# Peaked and Smooth Solitons forK*(4,1) Equation

Authors: Yongan Xie; Hualiang Fu; Shengqiang Tang
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

This paper is contributed to explore all possible single peak solutions for theK*(4,1) equation ut=uxu2+2α(uuxxx+2uxuxx). Our procedure shows that the K*(4,1) equation either has peakon, cuspon, and smooth soliton solutions when sitting on a nonzero constant pedestal limξ→±∞u=A≠0 or possesses compacton solutions only when limξ→±∞u=A=0. We present a new smooth soliton solution in an explicit form. Mathematical analysis and numeric graphs are provided for those soliton solutions of the K*(4,1) equation.


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## Body

## 1. Introduction

It is well known that the study of nonlinear wave equations and their solutions is of great importance in many areas of physics.In 1993, Cooper et al. [1] considered the following generalized KdV equation (GKdV):
(1)K*(l,p):ut=uxul-2+α[2uxxxup+4pup-1uxuxx+p(p-1)up-2(ux)3],
where l,p∈Z+. These equations are derived from Lagrangian
(2)L(l,p)=∫‍[12ψxψt-(ψx)ll(l-1)+α(ψx)p(ψxx)2]dx,
where u(x,t) is defined by u(x,t)=ψx(x,t). These equations have the same terms as the following equations, considered by Rosenau and Hyman [2]:
(3)K(m,n):ut+(um)x+(un)xxx=0,m,n>1,
but relative weights of the terms are quite different leading to the possibility that the integrability properties might be different. Cooper et al. [1] investigated Hamiltonian structure and integrability properties for this class of KdV equations. By using bifurcation theory of dynamical systems, when l,p≥2, Tang and Li [3] investigated the bifurcation behavior for traveling wave solutions of (1). In [4], by using sine-cosine method and extended tanh method, some new solitary patterns solutions and compactons solutions are formally derived. In [5], by using analytic methods from the dynamical systems theory, some new exact explicit parametric representations of breaking loop-solutions under some fixed parameter conditions are formally derived.In the development of soliton theory, there exist many different approaches to searching for exact solutions of nonlinear partial differential equations, such as mapping method [6], fan-expansion method [7], and (G′/G)-expansion method [8]. In particular, it is very interesting to investigate the traveling wave solutions on a constant pedestal. Qiao and Zhang [9] discussed the traveling wave solutions for the Camassa-Holm equation on the nonzero constant pedestal limξ→±∞u=A≠0 and found new soliton solutions, which are smooth and cusped. Later, Zhang and Qiao [10] investigated the Degasperis-Procesi equation under the boundary condition limξ→±∞u=A and obtained all possible single peak soliton solutions of the Degasperis-Procesi equation. Recently, Chen and Li [11] studied osmosis K(2,2) equation with a nonzero constant pedestal and obtained smooth, peaked, and cusped soliton solutions of the osmosis K(2,2) equation. More recently, Zhang et al. [12] studied the K(2,2) equation under an inhomogeneous boundary condition and obtained compacton solutions, loop soliton solutions, cusped soliton solutions, and smooth soliton solutions.In the literature [1, 3–5], the authors did not investigate the existence of peakon soliton for the K*(l,p) equation. We are thus interested in an important question that should be investigated: does the K*(l,p) equation have peakon soliton? We hope to answer this problem in this paper. Assume that l=4, p=1. Then, (1) becomes (simply called K*(4,1) equation)
(4)ut=uxu2+2α(uuxxx+2uxuxx).
We will study single peak solitary solutions of K*(4,1) equation under inhomogeneous boundary condition
(5)limξ→±∞u=A.
Peakon, compacton, cuspon, and smooth soliton solutions are obtained. Our method is based on phase portrait analysis technique under an inhomogeneous boundary condition.


