# Explicit Wave Solutions and Qualitative Analysis of the (1 + 2)-Dimensional Nonlinear Schrödinger Equation with Dual-Power Law Nonlinearity

Authors: Qing Meng; Bin He; Zhenyang Li
Journal: Advances in Mathematical Physics
 (2015)

---

## Abstract

The (1 + 2)-dimensional nonlinear Schrödinger equation with dual-power law nonlinearity is studied using the factorization technique, bifurcation theory of dynamical system, and phase portraits analysis. From a dynamic point of view, the existence of smooth solitary wave, and kink and antikink waves is proved and all possible explicit parametric representations of these waves are presented.


---

## Body

## 1. Introduction

The nonlinear Schrödinger equation (NLSE) is studied in various areas of applied mathematics, theoretical physics, and engineering. In particular, it appears in the study of nonlinear optics, plasma physics, fluid dynamics, biochemistry, and many other areas. This equation is completely integrable by the inverse scattering transform [1]. There have been various forms of this equation that arise in the study of different areas. They have different forms of nonlinearity. Some previous works concentrated on the Kerr law nonlinearity media [2–5]. However, as one increases the intensity of the incident light power to produce shorter (femtosecond) pulses, non-Kerr nonlinearity effects become prominent, and the dynamics of pulses should be described by the nonlinear Schrödinger family of equations with higher order nonlinear terms [6]. Hence, it is very important that all higher order effects are considered in the propagation of femtosecond pulses [7].The(1+2)-dimensional nonlinear Schrödinger equation with dual-power law nonlinearity is given in [8–13]: (1)iqt+12qxx+qyy+q2m+kq4mq=0,where q(x,y,t) is a complex function that indicates the complex amplitude of the wave form, i2=-1, m is an arbitrary nonzero constant, and k is a constant which indicates the saturation of the nonlinear refractive index. Biswas [8] obtained the 1-soliton solution and calculated a few conserved quantities. With the aid of symbolic computation, Zhang and Si [9] obtained soliton solutions, combined soliton solutions, triangular periodic solutions, and rational function solutions. Some soliton solutions are obtained in [10] using the G/G′-expansion method. For m=1, the dark 1-soliton solution is obtained by the ansatz method, the invariance, conservation laws, and double reductions in [11]. Bulut et al. [12] found some soliton solutions, rational and elliptic function solutions using the extended trial equation method. The bifurcation is considered in [13] and two implicit solutions are obtained (see (16) and (18) of [13]).In this paper, using the factorization technique [14], bifurcation theory of dynamical system, and phase portraits analysis [15–17], we present some explicit expressions of the smooth solitary, kink, and antikink waves of (1).Throughout this paper, we always suppose thatm≠-1 and -1/2, k≠0.


