p+γc-αp+1p+222-p2pc-αβ22p+1+γc-αp+1p+22.We useG2 to denote the numerator of (52), which means(53)G2=β22p+14-p+γc-αp+1p+222-p.From (53), we can obtain the following conclusion: if γ>0, c>α, p≥4, then G2<0; if 2<p<4 and c≥α+β2(2p+1)(4-p)/γ(p+1)(p+2)2(p-2), then G2≤0. So we can get the following Proposition 11 from (51).Proposition 11.
In the case ofγ>0, β<0, c>α, if p≥4, or 2<p<4 and c≥α+β2(2p+1)(4-p)/γ(p+1)(p+2)2(p-2), the stability discriminant of the solitary wave solution (6) satisfies d′′(c)<0.


### 3.1.3. The Orbital Stability of the Solitary Wave Solution (6) in the Case of 2<p<4, γ>0, β>0

From (43), we can get that -2/p(d1′-d1d2′/d2)>0 if β>0. So we only need to consider which value c takes; the first coefficient of (36) will be not less than 0; then we will have d′′(c)>0. From (52) and (53), the first coefficient of (36) can be rewritten as(54)-2pd2′d2+d3′p2d3=G22pc-αβ22p+1+γc-αp+1p+22.From (53) and (54), we can obtain that, if c satisfies(55)c≤α+β22p+14-pγp+1p+22p-2,then G2≥0. So we can deduce d′′(c)>0 from (36). And under this condition, we have Proposition 12.Proposition 12.
In the case of2<p<4, γ>0, β>0, if c satisfies(56)α<c≤α+β22p+14-pγp+1p+22p-2, then the stability discriminant of the solitary wave solution (6) satisfies d′′(c)>0.Due to(57)β2p2p+14-pγp+1p+22p-22>β22p+14-pγp+1p+22p-2,we can get Theorem 13 from Propositions 10–12 and Theorem 8.Theorem 13.
Assumep>2, γ>0, c>α.
(1) Ifp≥4, or 2<p<4 and c≥α+β2p(2p+1)(4-p)/γ(p+1)(p+2)2(p-2)2, the solitary wave solution (6) is orbitally instable no matter β>0 or β<0.
(2) In the case ofβ<0, if 2<p<4 and c≥α+β2(2p+1)(4-p)/γ(p+1)(p+2)2(p-2), the solitary wave solution (6) is orbitally instable.
(3) In the case of2<p<4,β>0, if α<c≤α+β2(2p+1)(4-p)/γ(p+1)(p+2)2(p-2), the solitary wave solution (6) is orbitally stable.


## 3.1.1. In the Case ofp>2, γ>0, β>0

Note that the orbital stability discriminant of the solitary wave solution (6) of (2) is (36); we can get (42)d2′=-2β22p+1+γc-αp+1p+22c-α2p+1p+22p+1β22p+1+γc-αp+1p+22,d1′=-12d2′+βc-αp+1p+2′=1c-α2p+1p+22β22p+1+γc-αp+1p+222p+1β22p+1+γc-αp+1p+22-β,(43)-2pd1′-d1d2′d2=βγp+2pc-αβ22p+1+γc-αp+1p+22From (43), we have that when β>0, γ>0, -2/p(d1′-d1d2′/d2)>0, and 0<d1+d2/d1+d2cosh2(d3x)<1(x≠0), then(44)∫0+∞1d1+d2cosh2d3x2/p+1dx<1d1+d2∫0+∞1d1+d2cosh2d3x2/pdx.Substituting (43) and (44) into (36), we can get when β>0, γ>0,(45)d′′c<-d3′d3-2pd1′+d2′d1+d2∫0+∞1d1+d2cosh2d3x2/pdx.In (45),(46)d1′+d2′d1+d2=-β2p+1+β22p+1+γc-αp+1p+222c-αβ22p+1+γc-αp+1p+22,d3′d3=12c-α>0,so the coefficients in the right hand of (45) satisfy(47)-d3′d3-d1′+d2′d1+d2=12c-α2pβ2p+1+β22p+1+γc-αp+1p+22β22p+1+γc-αp+1p+22-1=12pc-α-β2pp-42p+1-γc-αp+1p+22p-222β2p+1+p-2Fα,β,γ,c,pFα,β,γ,c,p, where F(α,β,γ,c,p) = β2(2p+1)+γ(c-α)(p+1)(p+2)2).We denote the numerator in the right hand of (47) as G1, which means(48)G1=-β2pp-42p+1-γc-αp+1p+22p-22.In (48), it can be shown that if γ>0, c>α, p≥4, then G1≤0, and d′′(c)<0 follows from (45) and (47). When 2<p<4, G1≤0 is equivalent to(49)c≥α+β2p4-p2p+1γp+1p+22p-22.So in the case of γ>0, β>0, c>α, if 2<p<4 and c≥α+β2p(4-p)(2p+1)/γ(p+1)p+22p-22, then G1≤0, and d′′(c)<0 follows from (45) and (47).We can get following Proposition10 from the above discussions.Proposition 10.
In the case ofγ>0, β>0, c>α, if p≥4, or 2<p<4 and c≥α+β2p(4-p)(2p+1)/γ(p+1)(p+2)2(p-2)2, the stability discriminant of the solitary wave solution (6) satisfies d′′(c)<0.


## 3.1.2. In the Case ofγ>0, β<0, c>α

Sinceβ<0, from (43), we have -2/p(d1′-d1d2′/d2)<0, which implies that(50)-2pd1′-d1d2′d2∫0+∞1d1+d2cosh2d3x2/p+1dx<0.Therefore, according to (36), we can get when β<0,(51)d′′c<-2pd2′d2+d3′p2d3∫0+∞1d1+d2cosh2d3x2/pdx<0.After calculating, we can get that the coefficients in the right hand of (51) satisfy(52)-2pd2′d2+d3′p2d3=β22p+14-p+γc-αp+1p+222-p2pc-αβ22p+1+γc-αp+1p+22.We useG2 to denote the numerator of (52), which means(53)G2=β22p+14-p+γc-αp+1p+222-p.From (53), we can obtain the following conclusion: if γ>0, c>α, p≥4, then G2<0; if 2<p<4 and c≥α+β2(2p+1)(4-p)/γ(p+1)(p+2)2(p-2), then G2≤0. So we can get the following Proposition 11 from (51).Proposition 11.
In the case ofγ>0, β<0, c>α, if p≥4, or 2<p<4 and c≥α+β2(2p+1)(4-p)/γ(p+1)(p+2)2(p-2), the stability discriminant of the solitary wave solution (6) satisfies d′′(c)<0.


## 3.1.3. The Orbital Stability of the Solitary Wave Solution (6) in the Case of 2<p<4, γ>0, β>0

From (43), we can get that -2/p(d1′-d1d2′/d2)>0 if β>0. So we only need to consider which value c takes; the first coefficient of (36) will be not less than 0; then we will have d′′(c)>0. From (52) and (53), the first coefficient of (36) can be rewritten as(54)-2pd2′d2+d3′p2d3=G22pc-αβ22p+1+γc-αp+1p+22.From (53) and (54), we can obtain that, if c satisfies(55)c≤α+β22p+14-pγp+1p+22p-2,then G2≥0. So we can deduce d′′(c)>0 from (36). And under this condition, we have Proposition 12.Proposition 12.
In the case of2<p<4, γ>0, β>0, if c satisfies(56)α<c≤α+β22p+14-pγp+1p+22p-2, then the stability discriminant of the solitary wave solution (6) satisfies d′′(c)>0.Due to(57)β2p2p+14-pγp+1p+22p-22>β22p+14-pγp+1p+22p-2,we can get Theorem 13 from Propositions 10–12 and Theorem 8.Theorem 13.
Assumep>2, γ>0, c>α.
(1) Ifp≥4, or 2<p<4 and c≥α+β2p(2p+1)(4-p)/γ(p+1)(p+2)2(p-2)2, the solitary wave solution (6) is orbitally instable no matter β>0 or β<0.
(2) In the case ofβ<0, if 2<p<4 and c≥α+β2(2p+1)(4-p)/γ(p+1)(p+2)2(p-2), the solitary wave solution (6) is orbitally instable.
(3) In the case of2<p<4,β>0, if α<c≤α+β2(2p+1)(4-p)/γ(p+1)(p+2)2(p-2), the solitary wave solution (6) is orbitally stable.


## 3.2. The Orbital Stability of the Solitary Wave Solution (8) When p>2

### 3.2.1. In the Case ofp>2, γ>0, β<0

Note thatd¯2=-d2<0, and cosh⁡x≥1 in [0,+∞); combining this with (38), we can deduce the following conclusion: no matter β>0 or β<0, d¯1+d¯2cosh2(d¯3x)≤d¯1+d¯2=d1<0, which means 0<d1/d¯1+d¯2cosh2(d¯3x)<1(x≠0),(58)∫0+∞1d¯1+d¯2cosh2d¯3x2/pdx>∫0+∞1d¯1+d¯2cosh2d¯3x2/pd1d¯1+d¯2cosh2d¯3xdx.And from (43), we can deduce that -2/pd1(d1′-d1d2′/d2)>0 when γ>0, β<0. So from (58), we have(59)-2pd1d1′-d1d2′d2∫0+∞1d¯1+d¯2cosh2⁡d¯3x2/pdx>-2pd1′-d1d2′d2∫0+∞1d¯1+d¯2cosh2⁡d¯3x2/p+1dxThen we substitute (59) into (41) and get(60)d¯′′c<-2pd1′d1-d3′d3∫0+∞1d¯1+d¯2cosh2d¯3x2/pdx.Since(61)d1′d1=β2p+1-β22p+1+γc-αp+1p+222c-αβ22p+1+γc-αp+1p+22,then the coefficient in the right hand of (60) satisfies(62)-2pd1′d1-d3′d3=12c-α2pβ22p+1+γc-αp+1p+22-β2p+1β22p+1+γc-αp+1p+22-1=-12pc-αβ2p2p+1p-4+γc-αp+1p+22p-22p-2Fα,β,γ,c,p-2β2p+1Fα,β,γ,c,p,where F(α,β,γ,c,p) = β2(2p+1)+γ(c-α)(p+1)(p+2)2).Here we set the numerator in the right hand of (62) (63)G3=β2p2p+1p-4+γc-αp+1p+22p-22.Sinceβ<0, similar to (48) and (49), from (63), we can deduce that if γ>0,c>α,p≥4, or 2<p<4 and c≥α+β2p(2p+1)(4-p)/γ(p+1)(p+2)2(p-2)2, then G3≥0. And we can get Proposition 14 from (60) and (62).Proposition 14.
In the case ofγ>0, c>α, β<0, if p≥4, or 2<p<4 and c≥α+β2p(2p+1)(4-p)/γ(p+1)p+22(p-2)2, the stability discriminant of the solitary wave solution (8) satisfies d¯′′(c)<0.