## 2. Asymptotic Behavior of Solutions

In this section, we first introduce some notations. LetCk(Ω) denote the set of all k times continuously differential functions on the open set Ω. Llocp(R) refers to the set of all functions whose restriction on any compact subset is Lp integrable. Hloc1(R) stands for Hloc1(R)={u∈Lloc2(R)∣u′∈Lloc2(R)}. Assume that sn(x,k), cn(x,k), dn(x,k) are the Jacobian elliptic functions with the modulus k,K(k) is the first kind of complete elliptic integral, E(ϕ,k) is the normal elliptic integral of the 2nd kind, and ∏‍(ϕ,α2,k) is the normal elliptic integral of the 3rd kind [13].Let us consider the traveling wave solution ofK*(4,1) equation (4) through a generic setting u(x,t)=u(ξ), ξ=x-ct, where c is wave speed. Substituting it into (4) yields
(6)-cu′=13(u3)′+α[2(uu′′)′+((u′)2)′],
where “′” is the derivative with respect to ξ. Taking the integration twice on both sides leads to
(7)(u′)2=u4+6cu2+12g1u+12g2-12αu,
where g1,g2∈R are two integration constants. Let us solve (7) with the boundary condition (5). From (7) we obtain that
(8)(u′)2=112α[-u3-6cu-3A2(A2+2c)u+4A(A2+3c)]=(u-A)2(u2+2Au+3A2+6c)-12αu.
If A2+3c<0, then (8) reduces to
(9)(u′)2=(u-A)2(u-B1)(u-B2)-12αu,
where
(10)B1=-A+-2(A2+3c),B2=-A--2(A2+3c).
Obviously, B1≥B2. We assume that α>0 throughout the paper since there are similar results for the case α<0.Definition 1.
A functionu(ξ) is said to be a single peak soliton solution for K*(4,1) equation (4) if u(ξ) satisfies the following conditions:(A1)
u(ξ) is continuous on R and has a unique peak point ξ0, where u(ξ) attains its global maximum or minimum value;(A2)
u(ξ)∈C3(R-{ξ0}) satisfies (8) on R-{ξ0};(A3)
limξ→±∞u(ξ)=A.Definition 2.
A wave functionu is called a peakon if u is smooth locally on either side of ξ0 and limξ↑ξ0u′(ξ)=-limξ↓ξ0u′(ξ)=a, a≠0, a≠±∞.Definition 3.
A wave functionu is called a cuspon if u is smooth locally on either side of ξ0 and limξ↑ξ0u′(ξ)=-limξ↓ξ0u′(ξ)=+∞(or-∞).
Without loss of generality, one assumes thatξ0=0.Lemma 4.
Equation (4) has trivial solution u≡A, if one of the following three conditions holds:(i)
A≥0, c>0;(ii)
A>-c, c<0;(iii)
--c≤A<0, c<0.Proof.
(i) IfA≥0, c>0, then u≥0, 2(A2+3c)>0, limξ→±∞u(ξ)=A, and
(11)(u′)2=(u-A)2[(u+A)2+2(A2+3c)]-12αu≤0.
The fact that (u′)2≥0 implies that u′=0 and u≡A.
IfA=0, c>0, then 2(A2+3c)>0, limξ→±∞u(ξ)=0, and (u′)2=u(u2+6c)/-12α≤0. The fact that (u′)2≥0 implies that u′=0 and u≡A.
(ii) IfA≥-3c, c<0, then u>0, 2(A2+3c)≥0, limξ→±∞u(ξ)=A>0, and
(12)(u′)2=(u-A)2[(u+A)2+2(A2+3c)]-12αu≤0.
The fact that (u′)2≥0 implies that u′=0 and u≡A.
If-c<A<-3c, c<0, then 2(A2+3c)<0, A>B1=-A+-2(A2+3c)>B2, limξ→±∞u(ξ)=A>0, and
(13)(u′)2=(u-A)2(u-B1)(u-B2)-12αu≤0.
The fact that (u′)2≥0 implies that u′=0 and u≡A.
(iii) If--c≤A<0, c<0, then 2(A2+3c)<0, B2≤A<B1, limξ→±∞u(ξ)=A<0, and
(14)(u′)2=(u-A)2(B1-u)(u-B2)12αu≤0.
The fact that (u′)2≥0 implies u′=0 and u≡A.Theorem 5.
Suppose thatu(ξ) is a single peak solitary wave solution for (4) at the peak point ξ0=0. If A<0, c>0 or 0≤A≤-c, c<0 or --3c≤A<--c, c<0, then u(0)=0 or u(0)=B1 or u(0)=B2.Proof.