## 2. Preliminaries

Using transformation(2)qx,y,t=eiηuξ,η=x+c1y+c2t,ξ=x+ω1y+ω2t,where c1, c2, ω1, and ω2 are constants, (1) can be rewritten as (3)121+ω12u′′+ku4m+1+u2m+1-121+c12+2c2u+i1+c1ω1+ω2u′eiη=0,where “′” is the derivative with respect to ξ.From (3), we have (4)1+c1ω1+ω2u′=0,(5)121+ω12u′′+ku4m+1+u2m+1-121+c12+2c2u=0.Obviously, if putting ω2=-(1+c1ω1), then solution of (5) must be solution of (1).Taking transformation(6)u=ϕ1/2m.Equation (5) is converted into the following equation: (7)8km2ϕ4+8m2ϕ3-41+c12+2c2m2ϕ2+21+ω12mϕϕ′′+1+ω121-2mϕ′2=0.Differentiating (7) once with respect to ξ, we have (8)21+ω12mϕϕ′′′+21+ω121-mϕ′ϕ′′+32km2ϕ3+24m2ϕ2-81+c12+2c2m2ϕϕ′=0.Using Proposition 2 of [14], we know that (8) has the factorization (9)fϕ∂ξ-ϕ1ϕϕ′-ϕ2ϕ∂ξξ-ϕ3ϕ∂ξ-ϕ4ϕϕ=0if and only if the following expressions are satisfied: (10)gϕ,ϕ′=-fϕϕ3ϕ-ϕ1ϕϕ′-ϕ2ϕ,ϕ1ϕϕ3ϕ-fϕdϕ3ϕdϕ=0,kϕ=ϕ2ϕϕ4ϕϕ,hϕ=ϕ2ϕϕ3ϕ-fϕdϕ4ϕdϕϕ-fϕϕ4ϕ+ϕ1ϕϕ4ϕϕ,wheref(ϕ)=2(1+ω12)mϕ, g(ϕ,ϕ′)=2(1+ω12)(1-m)ϕ′, and h(ϕ)=32km2ϕ3+24m2ϕ2-8(1+c12+2c2)m2ϕ, k(ϕ)=0.Solving (10) and setting the integral constant being zero, we have (11)ϕ1ϕ=21+ω12m-1,ϕ2ϕ=ϕ3ϕ=0,ϕ4ϕ=-4m24km+1ϕ2+32m+1ϕ-1+c12+2c2m+12m+11+ω12m+12m+1.Thus, (8) has the following factorization: (12)21+ω12mϕ∂ξ-21+ω12m-1ϕ′∂ξξ+4m2Aϕ2+Bϕ+C1+ω12m+12m+1ϕ=0,where A=4k(m+1), B=3(2m+1), and C=-1+c12+2c2m+12m+1.Some special solutions of (7) could be obtained by solving the following second-order differential equation: (13)ϕ′′=ρϕ2+Eϕ+Fϕ,where ρ=-16km2/(1+ω12)(2m+1), E=3(2m+1)/4k(m+1), and F=-(1+c12+2c2)(2m+1)/4k.Lettingy=dϕ/dξ, we get the following planar dynamical system: (14)dϕdξ=y,dydξ=ρϕ2+Eϕ+Fϕwith the first integral (15)Hϕ,y=y2-16ρ3ϕ2+4Eϕ+6Fϕ2=h.For a fixedh, the level curve H(ϕ,y)=h defined by (15) determines a set of invariant curves of (14) which contain different branches. As h is varied, it defines different families of orbits of (14) with different dynamical behaviors.The remainder of this paper is organized as follows. In Section3, we consider bifurcation sets and phase portraits of (14). Some explicit smooth solitary, kink, and antikink wave solutions of (1) are presented in Section 4. A short conclusion will be provided in Section 5.