### 3.2.2. In the Case ofp>2, γ>0, β>0

From (43), we can deduce that -2/p(d1′-d1d2′/d2)>0 when β>0, ∫0+∞(1/d¯1+d¯2cosh2(d¯3x))2/p+1dx<0. So from (41), we have(64)d¯′′c<-2pd2′d2+d3′p2d3∫0+∞1d¯1+d¯2cosh2d¯3x2/pdx.Then we can deduce the following proposition from (52) and (53).Proposition 15.
In the case ofγ>0, c>α, β>0, if p≥4, or 2<p<4 and c≥α+β2(2p+1)(4-p)/γ(p+1)p+22(p-2), the stability discriminant of the solitary wave solution (8) satisfies d¯′′(c)<0.


### 3.2.3. The Orbital Stability of the Solitary Wave Solution (8) When 2<p<4, γ>0, β<0

In the case ofγ>0, β<0, we have -2/p(d1′-d1d2′/d2)<0, d¯1+d¯2cosh2(d¯3x)<0, so the second term of (41) is greater than 0. And d¯′′(c)>0 only if the coefficient of the first term in (41) is greater than 0. That is, -2/pd2′/d2+d3′p/2d3≥0. And from the discussion in Section 3.1.3, we can deduce that if α<c≤α+β2(2p+1)(4-p)/γ(p+1)p+22(p-2), -2/p(d2′/d2+d3′p/2d3)≥0, so d¯′′(c)>0, which means the solitary wave solution (8) is orbitally stable.Combining with the above discussion and Propositions14 and 15, we can get the following theorem.Theorem 16.
Assumep>2,γ>0, and set p make q1/p (q is negative) meaningful in R.
(1) Ifp>4, or 2<p<4 and c≥α+β2p(2p+1)(4-p)/γ(p+1)p+22(p-2)2, the solitary wave solution (8) is orbitally instable no matter β>0 or β<0.
(2) In the case ofβ>0, if 2<p<4 and c≥α+β2(2p+1)(4-p)/γ(p+1)p+22(p-2), the solitary wave solution (8) is orbitally instable.
(3) In the case of2<p<4,β<0, if α<c≤α+β2(2p+1)(4-p)/γ(p+1)p+22(p-2), the solitary wave solution (8) is orbitally stable.


## 3.2.1. In the Case ofp>2, γ>0, β<0

Note thatd¯2=-d2<0, and cosh⁡x≥1 in [0,+∞); combining this with (38), we can deduce the following conclusion: no matter β>0 or β<0, d¯1+d¯2cosh2(d¯3x)≤d¯1+d¯2=d1<0, which means 0<d1/d¯1+d¯2cosh2(d¯3x)<1(x≠0),(58)∫0+∞1d¯1+d¯2cosh2d¯3x2/pdx>∫0+∞1d¯1+d¯2cosh2d¯3x2/pd1d¯1+d¯2cosh2d¯3xdx.And from (43), we can deduce that -2/pd1(d1′-d1d2′/d2)>0 when γ>0, β<0. So from (58), we have(59)-2pd1d1′-d1d2′d2∫0+∞1d¯1+d¯2cosh2⁡d¯3x2/pdx>-2pd1′-d1d2′d2∫0+∞1d¯1+d¯2cosh2⁡d¯3x2/p+1dxThen we substitute (59) into (41) and get(60)d¯′′c<-2pd1′d1-d3′d3∫0+∞1d¯1+d¯2cosh2d¯3x2/pdx.Since(61)d1′d1=β2p+1-β22p+1+γc-αp+1p+222c-αβ22p+1+γc-αp+1p+22,then the coefficient in the right hand of (60) satisfies(62)-2pd1′d1-d3′d3=12c-α2pβ22p+1+γc-αp+1p+22-β2p+1β22p+1+γc-αp+1p+22-1=-12pc-αβ2p2p+1p-4+γc-αp+1p+22p-22p-2Fα,β,γ,c,p-2β2p+1Fα,β,γ,c,p,where F(α,β,γ,c,p) = β2(2p+1)+γ(c-α)(p+1)(p+2)2).Here we set the numerator in the right hand of (62) (63)G3=β2p2p+1p-4+γc-αp+1p+22p-22.Sinceβ<0, similar to (48) and (49), from (63), we can deduce that if γ>0,c>α,p≥4, or 2<p<4 and c≥α+β2p(2p+1)(4-p)/γ(p+1)(p+2)2(p-2)2, then G3≥0. And we can get Proposition 14 from (60) and (62).Proposition 14.
In the case ofγ>0, c>α, β<0, if p≥4, or 2<p<4 and c≥α+β2p(2p+1)(4-p)/γ(p+1)p+22(p-2)2, the stability discriminant of the solitary wave solution (8) satisfies d¯′′(c)<0.


## 3.2.2. In the Case ofp>2, γ>0, β>0

From (43), we can deduce that -2/p(d1′-d1d2′/d2)>0 when β>0, ∫0+∞(1/d¯1+d¯2cosh2(d¯3x))2/p+1dx<0. So from (41), we have(64)d¯′′c<-2pd2′d2+d3′p2d3∫0+∞1d¯1+d¯2cosh2d¯3x2/pdx.Then we can deduce the following proposition from (52) and (53).Proposition 15.
In the case ofγ>0, c>α, β>0, if p≥4, or 2<p<4 and c≥α+β2(2p+1)(4-p)/γ(p+1)p+22(p-2), the stability discriminant of the solitary wave solution (8) satisfies d¯′′(c)<0.


## 3.2.3. The Orbital Stability of the Solitary Wave Solution (8) When 2<p<4, γ>0, β<0

In the case ofγ>0, β<0, we have -2/p(d1′-d1d2′/d2)<0, d¯1+d¯2cosh2(d¯3x)<0, so the second term of (41) is greater than 0. And d¯′′(c)>0 only if the coefficient of the first term in (41) is greater than 0. That is, -2/pd2′/d2+d3′p/2d3≥0. And from the discussion in Section 3.1.3, we can deduce that if α<c≤α+β2(2p+1)(4-p)/γ(p+1)p+22(p-2), -2/p(d2′/d2+d3′p/2d3)≥0, so d¯′′(c)>0, which means the solitary wave solution (8) is orbitally stable.Combining with the above discussion and Propositions14 and 15, we can get the following theorem.Theorem 16.
Assumep>2,γ>0, and set p make q1/p (q is negative) meaningful in R.
(1) Ifp>4, or 2<p<4 and c≥α+β2p(2p+1)(4-p)/γ(p+1)p+22(p-2)2, the solitary wave solution (8) is orbitally instable no matter β>0 or β<0.
(2) In the case ofβ>0, if 2<p<4 and c≥α+β2(2p+1)(4-p)/γ(p+1)p+22(p-2), the solitary wave solution (8) is orbitally instable.
(3) In the case of2<p<4,β<0, if α<c≤α+β2(2p+1)(4-p)/γ(p+1)p+22(p-2), the solitary wave solution (8) is orbitally stable.


## 4. The Orbital Stability of the Generalized Gardner Equation (2) When 0<p≤2

### 4.1. The Orbital Stability of the Solitary Wave Solution (6) of the Generalized Gardner Equation (2) When 0<p≤2

#### 4.1.1. The Simplification of the Orbital Stability Discriminantd′′(c) of the Solitary Wave Solution (6) When 0<p≤2

Since0<d1+d2cosh2(d3x)→∞(x→∞), in the case of 0<p≤2, (1/d1+d2cosh2(d3x))2/p+1 converges to the bounded function which converges to 0 when x→∞ or constant 1. So we can apply Mean Value Theorem of Integrals to the orbital stability discriminant (36) of the solitary wave solution (6). Then there exists x∗∈R, such that(65)d′′c=-2p1d1+d2cosh2d3x∗2/p-1·Ic,where(66)Ic=d2′d2+d3′p2d3∫0+∞dxd1+d2cosh2d3x+d1′-d1d2′d2∫0+∞dxd1+d2cosh2d3x2.Sincecosh2(d3x)=1/2cosh⁡(2d3x)+1/2, then we have(67)∫0+∞dxd1+d2cosh2d3x2=4d22∫0+∞dxcosh2d3x+2d1/d2+12.By making the substitution cosh⁡(2d3x)=z,zx=2d3z2-1 to the above formula, we can get(68)∫0+∞dxd1+d2cosh2d3x2=2d22d3∫1+∞dzz+2d1/d2+12z2-1=-12d1d3d1+d2-2d1+d24d3-d1d1+d23/2sgn⁡d2π2-arctan⁡2d1+d22-d1d1+d2. By the same substitution, we can get(69)∫0+∞dxd1+d2cosh2d3x=1d2d3∫1+∞dzz+2d1/d2+1z2-1=-12d3-d1d1+d2sgn⁡d2π2-arctan⁡2d1+d22-d1d1+d2.By substituting (68) and (69) into (66), and noticing that d2>0, then we have(70)Ic=-12d22d3d1′-d1d2′d2d22d1d2+d1+d2′d2+d3′p2d3+2d1+d22d1d2+d1d1′-d1d2′d21d3-4d1d2+d1π2-arctan⁡2d1+d22-d1d1+d2.By simple calculating, we have(71)d2′d2+d3′p2d3=-β22p+12c-αβ22p+1+γc-αp+1p+22+12c-αp2-1,(72)2d1+d2d1d1+d2=-2β2p+1γp+2,(73)d32d1+d2=βpc-αp+1p+2,(74)d1′-d1d2′d2=-βγp+22c-αβ22p+1+γc-αp+1p+22,(75)d22d1d2+d1=-4β22p+1+γc-αp+1p+22γc-αp+1p+22.Then by substituting (71)-(75) into (70), we have(76)Ic=-βd22d3p+1p+2c-α2-1-p2p+1p+22βpc-α2d1+d2-4d1d2+d1π2-arctan⁡2d1+d2-4d1d1+d2.Notice that(77)M1=2d1+d2-4d1d2+d1π2-arctan⁡2d1+d2-4d1d1+d2,k=β2p+1p+2γp+1,and then(78)2d1+d2-4d1d2+d1=β2p+1p+2γp+1·1c-α=kc-α,(79)M1=kc-απ2-arctan⁡kc-α.Substituting (77)-(79) into (76), then we have(80)Ic=-βp+1p+22p+12pc-αβ22p+1+γc-αp+1p+22-1-p2p+1p+22βpc-αM1=-p+1p+22pc-αβ2p+1β22p+1+γc-αp+1p+22+2-p2βM1,so (65) can be rewritten as(81)d′′c=p+1p+2βp2c-α1d1+d2cosh2d3x∗2/p-1β22p+1β22p+1+γc-αp+1p+22+2-p2M1.From Lemma1, we know that ψ1(ξ)=1/d1+d2cosh2d3x∗ is always positive in (81), and d1<0, d2>0, d3>0 can be given by (29), (30), and (31), respectively.