Ifu(0)≠0, then u(ξ)≠0 for any ξ∈R since u(ξ)∈C3(R-{0}). Differentiating both sides of (8) yields u(ξ)∈C∞(R).(i)
IfA≥0, c>0 or A>-c, c<0 or --c≤A<0, c<0, from Lemma 4, we know that (4) has trivial solution u≡A.(ii)
ForA=0, c<0, we have (u′)2=u(-6c-u)(u+-6c)/12α, if u(0)≠0, then, according to the definition of peak point, we have u′(0)=0; thus, u(0)=B1=-6c or u(0)=B2=--6c.(iii)
ForA<0, c>0 or 0<A≤-c, c<0 or --3c≤A<--c, c<0, if u(0)≠0, then u(ξ)∈C∞(R). According to the definition of peak point, we have u′(0)=0. Thus, u(0)=B1 or u(0)=B2 since u(0)=A contradicts the fact that 0 is the unique peak point.Theorem 6.
Suppose thatu(ξ) is a single peak solitary wave solution for (4) at the peak point ξ0=0. When A<0, c>0 or 0≤A≤-c, c<0 or --3c≤A<--c, c<0, then we have the following classification and asymptotic behavior of solutions.(i)
Ifu(0)=B2 and A<B2<0 or u(0)=B1 and 0<A<B1, then u(ξ) is a smooth solitary wave solution.(ii)
IfA=0,c<0, u(0)=B1, then u(ξ) is a compacton solution [2].(iii)
Ifu(0)=0 and A=--2c, then u(ξ) gives the peakon solution
(15)u(ξ)=-2c×[2-3tanh2(tanh-13+23-2+14-2cα|ξ|)].(iv)
Ifu(0)=0 and A≠--2c, c<0 and A<0, c>0 or 0<A≤-c or --3c≤A<--c, then u(ξ) is a cuspon soliton solution and
(16)u(ξ)=-(32h1(0))2/3|ξ|2/3+O(|ξ|),ξ⟶0,u′(ξ)=-23(32h1(0))2/3|ξ|-1/3+O(1),ξ⟶0,or(17)u(ξ)=(32h2(0))2/3|ξ|2/3+O(|ξ|),ξ⟶0,u′(ξ)=23(32h2(0))2/3|ξ|-1/3+O(1),ξ⟶0.Proof.
(i) From the process of proving Theorem5, we know that if u(0)=B1>0 or u(0)=B2<0, then u(0)≠0; thus, u(ξ) is a smooth solitary wave solution.
(ii) IfA=0, then B2=--6c, u(0)=B1=-6c, (9) becomes
(18)u′=u(-6c-u)(u--6c)12αsign(ξ).
Integrating both sides of (18) on the interval (0,-6c] leads to a compacton solution with compact support
(19)u(ξ)={-6c(1-sn2(142-6c3α|ξ|,22)),if|ξ|≤43α2-6cK(22),0,if|ξ|>43α2-6cK(22).
The profile of compacton is shown in Figure 2(b).
Remark
1. To the best of our knowledge, the solution (19) of (4) has not been reported in the literature.
(iii) Ifu(0)=0 and A=--2c, then (9) becomes
(20)u′=(u+-2c)2-6c-u23αsign(ξ).
Integrating both sides of (20) on the interval (--2c,0] leads to a peakon solution
(21)u(ξ)=-2c[2-3tanh2(tanh-123+14-2cα|ξ|)],
with the following properties:
(22)u(0)=0,u(±∞)=A,u′(0+)=c-2c-12cα,u′(0-)=-c-2c-12cα.
The profile of peakon solution is shown in Figure 2(e).
Remark
2. To the best of our knowledge, the solution (21) of (4) has not been reported in the literature.
(iv) Ifu(0)=0, A≠0, and A≠--2c, then u2+2Au+3(A2+2c) does not contain the factor u. From (9) we obtain
(23)u′=(u-A)u2+2Au+3(A2+2c)23α-usign(ξ),A<0,c>0,or--3c≤A<--2c,u′=-(u-A)-(u2+2Au+3(A2+2c))23αusign(ξ),iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii0<A≤-c.
