# Exact Periodic Wave, Bisoliton, and Various Breather Solutions for the Zakharov Equations

Authors: Heng Wang; Longwei Chen; Hongjiang Liu; Shuhua Zheng
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

The Zakharov equations, which involve the interactions between Langmuir and ion acoustic waves in plasma, are analytically studied. By using the method of Exp-function, the periodic wave, bisoliton, Akhmediev breather, Ma breather, and Peregrine breather of the Zakharov equations are obtained. These results presented in this paper enrich the diversity of solution structures of the Zakharov equations. Furthermore, based on the numerical simulations of these solutions, some physics analysis of bisolitons and various breathers are given.


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## Body

## 1. Introduction

In this paper, we consider the following Zakharov equations:(1)ntt-cs2nxx=βE2xx,iEt+αExx=δnE,where n is the perturbed number density of the ion (in the low-frequency response), E is the slow variation amplitude of the electric field intensity, cs is the thermal transportation velocity of the electron-ion, α≠0, β≠0, and δ≠0 [1]. The Zakharov equation is one of the classical models describing the interactions between high- and low-frequency waves and governing nonlinear dynamics of travelling waves. In the interaction of laser-plasma, (1) plays crucial roles. In recent years, many exact solutions of (1) have been successfully obtained by different methods in [2–7].Recently, some important mathematical physics equations have been widely studied [8–12]. In particular, the investigation of the rogue wave phenomenon of nonlinear partial differential equation achieved some significant advances. Many rogue waves have been obtained [13–18]. Here we continue to analyze the rogue wave phenomenon and study the existence of rogue waves of (1).Firstly, we setξ=k1x+k2t; k1 and k2 are arbitrary real constants. Equation (1) is transformed into the following equations:(2)n=βk12k22-cs2k12E2,iEt+αExx=δnE.Next (2) is reduced to the (1+1)-dimensional nonlinear Schrödinger equation:(3)iEt+αExx+γE2E=0,where γ=-δβk12/k22-cs2k12.Then, to find the exact solutions of the Zakharov equations, we assume that(4)Ex,t=reir2γt1+Ax,t+iBx,tCx,t,where r is a real constant. Substituting (4) into (3), we have (5)2γr2Ax,tCx,t2+2αAx,tCxx,t2+γr2Ax,t3-2αAxx,tCxx,tCx,t+αAxxx,tCx,t2-αAx,tCxxx,tCx,t+Bx,tCx,tCxx,t+3γr2Ax,t2Cx,t+γr2Bx,t2Cx,t-Btx,tCx,t2+γr2Ax,tBx,t2+iγrAx,t2Bx,t-Ax,tCx,tCxx,t-αBx,tCxxx,tCx,t+γr2Bx,t3+αBxx,tCx,t2+Axx,tCx,t2+2αBx,tCxx,t2+2γr2Ax,tBx,tCx,t-2αBxx,tCxx,tCx,t=0,where A(x,t), B(x,t), and C(x,t) are real functions. We separate the imaginary and real parts of (5) and have (6)2γr2Ax,tCx,t2+2αAx,tCxx,t2+γr2Ax,t3-2αAxx,tCxx,tCx,t+αAxxx,tCx,t2-αAx,tCxxx,tCx,t+Bx,tCx,tCxx,t+3γr2Ax,t2Cx,t+γr2Bx,t2Cx,t-Btx,tCx,t2+γr2Ax,tBx,t2=0,γrAx,t2Bx,t-Ax,tCx,tCxx,t-αBx,tCxxx,tCx,t+γr2Bx,t3+αBxx,tCx,t2+Axx,tCx,t2+2αBx,tCxx,t2+2γr2Ax,tBx,tCx,t-2αBxx,tCxx,tCx,t=0.We suppose that A(x,t), B(x,t), and C(x,t) are the following functions:(7)Ax,t=a1epVx+Kt+a2e-pVx+Kt+a3eqWx+Lt+a4e-qWx+Lt,Bx,t=b1epVx+Kt+b2e-pVx+Kt+b3eqWx+Lt+b4e-qWx+Lt,Cx,t=c1epVx+Kt+c2e-pVx+Kt+c3eqWx+Lt+c4e-qWx+Lt,where ai, bi, cii=1,…,4,p, q, V, K, W, and L are the constants to be determined later. We substitute (7) into (6) and equate all coefficients of emp(Vx+Kt)enq(Wx+Lt)(m,n=-3,…,3) to 0, and then we obtain a set of algebraic equations for ai, bi, ci(i=1,…,4),p, q, V, K, W, and L. With the aid of Maple, we have (8)a1=0,a2=0,a3=b424c22-b42,a4=b424c22-b42,b1=0,b2=0,b3=-b4,c1=c2,c3=-2c224c22-b42,c4=-2c224c22-b42,V=rb4-2γ/α2pc2,K=0,W=0,L=-γr2b44c22-b422c22q.Substituting (8) and (7) into (4), we can obtain the solution of (3) as follows: (9)Ex,t=reir2γt1+b42cosh⁡γr2b44c22-b42/2c22t+ib44c22-b42sinh⁡γr2b44c22-b42/2c22tc24c22-b42cosh⁡r-2γ/αb4/2c2x-2c22cosh⁡γr2b44c22-b42/2c22t.Finally, according ton=βk12/k22-cs2k12|E|2, we obtain the exact solution of (1) as follows:(10)Ex,t=reir2γt1+b42cosh⁡γr2b44c22-b42/2c22t+ib44c22-b42sinh⁡γr2b44c22-b42/2c22tc24c22-b42cosh⁡r-2γ/αb4/2c2x-2c22cosh⁡γr2b44c22-b42/2c22t,nx,t=βk12k22-cs2k12Ex,t2,where γ=-δβk12/k22-cs2k12. Here (10) is a unified solution formula of (1) which can produce periodic wave solution, bisoliton solution, and a series of breather solutions.