#### 4.1.2. The Orbital Stability of the Solitary Wave Solution (6)

(1) Case of p = 2. From (81), we can get the following: if β>0, then d′′(c)>0; if β<0, then d′′(c)<0. So the solitary wave solution (6) is orbitally stable when β>0 and orbitally instable when β<0 in the case of p=2.(2) Case of 0 < p < 2, β>0. Firstly, we consider M1 in (80) and (81). When β>0, we easily have k>0,M1>0 from (77) and (79). Then we can get d′′(c)>0 from (81), and the solitary wave solution (6) of (2) is orbitally stable.(3) Case of 0 < p < 2, β<0. From (77), it is easy to know k<0 when β<0. So M1∈(-∞,0) when c∈(α,+∞). Now we substitute (77) into (81) and set x=k/c-α; then (81) can be rewritten as(82)d′′c=p+1p+2kβp2c-α1d1+d2cosh2d3x∗2/p-1xx2+1+2-p2π2-arctan⁡x.From (82), we can deduce that, in the case of β<0 (now k<0), d′′(c)>0 is equivalent to h1(x)>0. Here(83)h1x=xx2+1+2-p2π2-arctan⁡x,x∈-∞,0,and(84)h1′x=12x2+12p-4x2+p,x∈-∞,0.From h1′(x)=0, we can get(85)x0=-p4-p. It is easy to obtain that if x∈(-∞,x0), then h1′(x)<0, and if x∈(x0,0), then h1′(x)>0. So h1(x) is monotone decreasing if x∈(-∞,x0), monotone increasing if x∈(x0,0), and h1(x) take the minimum value if x=x0. And we have(86)limx→0⁡h1x=limx→0⁡xx2+1+2-p2π2-arctan⁡x=2-p4π,limx→+∞⁡h1x=limx→+∞⁡xx2+1+2-p2π2-arctan⁡x=2-p2π.So according to Intermediate Value Theorem of the continuous function, we can get the following proposition.Proposition 17.
Set0<p<2, β<0.
(1) If the minimum valueh1(x0)>0, then h1(x)>0 for any x∈(-∞,0).
(2) If the minimumh1(x0)<0, then there exists x1,x2 (zero points of h1(x)=0); if x∈(-∞,x1) and x∈(x2,0), then h1(x)>0; if x∈(x1,x2), then h1(x)<0.Sincex=k/c-α(k<0), according to Proposition 17 and (82) and (83), we have the following proposition.Proposition 18.
Set0<p<2,β<0.
(1) If minimumh1(x0)<0, then d′′(c)>0 for any c∈(α,+∞); the solitary wave solution (6) of (2) is orbitally stable in (α,+∞).
(2) If minimumh1(x0)<0, then there exists x1,x2∈(-∞,0) (the zero points of h1(x)=0); if c∈(α,α+k2/x12)∪(α+k2/x22,+∞), then d′′(c)>0; the solitary wave solution (6) is orbitally stable; if c∈(α+k2/x12,α+k2/x22), then d′′(c)<0; the solitary wave solution (6) is orbitally instable.Now substitutingx0=-p/4-p into h(x), we can get the expression of h1(x0) about p,(87)h1x0=-p4-p4+2-p2π2+arctan⁡p4-p,p∈0,2.Sincelimp→0h1(x0)=π/2, limp→1h1(x0)=4π-33/12, limp→2⁡h1(x0)=-1/2, and h1(x0) is monotone decreasing about p in (0,2); from Intermediate Value Theorem of the continuous function, we can deduce that there exists p0∈(1,2) making the minimum h1(x0)=0 (from (82); we can know p0 also makes d′′(c)=0). Then for any p∈(0,p0), the minimum value h1(x0)>0, while, for any p∈(p0,2), the minimum value h1(x0)<0. Combining the above with Proposition 18, we can deduce the following proposition.Proposition 19.
We set0<p<2, β<0, and denote p0(1<p0<2) as the zero point of h1(x0)=0 in (87), and then we have the following.
(1) Whenp∈(0,p0) (now h1(x0)>0), for any c∈(α,+∞),d′′(c)>0, the solitary wave solution (6) of equation (2) is orbitally stable in (α,+∞).
(2) Whenp∈(p0,2) (now h1(x0)<0), there exists x1(p),x2(p) (the zero points of h1(x)=0); if c∈(α,α+k2/x12(p))∪(α+k2/x22(p),+∞), the solitary wave solution (6) is orbitally stable; if c∈(α+k2/x12(p),α+k2/x22(p)), the solitary wave solution (6) is orbitally instable.Now we setc1(p)=α+k2/x12(p), c2(p)=α+k2/x22(p). Through the analysis, we can get the following theorem.Theorem 20.
Set0<p≤2,γ>0,c>α, for the solitary wave solution (6), we have the following conclusions.
(1) In the case ofp=2, the solitary wave solution (6) is orbitally stable when β>0 and orbitally instable when β<0.
(2) In the case of0<p<2, β>0, the solitary wave solution (6) is orbitally stable for any c∈(α,+∞).
(3) In the case of0<p<2,β<0, there exists p0(1<p0<2) such that we have the following.
① Ifp∈(0,p0), for any c∈(α,+∞), the solitary wave solution (6) is orbitally stable.
② Whenp∈(p0,2), there exists c1(p),c2(p); if c∈(α,c1(p))∪(c2(p),+∞), the solitary wave solution (6) is orbitally stable; if c∈(c1(p),c2(p)), the solitary wave solution (6) is orbitally instable.


### 4.2. The Orbital Stability of the Solitary Wave Solution (8) of the Generalized Gardner Equation (2) When 0<p<2

#### 4.2.1. The Simplification of the Orbital Stability Discriminantd¯′′(c) of the Solitary Wave Solution (8) When 0<p<2

Sinceψ2(ξ) is always negative in the solitary wave solution (8), we assume that p makes q1/p (q is negative) meaningful in R (The case of p=2 is out of our discussion). In the case of 0<p<2, we make simplification for the discriminant d¯′′(c) of the solitary wave solution (8). Similar to the discussion of the orbital stability of u1(x), by applying Mean Value Theorem of Integrals to (41), we can get that there exists x¯∈R such that(88)d¯′′c<-2p1d¯1+d¯2cosh2d¯3x2/p-1·I¯c,x¯∈R,where(89)I¯c=d2′d2+d3′p2d3∫0+∞dxd¯1+d¯2cosh2d¯3x+d1′-d1d2′d2∫0+∞dxd¯1+d¯2cosh2d¯3x2.Sinced¯i and di(i=1,2,3) satisfy (38) and (40), according to (71)-(75), we can obtain(90)I¯c=-βd22d3p+1p+2+1-p2p+1p+22βpc-α2d1+d2-4d1d2+d1π2+arctan⁡2d1+d2-4d1d1+d2.Due to (78) and (79), we set(91)M2=2d1+d2-4d1d2+d1π2+arctan⁡2d1+d2-4d1d1+d2,and then we have(92)M2=kc-απ2+arctan⁡kc-α.So we have(93)I¯c=-βp+1p+22p+12pc-αβ22p+1+γc-αp+1p+22+1-p2p+1p+22βpc-αM2=-p+1p+22βpc-αβ22p+1β22p+1+γc-αp+1p+22-2-p2M2,(94)d¯′′c=p+1p+2βp2c-α1d¯1+d¯2cosh2d¯3x¯∗2/p-1β22p+1β22p+1+γc-αp+1p+22-2-p2M2.According to Lemma1, ψ2(ξ)=1/d¯1+d¯2cosh2d¯3x¯∗ in (94) is always negative. From the relationships between di and d¯i(i=1,2,3) in (38), we can know d¯1>0, d¯2<0, d¯3>0.


#### 4.2.2. The Orbital Stability of the Solitary Wave Solution (8) When 0<p<2

(1) The Orbital Stability of the Solitary Wave Solution (8) When 0 < p < 2 , β < 0. Firstly, from (77), it is easy to get k<0. Then we can substitute it into (92) and easily obtain M2<0. Furthermore, d¯′′(c)>0 can be deduced from (94), so the solitary wave solution (8) of (2) is orbitally stable.(2) The Orbital Stability of the Solitary Wave Solution (8) When 0 < p < 2, β>0. Firstly, from (77) and (92), it is easy to know that, in the case of β>0, if k>0 and c∈(α,+∞), M2∈(0,+∞).Similar to discussions in Section4.1.2, now we substitute (77) into (94) and set x=k/c-α; then (94) can be rewritten as(95)d¯′′c=p+1p+2kβp2c-α1d¯1+d¯2cosh2d¯3x¯∗2/p-1xx2+1-2-p2π2+arctan⁡x.Since ψ2(ξ)=1/d¯1+d¯2cosh2d¯3x¯∗ is always negative, from (95), it is easy to get that d¯′′(c)>0 is equivalent to h2(x)<0. Here(96)h2x=xx2+1-2-p2π2+arctan⁡x,x∈0,+∞.Since (97)h2′x=12x2+12p-4x2+p,x∈0,+∞, we have(98)x0=p4-p=-x0 from h2′(x)=0. So we can deduce that if h2′(x)>0, x∈(0,x¯0), and if x∈(x¯0,+∞), h2′(x)<0. So h2(x) is monotone increasing if x∈(0,x¯0) and monotone decreasing if x∈(x¯0,+∞), and h2(x) takes the maximum value when x=x¯0; meanwhile, (99)limx→0⁡h2x=limx→0⁡xx2+1-2-p2π2+arctan⁡x=-2-p4π,limx→+∞⁡h2x=limx→+∞⁡xx2+1-2-p2π2+arctan⁡x=-2-p2π.On the basis of Intermediate Value Theorem of continuous function, we have Proposition21 about h2(x¯0) and h2(x).Proposition 21.
(1) If the maximumh2(x¯0)<0, then h2(x)<0 for any x∈(0,+∞).
(2) If the maximumh2(x¯0)>0, there exists x¯1,x¯2; if x∈(0,x¯1) and x∈(x¯2,+∞), then h2(x)<0; if x∈(x¯1,x¯2), then h2(x)>0.Sinceβ>0, x=k/c-α(k>0); then we can deduce Proposition 22 from Proposition 21 and (95).Proposition 22.
Set0<p<2,β>0.
(1) If the maximumh2(x¯0)<0, then, for any c∈(α,+∞), d¯′′(c)>0, the solitary wave solution (8) of (2) is orbitally stable.
(2) If the maximumh2(x¯0)>0, then there exist x¯1,x¯2∈(0,+∞); if c∈(α,α+k2/x¯22)∪(α+k2/x¯12,+∞), d¯′′(c)>0, the solitary wave solution (8) is orbitally stable; if c∈(α+k2/x¯22,α+k2/x¯12), d¯′′(c)<0, the solitary wave solution (8) is orbitally instable.Now substitutingx¯0=p/4-p into h2(x), we can get the expression of h2(x¯0) as follows:(100)h2x¯0=p4-p4-2-p2π2+arctan⁡p4-p,p∈0,2.Sincelimp→0⁡h2(x¯0)=-π/2, limp→1⁡h2(x¯0)=33-4π/12, limp→2h2(x¯0)=1/2, and h2(x¯0) is monotone increasing in (0,2), from Intermediate Value Theorem of the continuous function, we can deduce, there exists p¯0∈(1,2) making the maximum h2(x¯0)=0 (from (95), we can know that p¯0 also makes d¯′′(c)=0). So for any p∈(0,p¯0), maximum h2(x¯0)<0, and for any p∈(p¯0,2), maximum h2(x¯0)>0. Summarizing the above discussion, we can get the following proposition.Proposition 23.
Set0<p<2, β>0, p¯0(1<p¯0<2) as the zero point of h2(x¯0)=0.
(1) Whenp∈(0,p¯0)(h2(x¯0)<0), for any c∈(α,+∞),d¯′′(c)>0, the solitary wave solution (8) of (2) is orbitally stable on (0,+∞).
(2) Whenp∈(p¯0,2)(h2(x¯0)>0), there exist x¯1(p), x¯2(p); if c∈(α,α+k2/x¯22(p))∪(α+k2/x¯12(p),+∞), the solitary wave solution (8) of (2) is orbitally stable; if c∈(α+k2/x¯22(p),α+k2/x¯12(p)), the solitary wave solution (8) is orbitally instable.We setc¯1(p)=α+k2/x¯22(p), c¯2(p)=α+k2/x¯12(p). From the above analysis, we can get the following theorem.Theorem 24.
Set p makeq1/p (q is negative) meaningful in R and 0<p<2, γ>0, c>α; then for the solitary wave solution (8); we have the following conclusions.
(1) In the case of0<p<2,β<0, for any c∈(α,+∞), the solitary wave solution (8) is orbitally stable.
(2) In the case of0<p<2,β>0, there exists p¯0(1<p¯0<2) with the following properties:
① Whenp∈(0,p¯0), for any c∈(α,+∞), the solitary wave solution (8) is orbitally stable.
② Whenp∈(p¯0,2), there exists c¯1(p), c¯2(p), if c∈(α,c¯1(p))∪(c¯2(p),+∞), such that the solitary wave solution (8) is orbitally stable; if c∈(c¯1(p),c¯2(p)), the solitary wave solution (8) is orbitally instable.