Let h1(u)=23α/(u-A)u2+2Au+3(A2+2c), then h1(0)=-(2/A)α/(A2+2c) and
(24)∫‍-uh1(u)du=∫‍sign(ξ)dξ.
Inserting h1(u)=h1(0)+O(u) into (24) and using the initial condition u(0)=0, we obtain
(25)-23(-u)3/2h1(0)(1+O(1))=|ξ|.
Thus,
(26)u=-|ξ|2/3(32h1(0))2/3(1+O(1))-2/3,ξ⟶0,
which implies that u=O(|ξ|3/2). Therefore, we have
(27)u(ξ)=-(32h1(0))2/3|ξ|2/3+O(|ξ|),ξ⟶0,(28)u′(ξ)=-23(32h1(0))2/3|ξ|-1/3+O(1),ξ⟶0.
Thus, u(ξ)∉Hloc1(R).
Leth2(u)=23α/(A-u)-(u2+2Au+3(A2+2c)); then h2(0)=(2/A)α/(-A2-2c) and
(29)∫‍uh2(u)du=∫‍sign(ξ)dξ.
Inserting h2(u)=h2(0)+O(u) into (29) and using the initial condition u(0)=0, we obtain
(30)23(u)3/2h2(0)(1+O(1))=|ξ|.
Thus,
(31)u=|ξ|2/3(32h2(0))2/3(1+O(1))-2/3,ξ⟶0,
which implies that u=O(|ξ|3/2). Therefore, we have
(32)u(ξ)=(32h2(0))2/3|ξ|2/3+O(|ξ|),ξ⟶0,u′(ξ)=23(32h2(0))2/3|ξ|-1/3+O(1),ξ⟶0.
Thus, u(ξ)∉Hloc1(R).


## 3. Smooth, Peaked, and Cusped Single Peak Solitary Wave Solutions

Theorem6 gives a classification for all single peak solitary wave solutions for (4). In this section, we will present all possible soliton solutions for (4). We shall discuss the four cases: Case 1: A<0, c>0; Case 2: A=0, c<0; Case 3: 0<A≤-c, c<0 or --3c≤A<--c, c<0.Case 1 (A<0, c>0, u(0)=0).
In this case, according to Theorem5 and standard phase portrait analytical technique (see Figure 1(a)), we have A<u≤0 and
(33)u′=-sign(A)u-Au×-u(u2+2Au+3A2+6c)12αsign(ξ).
Integrating both sides of (33) on the interval (A,0] leads to
(34)∫u0‍udu(u-A)-u[(u+A)2+2(A2+3c)]=-|ξ|23α.
Thus we obtain the implicit solutionu(ξ) defined by
(35)F(u)≡cn-1(Φ(u),k)-11-α1×[∏‍(ϕ,α12α12-1,k)-α1f1]=(A-δ)|ξ|23αδ,
where  Φ(u)=(δ+u)/(δ-u), ϕ=arccosΦ(u), f1=(1-α12)/(k2(1-α12)+α12)arctan((k2(1-α12)+α12)/(1-α12)sd(u1,k)),  α1=Φ(A), k2=(δ+A)/2δ, δ=3(A2+2c).
In view ofϕ(u)=((u-A)/u)-u(u2+2Au+3A2+6c)/12α<0, we know that F(u) is strictly decreasing on (A,0] with F1(u)=F(A,0](u), which gives a unique cuspon soliton solution u1(ξ) satisfying u1(0)=0, limξ→±∞u1(ξ)=A, u1′(0+)=-∞, u1′(0-)=+∞. The profile of cuspon soliton solution is shown in Figure 2(a).The phase portraits of (8) on the (u,u′) plane.
(a)
A
<
0, c>0
(b)
A
=
0, c<0
(c)
β
=
0
(d)
-
1
<
β
<
0
(e)
β
=
-
1
(f)
-
2
<
β
<
-
1
(g)
β
=
2
(h)
β
>
2The profiles of waves forα=1. (a) A=-2, c=4, (b) A=0, c=-2, (c) β=0, A=-3, c=-3, (d) -1<β<0, A=-2.1, c=-2, (e) β=-1, A=-2, c=-2, (f) -2<β<-1, A=-2.1, c=-4, (g) β=2, A=2, c=-4, (h) β>2, A=1, c=-4.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)Case 2 (A=0, c<0, u(0)=B1=-6c).