## 3. Bifurcation Sets and Phase Portraits of (14)

We will use the following notations:(16)Δ=E2-4F,ϕ1,2=12-E±Δ.Obviously, (14) has only one equilibrium point at (0,0) in ϕ-axis when Δ<0. Equation (14) has two equilibrium points at (-(1/2)E,0) and (0,0) in ϕ-axis when Δ=0. Equation (14) has three equilibrium points at (ϕ1,2,0) and (0,0) in ϕ-axis when Δ>0. From (15), the following conclusions hold: (17)h1=Hϕ1,0=148ρE-Δ2EE-Δ-6F,h2=Hϕ2,0=148ρE+Δ2EE+Δ-6F.LetM(ϕe,0) be the coefficient matrix of the linearized system of (14) at equilibrium point (ϕe,0). We have (18)Jϕe,0=det⁡Mϕe,0=-ρ3ϕe2+2Eϕe+F.For an equilibrium point(ϕe,0) of (14), we know that (ϕe,0) is a saddle point if J(ϕe,0)<0, a center point if J(ϕe,0)>0, and a cusp if J(ϕe,0)=0 and the Poincaré index of (ϕe,0) is zero.Using the properties of equilibrium points and the bifurcation method, we can obtain four bifurcation curves of (14) as follows: (19)C1:E=0,C2:F=0,C3:F=29E2,C4:F=14E2.The bifurcation curves C1, C2, C3, and C4 divide the (E,F)-parameter plane into 14 subregions. The bifurcation sets and phase portraits of (14) are drawn in Figures 1-2.Figure 1
Bifurcation sets and phase portraits of (14) when ρ<0.
(a)
E
<
0, F<0
(b)
E
>
0, F<0
(c)
E
<
0, F=0
(d)
E
>
0, F=0
(e)
E
<
0, 0<F<(2/9)E2
(f)
E
>
0, 0<F<(2/9)E2
(g)
E
<
0, F=(2/9)E2
(h)
E
>
0, F=(2/9)E2
(i)
E
<
0, (2/9)E2<F<(1/4)E2
(j)
E
>
0, (2/9)E2<F<(1/4)E2
(k)
E
<
0, F=(1/4)E2
(l)
E
>
0, F=(1/4)E2
(m)
E
<
0, F>(1/4)E2
(n)
E
>
0, F>(1/4)E2Figure 2
Bifurcation sets and phase portraits of (14) when ρ>0.
(a)
E
<
0, F<0
(b)
E
>
0, F<0
(c)
E
<
0, F=0
(d)
E
>
0, F=0
(e)
E
<
0, 0<F<(2/9)E2
(f)
E
>
0, 0<F<(2/9)E2
(g)
E
<
0, F=(2/9)E2
(h)
E
>
0, F=(2/9)E2
(i)
E
<
0, (2/9)E2<F<(1/4)E2
(j)
E
>
0, (2/9)E2<F<(1/4)E2
(k)
E
<
0, F=(1/4)E2
(l)
E
>
0, F=(1/4)E2
(m)
E
<
0, F>(1/4)E2
(n)
E
>
0, F>1/4E2