### 4.3. The Numerical Simulation of the Orbital Stability of the Solitary Wave Solution

For making the results in Sections3, 4.1, and 4.2 more intuitive, we use MAPLE to give the numerical simulations. To avoid duplication and save the space, we only give the illustrations for the stability of the solitary wave solution (6) for (2) (others can be similarly explained). We start from the orbital stability discriminant (36) of the solitary wave solution (6) for (2).We denote the first integral formula in (36) as I1; i.e.,(101)I1=∫0+∞d1+d2cosh2d3x-2/pdx. By using cosh2(d3x)=cosh(2d3x)/2+1/2 and setting σ=2d1/d2+1,τ=2/p, we can get(102)I1=2d2τ∫0+∞cosh⁡2d3x+σ-τdx.Then we set cosh(2d3x)=ω and have(103)I1=12d32d2τ∫1+∞dωω+στω2-1.With the help of MAPLE, the integral in (103) can be expressed as(104)Iσ,τ=∫1+∞dωω+στω2-1=122-τΓτΓτ22hypergeomτ2,τ2,12,σ2-2σΓτ+122hypergeomτ+12,τ+12,32,σ2,when hypergeom is Guass generalized hypergeometric function.Similarly, we set the second integral formula in (36) as I2; i.e.,(105)I2=∫o+∞d1+d2cosh2⁡d3x-2/p+1dx.With similar transformation, we have(106)I2=12d32d2τ+1·Iσ,τ+1,where I(σ,τ+1) can be obtained from (104) by replacing τ+1 with τ.By substitutingd1,d2,d3 in (29)-(31) and I1 and I2 into (36) and setting L1=β/p+1p+2, L2=γ/2p+1, we have(107)d′′c=1pc-αc-αp+1L12+c-αL22/p·Fc,p,where(108)Fc,p=2L12+c-αL2pc-αL12+c-αL2-12c-αIσ,τ+L1L2pL12+c-αL23/2Iσ,τ+1.We can use (107) to make the numerical simulation of the orbital stability of the solitary wave solution (6) on the basis of the expression d′′(c) which is expressed by Guass generalized hypergeometric function.


#### 4.3.1. The Numerical Simulation of the Orbital Stability of the Solitary Wave Solution (6) When p>2, γ>0, β>0

We setα=1, β=1, γ=3 in (107) and get the 3D image of d′′(c) about (c,p) shown in Figure 1, where dark blue surface is the image of d′′(c) and green flat surface is the horizontal plane of z=0.Figure 1
The 3D image ofd′′(c) when α=1, β=1, γ=3, 2<p<4.From Figure1, we can get that the 3D image of d′′(c) has both positive and negative area when α=1, β=1, γ=3, 2<p<4, so there will be wave velocity interval that makes d′′(c) both positive and negative. From Figure 2, we can get that when p>4, the 3D image of d′′(c) is under the surface z=0. So d′′(c) is always negative when p>4.Figure 2
The 3D image ofd′′(c) when α=1, β=1, γ=1, p>4.In Figure3, the first curve is given by c=α+β2p(2p+1)(4-p)/γ(p+1)p+22p-22, the second curve is d′′(c)=0, and the third curve is given by c=α+β2(2p+1)(4-p)/γ(p+1)(p+2)2(p-2). From Figure 3, we can obtain that the area above the curve d′′(c)=0 is the instable area, while the area below this curve is the stable area. But the instable area of the solitary wave solution (6) given by conclusion (1) of Theorem 13 lies above the first curve, and the stable region of the solitary wave solution (6) given by conclusion (3) lies below the third curve in Figure 3. The reason is we scale the inequalities in the analysis process, since it is hard to calculate the integral ∫0+∞(d1+d2cosh2(d3x))-2/pdx and ∫0+∞(d1+d2cosh2(d3x))-2/p-1dx. So conditions of the stability and instability of the solitary wave solution (6) given in Theorem 13 when 2<p<4 are only sufficient conditions.Figure 3
The stability region of the solitary wave solution (6) when α=1, β=1,γ=3,2<p<4.


#### 4.3.2. The Numerical Simulation of the Orbital Stability of the Solitary Wave Solution (6) When 0<p<2, γ>0, β<0

Using (107), by taking α=1, β=-1, γ=3, 0<p<2, we get the simulation results of d′′(c)=0 which are shown in Figure 4. From Figure 4, we can see clearly that the curve d′′(c)=0 takes the minimum point when p=p0. If p∈(0,p0), the curve d′′(c) will fall into the area below the curve d′′(c)=0. And if p∈(p0,2), the curve d′′(c) will fall into the area above the curve d′′(c)=0. From Figure 4, we can see that p0≈1.811 under this case.Figure 4
The curve ofd′′(c)=0 when α=1, β=-1, γ=3, 0<p<2.In the case ofα=1, β=-1, γ=3, when p=1.78, p=1.85, p=1.88, the curve d′′(c) is given in Figure 5. We can see that the curve d′′(c) is wholly above the straight line z=0 when p=1.78 (now p<p0), which means d′′(c)>0 for any c∈(0,+∞). The curve d′′(c) is above the straight line z=0 in (0,c1)∪(c2,+∞) and below the straight line z=0 in (c1,c2) when p=1.85, which means, for any c∈(0,c1)∪(c2,+∞),d′′(c)>0, while d′′(c)<0 for any c∈(c1,c2). Here c1≈1.1045454492, c2≈1.7549789051. The curve d′′(c) is above the straight line z=0 in (0,c¯1)∪(c¯2,+∞), and below the straight line z=0 in (c¯1,c¯2) when p=1.88, which means d′′(c)>0 for any c∈(0,c¯1)∪(c¯2,+∞) and d′′(c)<0 for any c∈(c¯1,c¯2). Here c1≈1.08300534093, c2≈2.410645553. In Figures 4 and 5, it can be shown that simulation results prove the correctness of Conclusion 3 in Theorem 20.Figure 5
The curve image ofd′′(c) when p=1.78,1.85,1.88.


## 4.1. The Orbital Stability of the Solitary Wave Solution (6) of the Generalized Gardner Equation (2) When 0<p≤2

### 4.1.1. The Simplification of the Orbital Stability Discriminantd′′(c) of the Solitary Wave Solution (6) When 0<p≤2

Since0<d1+d2cosh2(d3x)→∞(x→∞), in the case of 0<p≤2, (1/d1+d2cosh2(d3x))2/p+1 converges to the bounded function which converges to 0 when x→∞ or constant 1. So we can apply Mean Value Theorem of Integrals to the orbital stability discriminant (36) of the solitary wave solution (6). Then there exists x∗∈R, such that(65)d′′c=-2p1d1+d2cosh2d3x∗2/p-1·Ic,where(66)Ic=d2′d2+d3′p2d3∫0+∞dxd1+d2cosh2d3x+d1′-d1d2′d2∫0+∞dxd1+d2cosh2d3x2.Sincecosh2(d3x)=1/2cosh⁡(2d3x)+1/2, then we have(67)∫0+∞dxd1+d2cosh2d3x2=4d22∫0+∞dxcosh2d3x+2d1/d2+12.By making the substitution cosh⁡(2d3x)=z,zx=2d3z2-1 to the above formula, we can get(68)∫0+∞dxd1+d2cosh2d3x2=2d22d3∫1+∞dzz+2d1/d2+12z2-1=-12d1d3d1+d2-2d1+d24d3-d1d1+d23/2sgn⁡d2π2-arctan⁡2d1+d22-d1d1+d2. By the same substitution, we can get(69)∫0+∞dxd1+d2cosh2d3x=1d2d3∫1+∞dzz+2d1/d2+1z2-1=-12d3-d1d1+d2sgn⁡d2π2-arctan⁡2d1+d22-d1d1+d2.By substituting (68) and (69) into (66), and noticing that d2>0, then we have(70)Ic=-12d22d3d1′-d1d2′d2d22d1d2+d1+d2′d2+d3′p2d3+2d1+d22d1d2+d1d1′-d1d2′d21d3-4d1d2+d1π2-arctan⁡2d1+d22-d1d1+d2.By simple calculating, we have(71)d2′d2+d3′p2d3=-β22p+12c-αβ22p+1+γc-αp+1p+22+12c-αp2-1,(72)2d1+d2d1d1+d2=-2β2p+1γp+2,(73)d32d1+d2=βpc-αp+1p+2,(74)d1′-d1d2′d2=-βγp+22c-αβ22p+1+γc-αp+1p+22,(75)d22d1d2+d1=-4β22p+1+γc-αp+1p+22γc-αp+1p+22.Then by substituting (71)-(75) into (70), we have(76)Ic=-βd22d3p+1p+2c-α2-1-p2p+1p+22βpc-α2d1+d2-4d1d2+d1π2-arctan⁡2d1+d2-4d1d1+d2.Notice that(77)M1=2d1+d2-4d1d2+d1π2-arctan⁡2d1+d2-4d1d1+d2,k=β2p+1p+2γp+1,and then(78)2d1+d2-4d1d2+d1=β2p+1p+2γp+1·1c-α=kc-α,(79)M1=kc-απ2-arctan⁡kc-α.Substituting (77)-(79) into (76), then we have(80)Ic=-βp+1p+22p+12pc-αβ22p+1+γc-αp+1p+22-1-p2p+1p+22βpc-αM1=-p+1p+22pc-αβ2p+1β22p+1+γc-αp+1p+22+2-p2βM1,so (65) can be rewritten as(81)d′′c=p+1p+2βp2c-α1d1+d2cosh2d3x∗2/p-1β22p+1β22p+1+γc-αp+1p+22+2-p2M1.From Lemma1, we know that ψ1(ξ)=1/d1+d2cosh2d3x∗ is always positive in (81), and d1<0, d2>0, d3>0 can be given by (29), (30), and (31), respectively.