In this case, by the standard phase portrait analysis (see Figure1(b)), we have 0<u≤-6c and then obtain a compacton solution (see (19)). The profile of compacton is shown in Figure 2(b).Case 3 (0<A≤-c, c<0 or --3c≤A<--c, c<0).
By virtue of Theorem5, any single peak soliton solution for (4) must satisfy the following initial and boundary values problem:
(36)(u′)2=(u-A)2(B1-u)(u-B2)12αu,u(0)∈{0,B1,B2},limξ→±∞u(ξ)=A.
Equation (36) implies
(37)(B1-u)(u-B2)u≥0,(38)(B1-A)(A-B2)A≥0.(i)
WhenA>0, from (37) we obtain
(39)(-2+1A-2(A2+3c))(2+1A-2(A2+3c))≥0.(ii)
WhenA<0, from (37) we obtain
(40)(-2+1A-2(A2+3c))(2+1A-2(A2+3c))≤0.Since A2+3c≤0, A≠0, introducing the constant β=(1/A)-2(A2+3c) yields
(41)(-2+β)(2+β)≥0,A>0,(-2+β)(2+β)≤0,A<0,
which implies that for A>0(42)β≥2
and for A<0(43)-2≤β≤0.
From the standard phase portrait analysis (see Figures1(c)–1(h)), we know that if u(ξ) is a single peak soliton solution of (4), then
(44)u′=-sign(A)u-A2uu(B1-u)(u-B2)3αsign(ξ).
In the following part, let us separate seven cases to discuss.
(1)β=-1. In this case, by the standard phase portrait analysis (see Figure 1(e)), we have
(45)--2c=A<B2=0<B1=2-2c.(i)
u(0)=B2=0. If u(0)=B2, then A<u≤B2. In this case, by the standard phase portrait analysis (see Figure 1(e)), we obtain a peakon solution (see (21)). The profile of peakon solution is shown in Figure 2(e).(ii)
u(0)=B1. In this case there is no single peak solitary wave solution for the boundary condition u(±∞)=--2c.
(2)-2<β<-1. In this case, by the standard phase portrait analysis (see Figure 1(f)), we have
(46)A<B2<0<B1.(i)
u(0)=B2. If u(0)=B2, then A<u≤B2. Integrating both sides of (47) on the interval (A,B2] leads to
(47)Q(u)≡∏‍(ϕ,AA-B2,B1B1-B2)=A-B24B2B1-B23α|ξ|,whereϕ=arcsin(B2-u)/-u.Fromϕ(u)<0, we know that Q(u) is strictly decreasing on the interval (A,B2], Q1(u)=Q(A,B2](u) has the inverse denoted by
(48)u2(ξ)=B2sec2(Π-1(A-B24B2B1-B23α|ξ|,iiiiiiiiiiiiAA-B2,B1B1-B2)).Corresponding to the homoclinic orbit to the saddle point(A,0) shown in Figure 1(f), u2(ξ) gives a smooth soliton solution satisfying
(49)u2(0)=B2,limξ→±∞u2(ξ)=A,u2′(0)=0.The profile of smooth soliton solution is shown in Figure 2(f).
Remark
3.  To the best of our knowledge, the solution (48) of (4) has not been reported in the literature.(ii)
u(0)=B1 or u(0)=0. In this case there is no single peak solitary wave solution for the boundary condition u(±∞)=A.