## 4. Explicit Smooth Solitary, Kink and Anti-Kink Wave Solutions of (1)

In this section we present all possible explicit smooth solitary, kink, and antikink wave solutions of (1) in the following propositions.Proposition 1.
Whenρ<0, E≠0, and F<0, (1) has two smooth solitary wave solutions as follows: (20)qx,y,t=2ϕmϕM1/2meix+c1y+c2tϕm+ϕM+ϕM-ϕmcosh⁡ωx+ω1y-1+c1ω1t1/2m,(21)qx,y,t=2ϕmϕM1/2meix+c1y+c2tϕm+ϕM-ϕM-ϕmcosh⁡ωx+ω1y-1+c1ω1t1/2m,where ϕM,m=(1/3)(-2E±2(2E2-9F)), ω=(1/2)ρϕmϕM.
Whenρ<0, E<0, 0<F<(1/4)E2, and F≠(2/9)E2, (1) has two smooth solitary wave solutions as follows: (22)qx,y,t=ϕ2ϕM-ϕmcosh⁡ωx+ω1y-1+c1ω1t-ϕ2ϕm+ϕM-2ϕmϕM1/2meix+c1y+c2tϕM-ϕmcosh⁡ωx+ω1y-1+c1ω1t+ϕm+ϕM-2ϕ21/2m,(23)qx,y,t=ϕ2ϕM-ϕmcosh⁡ωx+ω1y-1+c1ω1t+ϕ2ϕm+ϕM-2ϕmϕM1/2meix+c1y+c2tϕM-ϕmcosh⁡ωx+ω1y-1+c1ω1t-ϕm+ϕM-2ϕ21/2m,where ϕM,m=(1/6)(3Δ-E±2E(E-3Δ)), ω=(1/2)ρ(ϕM-ϕ2)(ϕm-ϕ2).
Whenρ<0, E>0, 0<F<(1/4)E2, and F≠(2/9)E2, (1) has two smooth solitary wave solutions as follows: (24)qx,y,t=ϕ1ϕM-ϕmcosh⁡ωx+ω1y-1+c1ω1t-ϕ1ϕm+ϕM-2ϕmϕM1/2meix+c1y+c2tϕM-ϕmcosh⁡ωx+ω1y-1+c1ω1t+ϕm+ϕM-2ϕ11/2m,(25)qx,y,t=ϕ1ϕM-ϕmcosh⁡ωx+ω1y-1+c1ω1t+ϕ1ϕm+ϕM-2ϕmϕM1/2meix+c1y+c2tϕM-ϕmcosh⁡ωx+ω1y-1+c1ω1t-ϕm+ϕM-2ϕ11/2m,where ϕM,m=-(1/6)(3Δ+E∓2E(E+3Δ)), ω=(1/2)ρ(ϕM-ϕ1)(ϕm-ϕ1).
Whenρ<0, E≠0, and F=(2/9)E2, (1) has two smooth solitary wave solutions as follows: (26)qx,y,t=13E-1±2sech⁡ωx+ω1y-1+c1ω1t1/2meix+c1y+c2t,where ω=-(E/3)-ρ.Proof of Proposition1.
From Figures1(a) and 1(b), when ρ<0, E≠0, and F<0, there are two homoclinic orbits connecting with the saddle point (0,0) and passing through the points ϕm,0 and (ϕM,0), respectively, where ϕM,m=(1/3)(-2E±2(2E2-9F)). Their expressions are, respectively, (27)y=±-12ρϕϕM-ϕϕ-ϕm,ϕm≤ϕ<0,(28)y=±-12ρϕϕM-ϕϕ-ϕm,0<ϕ≤ϕM.Substituting (27) and (28) into dϕ/dξ=y, respectively, and integrating them along the homoclinic orbits, we have (29)∫ϕmϕdssϕM-ss-ϕm=--12ρξ,(30)∫ϕϕMdssϕM-ss-ϕm=-12ρξ.Completing the integrals in (29) and (30), we obtain two smooth solitary wave solutions of (7) as follows: (31)ϕξ=2ϕmϕMϕm+ϕM±ϕM-ϕmcosh⁡ωξ,where ω=(1/2)ρϕmϕM. The profiles of (31) are shown in Figures 3(a) and 3(b). From (2), (6), (31), and ω2=-(1+c1ω1), we obtain two smooth solitary wave solutions of (1) with ρ<0, E≠0, and F<0 given in (20) and (21).