### 4.1.2. The Orbital Stability of the Solitary Wave Solution (6)

(1) Case of p = 2. From (81), we can get the following: if β>0, then d′′(c)>0; if β<0, then d′′(c)<0. So the solitary wave solution (6) is orbitally stable when β>0 and orbitally instable when β<0 in the case of p=2.(2) Case of 0 < p < 2, β>0. Firstly, we consider M1 in (80) and (81). When β>0, we easily have k>0,M1>0 from (77) and (79). Then we can get d′′(c)>0 from (81), and the solitary wave solution (6) of (2) is orbitally stable.(3) Case of 0 < p < 2, β<0. From (77), it is easy to know k<0 when β<0. So M1∈(-∞,0) when c∈(α,+∞). Now we substitute (77) into (81) and set x=k/c-α; then (81) can be rewritten as(82)d′′c=p+1p+2kβp2c-α1d1+d2cosh2d3x∗2/p-1xx2+1+2-p2π2-arctan⁡x.From (82), we can deduce that, in the case of β<0 (now k<0), d′′(c)>0 is equivalent to h1(x)>0. Here(83)h1x=xx2+1+2-p2π2-arctan⁡x,x∈-∞,0,and(84)h1′x=12x2+12p-4x2+p,x∈-∞,0.From h1′(x)=0, we can get(85)x0=-p4-p. It is easy to obtain that if x∈(-∞,x0), then h1′(x)<0, and if x∈(x0,0), then h1′(x)>0. So h1(x) is monotone decreasing if x∈(-∞,x0), monotone increasing if x∈(x0,0), and h1(x) take the minimum value if x=x0. And we have(86)limx→0⁡h1x=limx→0⁡xx2+1+2-p2π2-arctan⁡x=2-p4π,limx→+∞⁡h1x=limx→+∞⁡xx2+1+2-p2π2-arctan⁡x=2-p2π.So according to Intermediate Value Theorem of the continuous function, we can get the following proposition.Proposition 17.
Set0<p<2, β<0.
(1) If the minimum valueh1(x0)>0, then h1(x)>0 for any x∈(-∞,0).
(2) If the minimumh1(x0)<0, then there exists x1,x2 (zero points of h1(x)=0); if x∈(-∞,x1) and x∈(x2,0), then h1(x)>0; if x∈(x1,x2), then h1(x)<0.Sincex=k/c-α(k<0), according to Proposition 17 and (82) and (83), we have the following proposition.Proposition 18.
Set0<p<2,β<0.
(1) If minimumh1(x0)<0, then d′′(c)>0 for any c∈(α,+∞); the solitary wave solution (6) of (2) is orbitally stable in (α,+∞).
(2) If minimumh1(x0)<0, then there exists x1,x2∈(-∞,0) (the zero points of h1(x)=0); if c∈(α,α+k2/x12)∪(α+k2/x22,+∞), then d′′(c)>0; the solitary wave solution (6) is orbitally stable; if c∈(α+k2/x12,α+k2/x22), then d′′(c)<0; the solitary wave solution (6) is orbitally instable.Now substitutingx0=-p/4-p into h(x), we can get the expression of h1(x0) about p,(87)h1x0=-p4-p4+2-p2π2+arctan⁡p4-p,p∈0,2.Sincelimp→0h1(x0)=π/2, limp→1h1(x0)=4π-33/12, limp→2⁡h1(x0)=-1/2, and h1(x0) is monotone decreasing about p in (0,2); from Intermediate Value Theorem of the continuous function, we can deduce that there exists p0∈(1,2) making the minimum h1(x0)=0 (from (82); we can know p0 also makes d′′(c)=0). Then for any p∈(0,p0), the minimum value h1(x0)>0, while, for any p∈(p0,2), the minimum value h1(x0)<0. Combining the above with Proposition 18, we can deduce the following proposition.Proposition 19.
We set0<p<2, β<0, and denote p0(1<p0<2) as the zero point of h1(x0)=0 in (87), and then we have the following.
(1) Whenp∈(0,p0) (now h1(x0)>0), for any c∈(α,+∞),d′′(c)>0, the solitary wave solution (6) of equation (2) is orbitally stable in (α,+∞).
(2) Whenp∈(p0,2) (now h1(x0)<0), there exists x1(p),x2(p) (the zero points of h1(x)=0); if c∈(α,α+k2/x12(p))∪(α+k2/x22(p),+∞), the solitary wave solution (6) is orbitally stable; if c∈(α+k2/x12(p),α+k2/x22(p)), the solitary wave solution (6) is orbitally instable.Now we setc1(p)=α+k2/x12(p), c2(p)=α+k2/x22(p). Through the analysis, we can get the following theorem.Theorem 20.
Set0<p≤2,γ>0,c>α, for the solitary wave solution (6), we have the following conclusions.
(1) In the case ofp=2, the solitary wave solution (6) is orbitally stable when β>0 and orbitally instable when β<0.
(2) In the case of0<p<2, β>0, the solitary wave solution (6) is orbitally stable for any c∈(α,+∞).
(3) In the case of0<p<2,β<0, there exists p0(1<p0<2) such that we have the following.
① Ifp∈(0,p0), for any c∈(α,+∞), the solitary wave solution (6) is orbitally stable.
② Whenp∈(p0,2), there exists c1(p),c2(p); if c∈(α,c1(p))∪(c2(p),+∞), the solitary wave solution (6) is orbitally stable; if c∈(c1(p),c2(p)), the solitary wave solution (6) is orbitally instable.


## 4.1.1. The Simplification of the Orbital Stability Discriminantd′′(c) of the Solitary Wave Solution (6) When 0<p≤2

Since0<d1+d2cosh2(d3x)→∞(x→∞), in the case of 0<p≤2, (1/d1+d2cosh2(d3x))2/p+1 converges to the bounded function which converges to 0 when x→∞ or constant 1. So we can apply Mean Value Theorem of Integrals to the orbital stability discriminant (36) of the solitary wave solution (6). Then there exists x∗∈R, such that(65)d′′c=-2p1d1+d2cosh2d3x∗2/p-1·Ic,where(66)Ic=d2′d2+d3′p2d3∫0+∞dxd1+d2cosh2d3x+d1′-d1d2′d2∫0+∞dxd1+d2cosh2d3x2.Sincecosh2(d3x)=1/2cosh⁡(2d3x)+1/2, then we have(67)∫0+∞dxd1+d2cosh2d3x2=4d22∫0+∞dxcosh2d3x+2d1/d2+12.By making the substitution cosh⁡(2d3x)=z,zx=2d3z2-1 to the above formula, we can get(68)∫0+∞dxd1+d2cosh2d3x2=2d22d3∫1+∞dzz+2d1/d2+12z2-1=-12d1d3d1+d2-2d1+d24d3-d1d1+d23/2sgn⁡d2π2-arctan⁡2d1+d22-d1d1+d2. By the same substitution, we can get(69)∫0+∞dxd1+d2cosh2d3x=1d2d3∫1+∞dzz+2d1/d2+1z2-1=-12d3-d1d1+d2sgn⁡d2π2-arctan⁡2d1+d22-d1d1+d2.By substituting (68) and (69) into (66), and noticing that d2>0, then we have(70)Ic=-12d22d3d1′-d1d2′d2d22d1d2+d1+d2′d2+d3′p2d3+2d1+d22d1d2+d1d1′-d1d2′d21d3-4d1d2+d1π2-arctan⁡2d1+d22-d1d1+d2.By simple calculating, we have(71)d2′d2+d3′p2d3=-β22p+12c-αβ22p+1+γc-αp+1p+22+12c-αp2-1,(72)2d1+d2d1d1+d2=-2β2p+1γp+2,(73)d32d1+d2=βpc-αp+1p+2,(74)d1′-d1d2′d2=-βγp+22c-αβ22p+1+γc-αp+1p+22,(75)d22d1d2+d1=-4β22p+1+γc-αp+1p+22γc-αp+1p+22.Then by substituting (71)-(75) into (70), we have(76)Ic=-βd22d3p+1p+2c-α2-1-p2p+1p+22βpc-α2d1+d2-4d1d2+d1π2-arctan⁡2d1+d2-4d1d1+d2.Notice that(77)M1=2d1+d2-4d1d2+d1π2-arctan⁡2d1+d2-4d1d1+d2,k=β2p+1p+2γp+1,and then(78)2d1+d2-4d1d2+d1=β2p+1p+2γp+1·1c-α=kc-α,(79)M1=kc-απ2-arctan⁡kc-α.Substituting (77)-(79) into (76), then we have(80)Ic=-βp+1p+22p+12pc-αβ22p+1+γc-αp+1p+22-1-p2p+1p+22βpc-αM1=-p+1p+22pc-αβ2p+1β22p+1+γc-αp+1p+22+2-p2βM1,so (65) can be rewritten as(81)d′′c=p+1p+2βp2c-α1d1+d2cosh2d3x∗2/p-1β22p+1β22p+1+γc-αp+1p+22+2-p2M1.From Lemma1, we know that ψ1(ξ)=1/d1+d2cosh2d3x∗ is always positive in (81), and d1<0, d2>0, d3>0 can be given by (29), (30), and (31), respectively.