(3)-1<β<0. In this case, by the standard phase portrait analysis (see Figure 1(d)), we have
(50)A<0<B2<B1.(i)
u(0)=0. If u(0)=0, then A<u≤0. Integrating both sides of (44) on the interval (A,0] leads to
(51)W(u)≡∏‍(arcsinω1,A-B2A,k)-sn-1(ω1,k)=B2-A4B2B13α|ξ|,whereω1=-u/(B2-u), k=(B1-B2)/B1.From((u-A)/2u)u(B1-u)(u-B2)/3α<0, we know that W(u) is strictly decreasing on (A,0],
(52)W1(u)=W(A,0](u)has the inverse denoted byu3(ξ)=W1-1(((B2-A)/4B2)B1/3α|ξ|), and u3(ξ) gives a cuspon soliton solution satisfying
(53)u3(0)=0,limξ→±∞u3(ξ)=A,u3′(0+)=-∞,u3′(0-)=+∞.The profile of cuspon soliton solution is shown in Figure2(d).(ii)
u(0)=B2 or u(0)=B1. In this case there is no single peak solitary wave solution for the boundary condition u(±∞)=A.
(4)β=-2. In this case, we have A=B1=--c, by virtue of Lemma 4(iii), (4) has trivial solution u≡A.
(5)β=0. In this case, by the standard phase portrait analysis (see Figure 1(c)), we have
(54)--3c=A<0<B2=B1=-3c.(i)
u(0)=0. If u(0)=0, then A<u≤0. Integrating both sides of (44) on the interval (A,0] leads to
(55)Z(u)≡(tanh-1-u(-3c)1/4-arctan-u(-3c)1/4)=(-3c)1/423α|ξ|.Fromϕ(u)<0, we know that Z(u) is strictly decreasing on (A,0],
(56)Z1(u)=Z(A,0](u)has the inverse denoted byu4(ξ)=Z1-1(((-3c)1/4/23α)|ξ|), and u4(ξ) gives a cuspon soliton solution satisfying
(57)u4(0)=0,limξ→±∞u4(ξ)=A,u4′(0+)=-∞,u4′(0-)=+∞.The profile of cuspon soliton solution is shown in Figure2(c).(ii)
u(0)=B2=B1. In this case, there is no single peak solitary wave solution for the boundary condition u(±∞)=A.
(6)β=2. In this case, by the standard phase portrait analysis (see Figure 1(g)), we have
(58)-3-c=B2<A=B1=-c.(i)
u(0)=0. If u(0)=0, then 0≤u<A. Integrating both sides of (44) on the interval [0,A) leads to
(59)Ψ(u)≡-dn(sn-1(-c-u-c,12),12)×cs(sn-1(-c-u-c,12),12)-E(arcsin(-c-u-c),12)=12(-c9α2)1/4|ξ|.In view ofϕ(u)<0, we know that Ψ(u) is strictly decreasing on [0,A)  and
(60)Ψ1(u)=Ψ[0,A)(u)has the inverse which is denoted byu5(ξ)=Ψ1-1(((-3c)1/4/23α)|ξ|), where u5(ξ) gives a cuspon soliton solution satisfying
(61)u5(0)=0,limξ→±∞u5(ξ)=A,u5′(0+)=+∞,u5′(0-)=-∞.The profile of cuspon soliton solution is shown in Figure2(g).(ii)
u(0)=B2. In this case there is no single peak solitary wave solution for the boundary condition u(±∞)=A.
(7)β>2. In this case, by the standard phase portrait analysis (see Figure 1(h)), we have
(62)B2<0<A<B1.(i)
u(0)=0. If u(0)=0, then 0≤u<A. This case is completely similar to the case of β=2,u(0)=0.(ii)
u(0)=B1. If u(0)=B1, then A<u≤B1. Integrating both sides of (44) on the interval (A,B1] leads to
(63)Γ(u)≡-sn-1(ω2,k)+AA-B1×∏‍(arcsinω2,B1B1-A,k)=14B1-B23α|ξ|,whereω2=(B1-u)/B1, k=B1/(B1-B2).In view ofϕ(u)>0, we know that Γ(u) is strictly increasing on [0,A) and
(64)Γ1(u)=Γ[0,A)(u)has the inverse which is denoted byu6(ξ)=Γ1-1(((-3c)1/4/23α)|ξ|), where u6(ξ) gives a smooth soliton solution satisfying
(65)u6(0)=0,limξ→±∞u6(ξ)=A,u6′(0)=0.The profile of smooth soliton solution is shown in Figure2(h).(iii)
u(0)=B2. In this case there is no single peak solitary wave solution for the boundary condition u(±∞)=A.Let us summarize our results in the following theorem.Theorem 7.