From Figures1(e) and 1(i), when ρ<0,  E<0,  0<F<(1/4)E2, and F≠(2/9)E2, there are two homoclinic orbits connecting with the saddle point (ϕ2,0) and passing through the points (ϕm,0) and (ϕM,0), respectively, where ϕM,m=(1/6)(3Δ-E±2E(E-3Δ)). Their expressions are, respectively, (32)y=±-12ρϕ2-ϕϕM-ϕϕ-ϕm,ϕm≤ϕ<ϕ2,(33)y=±-12ρϕ-ϕ2ϕM-ϕϕ-ϕm,ϕ2<ϕ≤ϕM.Substituting (32) and (33) into dϕ/dξ=y, respectively, and integrating them along the homoclinic orbits, we have (34)∫ϕmϕdsϕ2-sϕM-ss-ϕm=-12ρξ,(35)∫ϕϕMdss-ϕ2ϕM-ss-ϕm=-12ρξ.Completing the integrals in (34) and (35), we obtain two smooth solitary wave solutions of (7) as follows: (36)ϕξ=ϕ2ϕM-ϕmcosh⁡ωξ-ϕ2ϕm+ϕM-2ϕmϕMϕM-ϕmcosh⁡ωξ+ϕm+ϕM-2ϕ2,(37)ϕξ=ϕ2ϕM-ϕmcosh⁡ωξ+ϕ2ϕm+ϕM-2ϕmϕMϕM-ϕmcosh⁡ωξ-ϕm+ϕM-2ϕ2,where ω=-(1/2)ρ(ϕM-ϕ2)(ϕ2-ϕm). The profiles of (36) and (37) are shown in Figures 3(c) and 3(d), respectively. From (2), (6), (36), (37), and ω2=-(1+c1ω1), we obtain two smooth solitary wave solutions of (1) with ρ<0, E<0, 0<F<(1/4)E2, and F≠(2/9)E2 given in (22) and (23).
From Figures1(f) and 1(j), when ρ<0, E>0, 0<F<(1/4)E2, and F≠(2/9)E2, there are two homoclinic orbits connecting with the saddle point (ϕ1,0) and passing through the points (ϕm,0) and (ϕM,0), respectively, where ϕM,m=-(1/6)(3Δ+E∓2E(E+3Δ)). Their expressions are, respectively, (38)y=±-12ρϕ1-ϕϕM-ϕϕ-ϕm,ϕm≤ϕ<ϕ1,(39)y=±-12ρϕ-ϕ1ϕM-ϕϕ-ϕm,ϕ1<ϕ≤ϕM.Substituting (38) and (39) into dϕ/dξ=y, respectively, and integrating them along the homoclinic orbits, we have (40)∫ϕmϕdsϕ1-sϕM-ss-ϕm=-12ρξ,(41)∫ϕϕMdss-ϕ1ϕM-ss-ϕm=-12ρξ.Completing the integrals in (40) and (41), we obtain two smooth solitary wave solutions of (7) as follows: (42)ϕξ=ϕ1ϕM-ϕmcosh⁡ωξ-ϕ1ϕm+ϕM-2ϕmϕMϕM-ϕmcosh⁡ωξ+ϕm+ϕM-2ϕ1,(43)ϕξ=ϕ1ϕM-ϕmcosh⁡ωξ+ϕ1ϕm+ϕM-2ϕmϕMϕM-ϕmcosh⁡ωξ-ϕm+ϕM-2ϕ1,where ω=-(1/2)ρ(ϕM-ϕ1)(ϕ1-ϕm). The profiles of (42) and (43) are shown in Figures 3(e) and 3(f), respectively. From (2), (6), (42), (43), and ω2=-(1+c1ω1), we obtain two smooth solitary wave solutions of (1) with ρ<0, E>0, 0<F<(1/4)E2, and F≠(2/9)E2 given in (24) and (25).
From Figures1(g) and 1(h), when ρ<0, E≠0, and F=(2/9)E2, there are two homoclinic orbits connecting with the saddle point (-(1/3)E,0) and passing through the points ϕm,0 and (ϕM,0), respectively, where ϕM,m=-(1/3)(1±2)E when ρ<0, E<0, F=(2/9)E2, and ϕM,m=-(1/3)(1∓2)E when ρ<0, E>0, and F=(2/9)E2. Their expressions are, respectively, (44)y=±-12ρ-13E-ϕϕM-ϕϕ-ϕm,ϕm≤ϕ<-13E,(45)y=±-12ρϕ+13EϕM-ϕϕ-ϕm,-13E<ϕ≤ϕM.Substituting (44) and (45) into dϕ/dξ=y, respectively, and integrating them along the homoclinic orbits, we have (46)∫ϕmϕds-1/3E-sϕM-ss-ϕm=-12ρξ,(47)∫ϕϕMdss+1/3EϕM-ss-ϕm=-12ρξ.Completing the integrals in (46) and (47), we obtain two smooth solitary wave solutions of (7) as follows: (48)ϕξ=13E-1±2sech⁡ωξ,where ω=-(E/3)-ρ. The profiles of (48) are shown in Figures 3(g) and 3(h). From (2), (6), (48), and ω2=-(1+c1ω1), we obtain two smooth solitary wave solutions of (1) with ρ<0, E≠0, and F=(2/9)E2 given in (26).