## 4.1.2. The Orbital Stability of the Solitary Wave Solution (6)

(1) Case of p = 2. From (81), we can get the following: if β>0, then d′′(c)>0; if β<0, then d′′(c)<0. So the solitary wave solution (6) is orbitally stable when β>0 and orbitally instable when β<0 in the case of p=2.(2) Case of 0 < p < 2, β>0. Firstly, we consider M1 in (80) and (81). When β>0, we easily have k>0,M1>0 from (77) and (79). Then we can get d′′(c)>0 from (81), and the solitary wave solution (6) of (2) is orbitally stable.(3) Case of 0 < p < 2, β<0. From (77), it is easy to know k<0 when β<0. So M1∈(-∞,0) when c∈(α,+∞). Now we substitute (77) into (81) and set x=k/c-α; then (81) can be rewritten as(82)d′′c=p+1p+2kβp2c-α1d1+d2cosh2d3x∗2/p-1xx2+1+2-p2π2-arctan⁡x.From (82), we can deduce that, in the case of β<0 (now k<0), d′′(c)>0 is equivalent to h1(x)>0. Here(83)h1x=xx2+1+2-p2π2-arctan⁡x,x∈-∞,0,and(84)h1′x=12x2+12p-4x2+p,x∈-∞,0.From h1′(x)=0, we can get(85)x0=-p4-p. It is easy to obtain that if x∈(-∞,x0), then h1′(x)<0, and if x∈(x0,0), then h1′(x)>0. So h1(x) is monotone decreasing if x∈(-∞,x0), monotone increasing if x∈(x0,0), and h1(x) take the minimum value if x=x0. And we have(86)limx→0⁡h1x=limx→0⁡xx2+1+2-p2π2-arctan⁡x=2-p4π,limx→+∞⁡h1x=limx→+∞⁡xx2+1+2-p2π2-arctan⁡x=2-p2π.So according to Intermediate Value Theorem of the continuous function, we can get the following proposition.Proposition 17.
Set0<p<2, β<0.
(1) If the minimum valueh1(x0)>0, then h1(x)>0 for any x∈(-∞,0).
(2) If the minimumh1(x0)<0, then there exists x1,x2 (zero points of h1(x)=0); if x∈(-∞,x1) and x∈(x2,0), then h1(x)>0; if x∈(x1,x2), then h1(x)<0.Sincex=k/c-α(k<0), according to Proposition 17 and (82) and (83), we have the following proposition.Proposition 18.
Set0<p<2,β<0.
(1) If minimumh1(x0)<0, then d′′(c)>0 for any c∈(α,+∞); the solitary wave solution (6) of (2) is orbitally stable in (α,+∞).
(2) If minimumh1(x0)<0, then there exists x1,x2∈(-∞,0) (the zero points of h1(x)=0); if c∈(α,α+k2/x12)∪(α+k2/x22,+∞), then d′′(c)>0; the solitary wave solution (6) is orbitally stable; if c∈(α+k2/x12,α+k2/x22), then d′′(c)<0; the solitary wave solution (6) is orbitally instable.Now substitutingx0=-p/4-p into h(x), we can get the expression of h1(x0) about p,(87)h1x0=-p4-p4+2-p2π2+arctan⁡p4-p,p∈0,2.Sincelimp→0h1(x0)=π/2, limp→1h1(x0)=4π-33/12, limp→2⁡h1(x0)=-1/2, and h1(x0) is monotone decreasing about p in (0,2); from Intermediate Value Theorem of the continuous function, we can deduce that there exists p0∈(1,2) making the minimum h1(x0)=0 (from (82); we can know p0 also makes d′′(c)=0). Then for any p∈(0,p0), the minimum value h1(x0)>0, while, for any p∈(p0,2), the minimum value h1(x0)<0. Combining the above with Proposition 18, we can deduce the following proposition.Proposition 19.
We set0<p<2, β<0, and denote p0(1<p0<2) as the zero point of h1(x0)=0 in (87), and then we have the following.
(1) Whenp∈(0,p0) (now h1(x0)>0), for any c∈(α,+∞),d′′(c)>0, the solitary wave solution (6) of equation (2) is orbitally stable in (α,+∞).
(2) Whenp∈(p0,2) (now h1(x0)<0), there exists x1(p),x2(p) (the zero points of h1(x)=0); if c∈(α,α+k2/x12(p))∪(α+k2/x22(p),+∞), the solitary wave solution (6) is orbitally stable; if c∈(α+k2/x12(p),α+k2/x22(p)), the solitary wave solution (6) is orbitally instable.Now we setc1(p)=α+k2/x12(p), c2(p)=α+k2/x22(p). Through the analysis, we can get the following theorem.Theorem 20.
Set0<p≤2,γ>0,c>α, for the solitary wave solution (6), we have the following conclusions.
(1) In the case ofp=2, the solitary wave solution (6) is orbitally stable when β>0 and orbitally instable when β<0.
(2) In the case of0<p<2, β>0, the solitary wave solution (6) is orbitally stable for any c∈(α,+∞).
(3) In the case of0<p<2,β<0, there exists p0(1<p0<2) such that we have the following.
① Ifp∈(0,p0), for any c∈(α,+∞), the solitary wave solution (6) is orbitally stable.
② Whenp∈(p0,2), there exists c1(p),c2(p); if c∈(α,c1(p))∪(c2(p),+∞), the solitary wave solution (6) is orbitally stable; if c∈(c1(p),c2(p)), the solitary wave solution (6) is orbitally instable.


## 4.2. The Orbital Stability of the Solitary Wave Solution (8) of the Generalized Gardner Equation (2) When 0<p<2

### 4.2.1. The Simplification of the Orbital Stability Discriminantd¯′′(c) of the Solitary Wave Solution (8) When 0<p<2

Sinceψ2(ξ) is always negative in the solitary wave solution (8), we assume that p makes q1/p (q is negative) meaningful in R (The case of p=2 is out of our discussion). In the case of 0<p<2, we make simplification for the discriminant d¯′′(c) of the solitary wave solution (8). Similar to the discussion of the orbital stability of u1(x), by applying Mean Value Theorem of Integrals to (41), we can get that there exists x¯∈R such that(88)d¯′′c<-2p1d¯1+d¯2cosh2d¯3x2/p-1·I¯c,x¯∈R,where(89)I¯c=d2′d2+d3′p2d3∫0+∞dxd¯1+d¯2cosh2d¯3x+d1′-d1d2′d2∫0+∞dxd¯1+d¯2cosh2d¯3x2.Sinced¯i and di(i=1,2,3) satisfy (38) and (40), according to (71)-(75), we can obtain(90)I¯c=-βd22d3p+1p+2+1-p2p+1p+22βpc-α2d1+d2-4d1d2+d1π2+arctan⁡2d1+d2-4d1d1+d2.Due to (78) and (79), we set(91)M2=2d1+d2-4d1d2+d1π2+arctan⁡2d1+d2-4d1d1+d2,and then we have(92)M2=kc-απ2+arctan⁡kc-α.So we have(93)I¯c=-βp+1p+22p+12pc-αβ22p+1+γc-αp+1p+22+1-p2p+1p+22βpc-αM2=-p+1p+22βpc-αβ22p+1β22p+1+γc-αp+1p+22-2-p2M2,(94)d¯′′c=p+1p+2βp2c-α1d¯1+d¯2cosh2d¯3x¯∗2/p-1β22p+1β22p+1+γc-αp+1p+22-2-p2M2.According to Lemma1, ψ2(ξ)=1/d¯1+d¯2cosh2d¯3x¯∗ in (94) is always negative. From the relationships between di and d¯i(i=1,2,3) in (38), we can know d¯1>0, d¯2<0, d¯3>0.


### 4.2.2. The Orbital Stability of the Solitary Wave Solution (8) When 0<p<2

(1) The Orbital Stability of the Solitary Wave Solution (8) When 0 < p < 2 , β < 0. Firstly, from (77), it is easy to get k<0. Then we can substitute it into (92) and easily obtain M2<0. Furthermore, d¯′′(c)>0 can be deduced from (94), so the solitary wave solution (8) of (2) is orbitally stable.(2) The Orbital Stability of the Solitary Wave Solution (8) When 0 < p < 2, β>0. Firstly, from (77) and (92), it is easy to know that, in the case of β>0, if k>0 and c∈(α,+∞), M2∈(0,+∞).Similar to discussions in Section4.1.2, now we substitute (77) into (94) and set x=k/c-α; then (94) can be rewritten as(95)d¯′′c=p+1p+2kβp2c-α1d¯1+d¯2cosh2d¯3x¯∗2/p-1xx2+1-2-p2π2+arctan⁡x.Since ψ2(ξ)=1/d¯1+d¯2cosh2d¯3x¯∗ is always negative, from (95), it is easy to get that d¯′′(c)>0 is equivalent to h2(x)<0. Here(96)h2x=xx2+1-2-p2π2+arctan⁡x,x∈0,+∞.Since (97)h2′x=12x2+12p-4x2+p,x∈0,+∞, we have(98)x0=p4-p=-x0 from h2′(x)=0. So we can deduce that if h2′(x)>0, x∈(0,x¯0), and if x∈(x¯0,+∞), h2′(x)<0. So h2(x) is monotone increasing if x∈(0,x¯0) and monotone decreasing if x∈(x¯0,+∞), and h2(x) takes the maximum value when x=x¯0; meanwhile, (99)limx→0⁡h2x=limx→0⁡xx2+1-2-p2π2+arctan⁡x=-2-p4π,limx→+∞⁡h2x=limx→+∞⁡xx2+1-2-p2π2+arctan⁡x=-2-p2π.On the basis of Intermediate Value Theorem of continuous function, we have Proposition21 about h2(x¯0) and h2(x).Proposition 21.
(1) If the maximumh2(x¯0)<0, then h2(x)<0 for any x∈(0,+∞).
(2) If the maximumh2(x¯0)>0, there exists x¯1,x¯2; if x∈(0,x¯1) and x∈(x¯2,+∞), then h2(x)<0; if x∈(x¯1,x¯2), then h2(x)>0.Sinceβ>0, x=k/c-α(k>0); then we can deduce Proposition 22 from Proposition 21 and (95).Proposition 22.
Set0<p<2,β>0.
(1) If the maximumh2(x¯0)<0, then, for any c∈(α,+∞), d¯′′(c)>0, the solitary wave solution (8) of (2) is orbitally stable.
(2) If the maximumh2(x¯0)>0, then there exist x¯1,x¯2∈(0,+∞); if c∈(α,α+k2/x¯22)∪(α+k2/x¯12,+∞), d¯′′(c)>0, the solitary wave solution (8) is orbitally stable; if c∈(α+k2/x¯22,α+k2/x¯12), d¯′′(c)<0, the solitary wave solution (8) is orbitally instable.Now substitutingx¯0=p/4-p into h2(x), we can get the expression of h2(x¯0) as follows:(100)h2x¯0=p4-p4-2-p2π2+arctan⁡p4-p,p∈0,2.Sincelimp→0⁡h2(x¯0)=-π/2, limp→1⁡h2(x¯0)=33-4π/12, limp→2h2(x¯0)=1/2, and h2(x¯0) is monotone increasing in (0,2), from Intermediate Value Theorem of the continuous function, we can deduce, there exists p¯0∈(1,2) making the maximum h2(x¯0)=0 (from (95), we can know that p¯0 also makes d¯′′(c)=0). So for any p∈(0,p¯0), maximum h2(x¯0)<0, and for any p∈(p¯0,2), maximum h2(x¯0)>0. Summarizing the above discussion, we can get the following proposition.Proposition 23.
Set0<p<2, β>0, p¯0(1<p¯0<2) as the zero point of h2(x¯0)=0.
(1) Whenp∈(0,p¯0)(h2(x¯0)<0), for any c∈(α,+∞),d¯′′(c)>0, the solitary wave solution (8) of (2) is orbitally stable on (0,+∞).
(2) Whenp∈(p¯0,2)(h2(x¯0)>0), there exist x¯1(p), x¯2(p); if c∈(α,α+k2/x¯22(p))∪(α+k2/x¯12(p),+∞), the solitary wave solution (8) of (2) is orbitally stable; if c∈(α+k2/x¯22(p),α+k2/x¯12(p)), the solitary wave solution (8) is orbitally instable.We setc¯1(p)=α+k2/x¯22(p), c¯2(p)=α+k2/x¯12(p). From the above analysis, we can get the following theorem.Theorem 24.
Set p makeq1/p (q is negative) meaningful in R and 0<p<2, γ>0, c>α; then for the solitary wave solution (8); we have the following conclusions.
(1) In the case of0<p<2,β<0, for any c∈(α,+∞), the solitary wave solution (8) is orbitally stable.
(2) In the case of0<p<2,β>0, there exists p¯0(1<p¯0<2) with the following properties:
① Whenp∈(0,p¯0), for any c∈(α,+∞), the solitary wave solution (8) is orbitally stable.
② Whenp∈(p¯0,2), there exists c¯1(p), c¯2(p), if c∈(α,c¯1(p))∪(c¯2(p),+∞), such that the solitary wave solution (8) is orbitally stable; if c∈(c¯1(p),c¯2(p)), the solitary wave solution (8) is orbitally instable.