Suppose thatu(ξ) is a single peak soliton for (4) at the peak point ξ0=0, which satisfies the boundary condition (5). Then we have the following conclusions.
(i)A<0, c>0. If u(0)=0, (4) has cuspon soliton solution which can be expressed as
(66)u(ξ)=F(A,0]-1((A-δ)|ξ|23αδ),
where δ=3(A2+2c). It satisfies
(67)u(0)=0,limξ→±∞u(ξ)=A,(68)u′(0+)=-∞,u′(0-)=+∞.
(ii)A=0, c<0. If u(0)=B1=-6c, the only possible quasi single peak soliton is compacton:
(69)u(ξ)={-6c(1-sn2(142-6c3α|ξ|,22)),if|ξ|≤43α2-6cK(22),0,if|ξ|>43α2-6cK(22),
with the following properties:
(70)u(0)=-6c,u′(0)=0.
(iii)0<A≤-c, c<0, or --3c≤A<--c,c<0.(1)
β=-1. If β=-1, then u(0)=0=B2, and (4) has peakon soliton solution which can be expressed as
(71)u(ξ)=-2c[2-3tanh2(tanh-123+14-2cα|ξ|)],with the following properties:(72)u(0)=0,u(±∞)=A,u′(0+)=c-2c-12cα,u′(0-)=-c-2c-12cα.(2)
-2<β<-1. If u(0)=B2, (4) has smooth soliton solution which can be expressed as
(73)u(ξ)=B2sec2(Π-1(A-B24B2B1-B23α|ξ|,iiiiiiiiiiiiiAA-B2,B1B1-B2)),with the following properties:(74)u(0)=B2,limξ→±∞u(ξ)=A,u′(0)=0.(3)
-1<β<0. If u(0)=0, (4) has cuspon soliton solution which can be expressed as
(75)u(ξ)=W1-1(B2-A4B2B13α|ξ|),with the following properties:(76)u(0)=0,limξ→±∞u(ξ)=A,u′(0+)=-∞,u′(0-)=+∞.(4)
β=0. If u(0)=0, (4) has cuspon soliton solution which can be expressed as
(77)u(ξ)=Z1-1((-3c)1/423α|ξ|),with the following properties:(78)u(0)=0,limξ→±∞u(ξ)=A,u′(0+)=-∞,u′(0-)=+∞.(5)
β=2. If u(0)=0, (4) has cuspon soliton solution which can be expressed as
(79)u(ξ)=Ψ1-1((-3c)1/423α|ξ|),with the following properties:(80)u(0)=0,limξ→±∞u(ξ)=A,u′(0+)=+∞,u′(0-)=-∞.(6)
β>2.
(i)
u(0)=0. If u(0)=0, then 0≤u<A. This case is completely similar to the case of β=2,u(0)=0.(ii)
u(0)=B1. If u(0)=B1, (4) has cuspon soliton solution which can be expressed as
(81)u(ξ)=Γ1-1((-3c)1/423α|ξ|),with the following properties:(82)u(0)=0,limξ→±∞u(ξ)=A,u′(0)=0.


## 4. Conclusion

In this paper, we study the single peak solitary wave solutions ofK*(4,1) equation under inhomogeneous boundary condition. The conditions of the existence of peakon, compacton, cuspon, and smooth soliton solutions are given by using phase portrait analytical technique. We have obtained all peakon, compacton, cuspon, and smooth soliton solutions of K*(4,1) equation and analyzed their analytic and dynamical behavior. We have gotten a new type of smooth soliton, which is expressed in terms of trigonometric functions (see (48)), for K*(4,1) equation. New peaked solitons and new type of smooth soliton solutions are expected to apply in nonlinear shallow-water wave theory and Newton motion theory because they have a very close relation. Actually, the ODE (8) has a physical meaning and coincides with the Newton equation of a particle in the potential
(83)V(u)=u3+(6c-A2)u-12α+A2(3A2+6c)12αu.
We solve the Newton equation (u′)2=V(u)-V(A), for all possible single peak solitary wave solutions, where V(A)=-A(3A2+6c)/6α.



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