The proof of the Proposition1 is completed.Figure 3
Smooth solitary waves of (7).
(a)
ρ
=
-
1.5, E=1.0, and F=-1.2
(b)
ρ
=
-
1.5, E=1.0, and F=-1.2
(c)
ρ
=
-
1.5, E=-1.0, and F=0.1
(d)
ρ
=
-
1.5, E=-1.0, and F=0.1
(e)
ρ
=
-
1.5, E=1.0, and F=0.1
(f)
ρ
=
-
1.5, E=1.0, and F=0.1
(g)
ρ
=
-
1.5, E=-1.0, and F=2/9
(h)
ρ
=
-
1.5, E=-1.0, and F=2/9
(i)
ρ
=
-
1.5, E=-1.0, and F=0
(j)
ρ
=
-
1.5, E=1.0, and F=0
(k)
ρ
=
-
1.5, E=-1.0, and F=1/4
(l)
ρ
=
-
1.5
,
E
=
1.0
,
F
=
1
/
4Proposition 2.
Whenρ<0, E≠0, and F=0, (1) has one smooth solitary wave solution as follows: (49)qx,y,t=-4E1/2meix+c1y+c2t31+ωx+ω1y-1+c1ω1t21/2m,where ω=(2E/3)-(1/2)ρ.
Whenρ<0, E≠0, and F=(1/4)E2, (1) has one smooth solitary wave solution as follows: (50)qx,y,t=E1-3ωx+ω1y-1+c1ω1t21/2meix+c1y+c2t61+ωx+ω1y-1+c1ω1t21/2m,where ω=(E/3)-(1/2)ρ.
Whenρ>0, E<0, and F<0 (or ρ>0, E>0, and (2/9)E2<F<(1/4)E2), (1) has one smooth solitary wave solution the same as in (22).
Whenρ>0, E>0, and F<0 (or ρ>0, E<0, and (2/9)E2<F<(1/4)E2), (1) has one smooth solitary wave solution the same as in (25).
Whenρ>0, E<0, and 0<F<(2/9)E2, (1) has one smooth solitary wave solution the same as in (20).
Whenρ>0, E>0, and 0<F<(2/9)E2, (1) has one smooth solitary wave solution the same as in (21).Proof of Proposition2.
From Figures1(c) and 1(d), when ρ<0, E≠0, and F=0, there is a homoclinic orbit connecting with the cusp (0,0) and passing through the point (-(4/3)E,0). When ρ<0, E<0, and F=0, its expression is (51)y=±-12ρϕϕ-43E-ϕ,0<ϕ≤-43E.When ρ<0, E>0, and F=0, its expression is (52)y=±-12ρϕ-ϕϕ+43E,-43E≤ϕ<0.Substituting (51) and (52) into dϕ/dξ=y, respectively, and integrating them along the homoclinic orbits, we have (53)∫ϕ-4E/3dsss-4/3E-s=-12ρξ,(54)∫-4E/3ϕdss-ss+4/3E=--12ρξ.Completing the integral in (53), we obtain a smooth solitary wave solution of (7) as follows: (55)ϕξ=-4E31+ωξ2,where ω=(2E/3)-(1/2)ρ. Completing the integral in (54), we obtain a smooth solitary wave solution of (7) the same as in (55). The profiles of (55) are shown in Figures 3(i) and 3(j). From (2), (6), (55), and ω2=-(1+c1ω1), we obtain a smooth solitary wave solution of (1) with ρ<0, E≠0, and F=0 given in (49).