## 4.2.1. The Simplification of the Orbital Stability Discriminantd¯′′(c) of the Solitary Wave Solution (8) When 0<p<2

Sinceψ2(ξ) is always negative in the solitary wave solution (8), we assume that p makes q1/p (q is negative) meaningful in R (The case of p=2 is out of our discussion). In the case of 0<p<2, we make simplification for the discriminant d¯′′(c) of the solitary wave solution (8). Similar to the discussion of the orbital stability of u1(x), by applying Mean Value Theorem of Integrals to (41), we can get that there exists x¯∈R such that(88)d¯′′c<-2p1d¯1+d¯2cosh2d¯3x2/p-1·I¯c,x¯∈R,where(89)I¯c=d2′d2+d3′p2d3∫0+∞dxd¯1+d¯2cosh2d¯3x+d1′-d1d2′d2∫0+∞dxd¯1+d¯2cosh2d¯3x2.Sinced¯i and di(i=1,2,3) satisfy (38) and (40), according to (71)-(75), we can obtain(90)I¯c=-βd22d3p+1p+2+1-p2p+1p+22βpc-α2d1+d2-4d1d2+d1π2+arctan⁡2d1+d2-4d1d1+d2.Due to (78) and (79), we set(91)M2=2d1+d2-4d1d2+d1π2+arctan⁡2d1+d2-4d1d1+d2,and then we have(92)M2=kc-απ2+arctan⁡kc-α.So we have(93)I¯c=-βp+1p+22p+12pc-αβ22p+1+γc-αp+1p+22+1-p2p+1p+22βpc-αM2=-p+1p+22βpc-αβ22p+1β22p+1+γc-αp+1p+22-2-p2M2,(94)d¯′′c=p+1p+2βp2c-α1d¯1+d¯2cosh2d¯3x¯∗2/p-1β22p+1β22p+1+γc-αp+1p+22-2-p2M2.According to Lemma1, ψ2(ξ)=1/d¯1+d¯2cosh2d¯3x¯∗ in (94) is always negative. From the relationships between di and d¯i(i=1,2,3) in (38), we can know d¯1>0, d¯2<0, d¯3>0.


## 4.2.2. The Orbital Stability of the Solitary Wave Solution (8) When 0<p<2

(1) The Orbital Stability of the Solitary Wave Solution (8) When 0 < p < 2 , β < 0. Firstly, from (77), it is easy to get k<0. Then we can substitute it into (92) and easily obtain M2<0. Furthermore, d¯′′(c)>0 can be deduced from (94), so the solitary wave solution (8) of (2) is orbitally stable.(2) The Orbital Stability of the Solitary Wave Solution (8) When 0 < p < 2, β>0. Firstly, from (77) and (92), it is easy to know that, in the case of β>0, if k>0 and c∈(α,+∞), M2∈(0,+∞).Similar to discussions in Section4.1.2, now we substitute (77) into (94) and set x=k/c-α; then (94) can be rewritten as(95)d¯′′c=p+1p+2kβp2c-α1d¯1+d¯2cosh2d¯3x¯∗2/p-1xx2+1-2-p2π2+arctan⁡x.Since ψ2(ξ)=1/d¯1+d¯2cosh2d¯3x¯∗ is always negative, from (95), it is easy to get that d¯′′(c)>0 is equivalent to h2(x)<0. Here(96)h2x=xx2+1-2-p2π2+arctan⁡x,x∈0,+∞.Since (97)h2′x=12x2+12p-4x2+p,x∈0,+∞, we have(98)x0=p4-p=-x0 from h2′(x)=0. So we can deduce that if h2′(x)>0, x∈(0,x¯0), and if x∈(x¯0,+∞), h2′(x)<0. So h2(x) is monotone increasing if x∈(0,x¯0) and monotone decreasing if x∈(x¯0,+∞), and h2(x) takes the maximum value when x=x¯0; meanwhile, (99)limx→0⁡h2x=limx→0⁡xx2+1-2-p2π2+arctan⁡x=-2-p4π,limx→+∞⁡h2x=limx→+∞⁡xx2+1-2-p2π2+arctan⁡x=-2-p2π.On the basis of Intermediate Value Theorem of continuous function, we have Proposition21 about h2(x¯0) and h2(x).Proposition 21.
(1) If the maximumh2(x¯0)<0, then h2(x)<0 for any x∈(0,+∞).
(2) If the maximumh2(x¯0)>0, there exists x¯1,x¯2; if x∈(0,x¯1) and x∈(x¯2,+∞), then h2(x)<0; if x∈(x¯1,x¯2), then h2(x)>0.Sinceβ>0, x=k/c-α(k>0); then we can deduce Proposition 22 from Proposition 21 and (95).Proposition 22.
Set0<p<2,β>0.
(1) If the maximumh2(x¯0)<0, then, for any c∈(α,+∞), d¯′′(c)>0, the solitary wave solution (8) of (2) is orbitally stable.
(2) If the maximumh2(x¯0)>0, then there exist x¯1,x¯2∈(0,+∞); if c∈(α,α+k2/x¯22)∪(α+k2/x¯12,+∞), d¯′′(c)>0, the solitary wave solution (8) is orbitally stable; if c∈(α+k2/x¯22,α+k2/x¯12), d¯′′(c)<0, the solitary wave solution (8) is orbitally instable.Now substitutingx¯0=p/4-p into h2(x), we can get the expression of h2(x¯0) as follows:(100)h2x¯0=p4-p4-2-p2π2+arctan⁡p4-p,p∈0,2.Sincelimp→0⁡h2(x¯0)=-π/2, limp→1⁡h2(x¯0)=33-4π/12, limp→2h2(x¯0)=1/2, and h2(x¯0) is monotone increasing in (0,2), from Intermediate Value Theorem of the continuous function, we can deduce, there exists p¯0∈(1,2) making the maximum h2(x¯0)=0 (from (95), we can know that p¯0 also makes d¯′′(c)=0). So for any p∈(0,p¯0), maximum h2(x¯0)<0, and for any p∈(p¯0,2), maximum h2(x¯0)>0. Summarizing the above discussion, we can get the following proposition.Proposition 23.
Set0<p<2, β>0, p¯0(1<p¯0<2) as the zero point of h2(x¯0)=0.
(1) Whenp∈(0,p¯0)(h2(x¯0)<0), for any c∈(α,+∞),d¯′′(c)>0, the solitary wave solution (8) of (2) is orbitally stable on (0,+∞).
(2) Whenp∈(p¯0,2)(h2(x¯0)>0), there exist x¯1(p), x¯2(p); if c∈(α,α+k2/x¯22(p))∪(α+k2/x¯12(p),+∞), the solitary wave solution (8) of (2) is orbitally stable; if c∈(α+k2/x¯22(p),α+k2/x¯12(p)), the solitary wave solution (8) is orbitally instable.We setc¯1(p)=α+k2/x¯22(p), c¯2(p)=α+k2/x¯12(p). From the above analysis, we can get the following theorem.Theorem 24.
Set p makeq1/p (q is negative) meaningful in R and 0<p<2, γ>0, c>α; then for the solitary wave solution (8); we have the following conclusions.
(1) In the case of0<p<2,β<0, for any c∈(α,+∞), the solitary wave solution (8) is orbitally stable.
(2) In the case of0<p<2,β>0, there exists p¯0(1<p¯0<2) with the following properties:
① Whenp∈(0,p¯0), for any c∈(α,+∞), the solitary wave solution (8) is orbitally stable.
② Whenp∈(p¯0,2), there exists c¯1(p), c¯2(p), if c∈(α,c¯1(p))∪(c¯2(p),+∞), such that the solitary wave solution (8) is orbitally stable; if c∈(c¯1(p),c¯2(p)), the solitary wave solution (8) is orbitally instable.


## 4.3. The Numerical Simulation of the Orbital Stability of the Solitary Wave Solution

For making the results in Sections3, 4.1, and 4.2 more intuitive, we use MAPLE to give the numerical simulations. To avoid duplication and save the space, we only give the illustrations for the stability of the solitary wave solution (6) for (2) (others can be similarly explained). We start from the orbital stability discriminant (36) of the solitary wave solution (6) for (2).We denote the first integral formula in (36) as I1; i.e.,(101)I1=∫0+∞d1+d2cosh2d3x-2/pdx. By using cosh2(d3x)=cosh(2d3x)/2+1/2 and setting σ=2d1/d2+1,τ=2/p, we can get(102)I1=2d2τ∫0+∞cosh⁡2d3x+σ-τdx.Then we set cosh(2d3x)=ω and have(103)I1=12d32d2τ∫1+∞dωω+στω2-1.With the help of MAPLE, the integral in (103) can be expressed as(104)Iσ,τ=∫1+∞dωω+στω2-1=122-τΓτΓτ22hypergeomτ2,τ2,12,σ2-2σΓτ+122hypergeomτ+12,τ+12,32,σ2,when hypergeom is Guass generalized hypergeometric function.Similarly, we set the second integral formula in (36) as I2; i.e.,(105)I2=∫o+∞d1+d2cosh2⁡d3x-2/p+1dx.With similar transformation, we have(106)I2=12d32d2τ+1·Iσ,τ+1,where I(σ,τ+1) can be obtained from (104) by replacing τ+1 with τ.By substitutingd1,d2,d3 in (29)-(31) and I1 and I2 into (36) and setting L1=β/p+1p+2, L2=γ/2p+1, we have(107)d′′c=1pc-αc-αp+1L12+c-αL22/p·Fc,p,where(108)Fc,p=2L12+c-αL2pc-αL12+c-αL2-12c-αIσ,τ+L1L2pL12+c-αL23/2Iσ,τ+1.We can use (107) to make the numerical simulation of the orbital stability of the solitary wave solution (6) on the basis of the expression d′′(c) which is expressed by Guass generalized hypergeometric function.


### 4.3.1. The Numerical Simulation of the Orbital Stability of the Solitary Wave Solution (6) When p>2, γ>0, β>0

We setα=1, β=1, γ=3 in (107) and get the 3D image of d′′(c) about (c,p) shown in Figure 1, where dark blue surface is the image of d′′(c) and green flat surface is the horizontal plane of z=0.Figure 1
The 3D image ofd′′(c) when α=1, β=1, γ=3, 2<p<4.From Figure1, we can get that the 3D image of d′′(c) has both positive and negative area when α=1, β=1, γ=3, 2<p<4, so there will be wave velocity interval that makes d′′(c) both positive and negative. From Figure 2, we can get that when p>4, the 3D image of d′′(c) is under the surface z=0. So d′′(c) is always negative when p>4.Figure 2
The 3D image ofd′′(c) when α=1, β=1, γ=1, p>4.In Figure3, the first curve is given by c=α+β2p(2p+1)(4-p)/γ(p+1)p+22p-22, the second curve is d′′(c)=0, and the third curve is given by c=α+β2(2p+1)(4-p)/γ(p+1)(p+2)2(p-2). From Figure 3, we can obtain that the area above the curve d′′(c)=0 is the instable area, while the area below this curve is the stable area. But the instable area of the solitary wave solution (6) given by conclusion (1) of Theorem 13 lies above the first curve, and the stable region of the solitary wave solution (6) given by conclusion (3) lies below the third curve in Figure 3. The reason is we scale the inequalities in the analysis process, since it is hard to calculate the integral ∫0+∞(d1+d2cosh2(d3x))-2/pdx and ∫0+∞(d1+d2cosh2(d3x))-2/p-1dx. So conditions of the stability and instability of the solitary wave solution (6) given in Theorem 13 when 2<p<4 are only sufficient conditions.Figure 3
The stability region of the solitary wave solution (6) when α=1, β=1,γ=3,2<p<4.


### 4.3.2. The Numerical Simulation of the Orbital Stability of the Solitary Wave Solution (6) When 0<p<2, γ>0, β<0

Using (107), by taking α=1, β=-1, γ=3, 0<p<2, we get the simulation results of d′′(c)=0 which are shown in Figure 4. From Figure 4, we can see clearly that the curve d′′(c)=0 takes the minimum point when p=p0. If p∈(0,p0), the curve d′′(c) will fall into the area below the curve d′′(c)=0. And if p∈(p0,2), the curve d′′(c) will fall into the area above the curve d′′(c)=0. From Figure 4, we can see that p0≈1.811 under this case.Figure 4
The curve ofd′′(c)=0 when α=1, β=-1, γ=3, 0<p<2.In the case ofα=1, β=-1, γ=3, when p=1.78, p=1.85, p=1.88, the curve d′′(c) is given in Figure 5. We can see that the curve d′′(c) is wholly above the straight line z=0 when p=1.78 (now p<p0), which means d′′(c)>0 for any c∈(0,+∞). The curve d′′(c) is above the straight line z=0 in (0,c1)∪(c2,+∞) and below the straight line z=0 in (c1,c2) when p=1.85, which means, for any c∈(0,c1)∪(c2,+∞),d′′(c)>0, while d′′(c)<0 for any c∈(c1,c2). Here c1≈1.1045454492, c2≈1.7549789051. The curve d′′(c) is above the straight line z=0 in (0,c¯1)∪(c¯2,+∞), and below the straight line z=0 in (c¯1,c¯2) when p=1.88, which means d′′(c)>0 for any c∈(0,c¯1)∪(c¯2,+∞) and d′′(c)<0 for any c∈(c¯1,c¯2). Here c1≈1.08300534093, c2≈2.410645553. In Figures 4 and 5, it can be shown that simulation results prove the correctness of Conclusion 3 in Theorem 20.Figure 5
The curve image ofd′′(c) when p=1.78,1.85,1.88.


## 4.3.1. The Numerical Simulation of the Orbital Stability of the Solitary Wave Solution (6) When p>2, γ>0, β>0

We setα=1, β=1, γ=3 in (107) and get the 3D image of d′′(c) about (c,p) shown in Figure 1, where dark blue surface is the image of d′′(c) and green flat surface is the horizontal plane of z=0.Figure 1
The 3D image ofd′′(c) when α=1, β=1, γ=3, 2<p<4.From Figure1, we can get that the 3D image of d′′(c) has both positive and negative area when α=1, β=1, γ=3, 2<p<4, so there will be wave velocity interval that makes d′′(c) both positive and negative. From Figure 2, we can get that when p>4, the 3D image of d′′(c) is under the surface z=0. So d′′(c) is always negative when p>4.Figure 2
The 3D image ofd′′(c) when α=1, β=1, γ=1, p>4.In Figure3, the first curve is given by c=α+β2p(2p+1)(4-p)/γ(p+1)p+22p-22, the second curve is d′′(c)=0, and the third curve is given by c=α+β2(2p+1)(4-p)/γ(p+1)(p+2)2(p-2). From Figure 3, we can obtain that the area above the curve d′′(c)=0 is the instable area, while the area below this curve is the stable area. But the instable area of the solitary wave solution (6) given by conclusion (1) of Theorem 13 lies above the first curve, and the stable region of the solitary wave solution (6) given by conclusion (3) lies below the third curve in Figure 3. The reason is we scale the inequalities in the analysis process, since it is hard to calculate the integral ∫0+∞(d1+d2cosh2(d3x))-2/pdx and ∫0+∞(d1+d2cosh2(d3x))-2/p-1dx. So conditions of the stability and instability of the solitary wave solution (6) given in Theorem 13 when 2<p<4 are only sufficient conditions.Figure 3
The stability region of the solitary wave solution (6) when α=1, β=1,γ=3,2<p<4.


## 4.3.2. The Numerical Simulation of the Orbital Stability of the Solitary Wave Solution (6) When 0<p<2, γ>0, β<0

Using (107), by taking α=1, β=-1, γ=3, 0<p<2, we get the simulation results of d′′(c)=0 which are shown in Figure 4. From Figure 4, we can see clearly that the curve d′′(c)=0 takes the minimum point when p=p0. If p∈(0,p0), the curve d′′(c) will fall into the area below the curve d′′(c)=0. And if p∈(p0,2), the curve d′′(c) will fall into the area above the curve d′′(c)=0. From Figure 4, we can see that p0≈1.811 under this case.Figure 4
The curve ofd′′(c)=0 when α=1, β=-1, γ=3, 0<p<2.In the case ofα=1, β=-1, γ=3, when p=1.78, p=1.85, p=1.88, the curve d′′(c) is given in Figure 5. We can see that the curve d′′(c) is wholly above the straight line z=0 when p=1.78 (now p<p0), which means d′′(c)>0 for any c∈(0,+∞). The curve d′′(c) is above the straight line z=0 in (0,c1)∪(c2,+∞) and below the straight line z=0 in (c1,c2) when p=1.85, which means, for any c∈(0,c1)∪(c2,+∞),d′′(c)>0, while d′′(c)<0 for any c∈(c1,c2). Here c1≈1.1045454492, c2≈1.7549789051. The curve d′′(c) is above the straight line z=0 in (0,c¯1)∪(c¯2,+∞), and below the straight line z=0 in (c¯1,c¯2) when p=1.88, which means d′′(c)>0 for any c∈(0,c¯1)∪(c¯2,+∞) and d′′(c)<0 for any c∈(c¯1,c¯2). Here c1≈1.08300534093, c2≈2.410645553. In Figures 4 and 5, it can be shown that simulation results prove the correctness of Conclusion 3 in Theorem 20.Figure 5
The curve image ofd′′(c) when p=1.78,1.85,1.88.


## 5. The Discussion of the Influence Caused by Interaction between Two Nonlinear Terms on the Stability of the Solitary Wave Solution

In this paper, we study the orbital stability of the solitary wave solution of the generalized Gardner equation (2). And we obtain Theorems 13 and 20 for the solitary wave solution (6) and obtain Theorems 16 and 24 for the solitary wave solution (8).Comparing with results that the solitary wave solution of the KdV equation (3) is orbitally instable when p≥4 and orbitally stable when 0<p<4, it can be deduced that, when the order of the nonlinear terms in (3) and (2) is wholly high (in this paper p≥4), the solitary wave solutions of both equations are orbitally instable. However when 0<p<4, the influence caused by interaction between two nonlinear terms on the orbital stability of the solitary wave solution for the generalized Gardner equation is obvious. Details are as follows.(1) In the case of2<p<4,β>0, the order of the higher order nonlinear term in (2) is greater than 4, and the effect of the lower order nonlinear term in (2) is obvious. According to Theorem 13, we can get that both orbitally stable and instable intervals exist for the solitary wave solution (6). If β is larger, the interval which makes the solitary wave solution (6) stable is larger. If γ is larger, the interval which makes the solitary wave solution (6) stable is smaller, and the interval that makes the solitary wave solution (6) instable is larger. This seems that the lower order nonlinear term βupux plays a role in promoting stability in (2) when β>0.(2) In the case ofβ>0, 0<p<2, the order of the higher order nonlinear term in (2) is smaller than 4. And both the solitary wave solution (6) of (2) and the solitary wave solution of (3) are orbitally stable. But if p=2, α=0, from Theorem 20 in this paper, we can know that (2) can be rewritten as(109)ut+βu2ux+γu4ux+uxxx=0,and its corresponding solitary wave solution (6) is orbitally stable for any c∈(0,+∞) when β>0. But according to [20], we can get that if we let β=0 in (109), then the solitary wave solution of the KdV equation(110)ut+γu4ux+uxxx=0is orbitally instable. By comparing, when β>0, the lower order nonlinear term in (109) seems to play a role in promoting the stability of the solitary wave solution (6).(3) In the case of0<p<2,β<0, we consider the solitary wave solution (6). From Theorem 20, we can get that there exists p0(1<p0<2), if 0<p<p0, the solitary wave solution (6) is orbitally stable in (α,+∞), but if p0<p<2, although the order of the higher order nonlinear term in (2) is less than 4, there is still part of wave velocity interval (α,+∞) in which the solution is orbitally instable. It seems that, for the solitary wave solution (6), when β<0, the term βupux in (2) plays a role against the stability in equation.(4) For the solitary wave solution (8) of (2), we can make the similar analysis as conclusions (1)-(3). From Theorems 16 and 24, we can get that, for the solitary wave solution (8), the term βupux in (2) plays a role promoting the stability in the equation if β<0 and plays a role against the stability in equation if β>0.The method in this paper is an extension of the theory of orbital stability of Grillakis-Shatah-Strauss. The most fundamental result is that we find and prove,∫0+∞d/dc(1/d1+d2cosh2(d3x))2/pdx converges for any fixed c when p>0 and uniformly converges in any subintervals [a,b] in (α,+∞). Then we apply our method to calculate the integral with a parameter to establish the explicit expressions (36) and (41) of the orbital stability discriminant of the solitary wave solutions (6) and (8), respectively. All conclusions in this paper are obtained on the basis of the above two formulas and by the analysis method.From the study in this paper, we can find that if an equation satisfies the conditions of the theory of orbital stability of Grillakis-Shatah-Strauss and the solitary wave solutions of equation can be solved, then the method in this paper can be referred to study the stability and instability of its solitary wave solutions. Since the exact explicit solitary wave solutions of a large number of soliton equations have been obtained in the field of Mathematics and Physics since 1980s, in order to further study the stability of these solitary wave solutions with explicit expressions, we think the method in this paper is worth being referred.



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