From Figures1(k) and 1(l), when ρ<0, E≠0, and F=(1/4)E2, there is a homoclinic orbit connecting with the cusp (-(1/2)E,0) and passing through the point ((1/6)E,0). When ρ<0, E<0, and F=(1/4)E2, its expression is (56)y=±-12ρ-12E-ϕ-12E-ϕϕ-16E,16E≤ϕ<-12E.When ρ<0, E>0, and F=(1/4)E2, its expression is (57)y=±-12ρϕ+12Eϕ+12E16E-ϕ,-12E<ϕ≤16E.Substituting (56) and (57) into dϕ/dξ=y, respectively, and integrating them along the homoclinic orbits, we have (58)∫E/6ϕds-1/2E-s-1/2E-ss-1/6E=-12ρξ,(59)∫ϕE/6dss+1/2Es+1/2E1/6E-s=-12ρξ.Completing the integral in (58), we obtain a smooth solitary wave solution of (7) as follows: (60)ϕξ=E1-3ωξ261+ωξ2,where ω=(E/3)-(1/2)ρ. Completing the integral in (59), we obtain a smooth solitary wave solution of (7) the same as in (60). The profiles of (60) are shown in Figures 3(k) and 3(l). From (2), (6), (60), and ω2=-(1+c1ω1), we obtain a smooth solitary wave solution of (1) with ρ<0, E≠0, and F=(1/4)E2 given in (50).
From Figures2(a) and 2(j), when ρ>0, E<0, and F<0 (or ρ>0, E>0, and (2/9)E2<F<(1/4)E2), there is a homoclinic orbit connecting with the saddle point (ϕ2,0) and passing through the point (ϕm,0), and its expression is (61)y=±12ρϕ-ϕ2ϕM-ϕϕm-ϕ,ϕ2<ϕ≤ϕm,where ϕM,m=(1/6)(3Δ-E±2E(E-3Δ)). Substituting (61) into dϕ/dξ=y and integrating it along the homoclinic orbit, we have (62)∫ϕϕmdss-ϕ2ϕM-sϕm-s=12ρξ.Completing the integral in (62), and using (2), (6), and ω2=-(1+c1ω1), we obtain a smooth solitary wave solution of (1) with ρ>0, E<0, and F<0 (or ρ>0, E>0, and (2/9)E2<F<(1/4)E2) given in (22).
From Figures2(b) and 2(i), when ρ>0, E>0, and F<0 (or ρ>0, E<0, and (2/9)E2<F<(1/4)E2), there is a homoclinic orbit connecting with the saddle point (ϕ1,0) and passing through the point (ϕM,0), and its expression is (63)y=±12ρϕ1-ϕϕ-ϕMϕ-ϕm,ϕM≤ϕ<ϕ1,where ϕM,m=-(1/6)(3Δ+E∓2E(E+3Δ)). Substituting (63) into dϕ/dξ=y and integrating it along the homoclinic orbit, we have (64)∫ϕMϕdsϕ1-ss-ϕMs-ϕm=12ρξ.Completing the integral in (64), and using (2), (6), and ω2=-(1+c1ω1), we obtain a smooth solitary wave solution of (1) with ρ>0, E>0, and F<0 (or ρ>0, E<0, and (2/9)E2<F<(1/4)E2) given in (25).
From Figure2(e), when ρ>0, E<0, and 0<F<(2/9)E2, there is a homoclinic orbit connecting with the saddle point (0,0) and passing through the point (ϕm,0), and its expression is (65)y=±12ρϕϕM-ϕϕm-ϕ,0<ϕ≤ϕm,where ϕM,m=(1/3)(-2E±2(2E2-9F)). Substituting (65) into dϕ/dξ=y and integrating it along the homoclinic orbit, we have (66)∫ϕϕmdssϕM-sϕm-s=12ρξ.Completing the integral in (66), and using (2), (6), and ω2=-(1+c1ω1), we obtain a smooth solitary wave solution of (1) with ρ>0, E<0, and 0<F<(2/9)E2 given in (20).
From Figure2(f), when ρ>0, E>0, and 0<F<(2/9)E2, there is a homoclinic orbit connecting with the saddle point (0,0) and passing through the point (ϕM,0), and its expression is (67)y=±12ρϕϕ-ϕMϕ-ϕm,ϕM≤ϕ<0,where ϕM,m=(1/3)(-2E±2(2E2-9F)). Substituting (67) into dϕ/dξ=y and integrating it along the homoclinic orbit, we have (68)∫ϕMϕdsss-ϕMs-ϕm=-12ρξ.Completing the integral in (68), and using (2), (6), and ω2=-(1+c1ω1), we obtain a smooth solitary wave solution of (1) with ρ>0, E>0, and 0<F<(2/9)E2 given in (21).
The proof of the Proposition2 is completed.Proposition 3.
Whenρ>0, E≠0, and F=(2/9)E2, (1) has the kink and antikink wave solutions as follows: (69)qx,y,t=-2E1/2meix+c1y+c2t31+e±ωx+ω1y-1+c1ω1t1/2m,where ω=(2E/3)(1/2)ρ.Proof of Proposition3.
From Figures2(g) and 2(h), when ρ>0, E≠0, and F=(2/9)E2, there are two heteroclinic orbits connecting with the saddle points (0,0) and (-(2/3)E,0). When ρ>0, E<0, and F=(2/9)E2, its expression is (70)y=±12ρϕ-23E-ϕ,0<ϕ<-23E.When ρ>0, E>0, and F=(2/9)E2, its expression is (71)y=±12ρϕϕ+23E,-23E<ϕ<0.Substituting (70) and (71) into dϕ/dξ=y, respectively, and integrating them along the heteroclinic orbits, we have (72)∫ϕ-E/3dss-2/3E-s=±12ρξ,(73)∫-E/3ϕdsss+2/3E=±12ρξ.Completing the integral in (72), we obtain the kink and antikink wave solutions of (7) as follows: (74)ϕξ=-2E31+e±ωξ,where ω=(2E/3)(1/2)ρ. Completing the integral in (73), we obtain the kink and antikink wave solutions of (7) the same as in (74). The profiles of (74) are shown in Figures 4(a) and 4(b). From (2), (6), (74), and ω2=-(1+c1ω1), we obtain the kink and antikink wave solutions of (1) with ρ>0, E≠0, and F=(2/9)E2 given in (69).
The proof of Proposition3 is completed.Figure 4
Kink and antikink waves of (7).
(a)
ρ
=
1.5, E=-1.0, and F=2/9
(b)
ρ
=
1.5, E=1.0, and F=2/9Remark 4.
The implicit solutions in [13] can be deduced if setting that concrete values of a set of the parameters. For instance, if one set a1→1, a2→c1, a3→c2, a4→0, A1→1, A2→ω1, A3→ω2=-(1+c1ω1), A4→0, m→2m, and v→ϕ in [13], (16) and (18) in [13], respectively, can be deduced as follows: (75)4m4m21+2c2+c12+2m1+2c2+c12-2mϕ2+2m+1Φ1+2c2+c121+ω122m+11+ω12ϕ=e∓4m1+2c2+c12ξ/1+ω12,(76)ξ=±∫4m2ϕ21+2c2+c128m2+6m+1-21+4mϕ2-2k2m+1ϕ4+2h8m2+6m+1ϕ2-1/m8m2+6m+11+ω12-1/2dϕ,where (77)ξ=x+ω1y-1+c1ω1t,Φ=2m21+2c2+c122m+14m+1-2k2m+1ϕ4-24m+1ϕ22m+14m+11+ω12and h is given in (10) of [13]. Equations (75) and (76) maybe can be reduced to some of results in this paper under concrete values of some sets of the parameters. We omit the discussions here.


## 5. Conclusion

In this paper, we present some explicit smooth solitary wave solutions for (1) expressed in (20)–(26), (49), and (50), and also some explicit kink and antikink wave solutions shown in (69). We will continue to discuss the properties of (1), and more results will be obtained.



---

*Source: 408630-2015-10-21.xml*