## 2. Exact Periodic Wave, Bisoliton, and Various Breather Solutions for (1)

In this section, according to (10), when we select different parameters in (10), we can obtain different structure exact solutions of (1) as follows.Case 1 (the periodic wave solution).
Whenαγ<0, we set b4=ib, and we obtain the following periodic solution:(11)Ex,t=reir2γt1-b2cos⁡γr2b4c22+b2/2c22t+ib4c22+b2sin⁡γr2b4c22+b2/2c22tc24c22+b2cos⁡r-2γ/αb/2c2x-2c22cos⁡γr2b4c22+b2/2c22t,nx,t=βk12k22-cs2k12Ex,t2.Case 2 (the bisoliton solution).
Whenαγ<0 and 4c22-b42>0, we obtain the following bisoliton solution:(12)Ex,t=reir2γt1+b42cosh⁡γr2b44c22-b42/2c22t+ib44c22-b42sinh⁡γr2b44c22-b42/2c22tc24c22-b42cosh⁡r-2γ/αb4/2c2x-2c22cosh⁡γr2b44c22-b42/2c22t,nx,t=βk12k22-cs2k12Ex,t2.Case 3 (various breather solutions).
(1) Whenαγ>0 and 4c22-b42>0, we obtain the following Akhmediev breather solution:(13)Ex,t=reir2γt1+b42cosh⁡γr2b44c22-b42/2c22t+ib44c22-b42sinh⁡γr2b44c22-b42/2c22tc24c22-b42cos⁡r-2γ/αb4/2c2x-2c22cosh⁡γr2b44c22-b42/2c22t,nx,t=βk12k22-cs2k12Ex,t2.
(2) Whenαγ>0, we set b4=ib, and we obtain the following Ma breather solution:(14)Ex,t=reir2γt1-b2cos⁡γr2b4c22+b2/2c22t+ib4c22+b2sin⁡γr2b4c22+b2/2c22tc24c22+b2cosh⁡r-2γ/αb/2c2x-2c22cos⁡γr2b4c22+b2/2c22t,nx,t=βk12k22-cs2k12Ex,t2.
(3) Whenαγ>0, we set c2>0 and b4→0, and we obtain the following Peregrine breather solution:(15)Ex,t=reir2γt1-α+i2αγr2tr2γx2+αγ2r4t2,nx,t=βk12k22-cs2k12Ex,t2.To verify the correctness of our results, by using the numerical simulation method, the 3D graphics of the periodic waves, bisolitons, and various breathers of (1) are given in Figures 1–10.Figure 1
The 3D graphics of|E(x,t)| in (11) (α=-2, β=1, δ=-1, cs=1, γ=4, k1=2, k2=5, r=2, b=2, and c2=0.2).Figure 2
The 3D graphics ofn(x,t) in (11) (α=-2, β=1, δ=-1, cs=1, γ=4, k1=2, k2=5, r=2, b=2, and c2=0.2).Figure 3
The 3D graphics of|E(x,t)| in (12) (α=-2, β=1, δ=-1, cs=1, γ=4, k1=2, k2=5, r=2, b4=2, and c2=5).Figure 4
The 3D graphics ofn(x,t) in (12) (α=-2, β=1, δ=-1, cs=1, γ=4, k1=2, k2=5, r=2, b4=2, and c2=5).Figure 5
The 3D graphics of|E(x,t)| in (13) (α=-2, β=1, δ=1, cs=1, γ=-4, k1=2, k2=5, r=2, b4=2, and c2=5).Figure 6
The 3D graphics ofn(x,t) in (13) (α=-2, β=1, δ=1, cs=1, γ=-4, k1=2, k2=5, r=2, b4=2, and c2=5).Figure 7
The 3D graphics of|E(x,t)| in (14) (α=-2, β=1, δ=1, cs=1, γ=-4, k1=2, k2=5, r=2, b=2, and c2=5).Figure 8
The 3D graphics ofn(x,t) in (14) (α=-2, β=1, δ=1, cs=1, γ=-4, k1=2, k2=5, r=2, b=2, and c2=5).Figure 9
The 3D graphics of|E(x,t)| in (15) (α=-2, β=1, δ=1, cs=1, γ=-4, k1=2, k2=5, r=2, and c2=5).Figure 10
The 3D graphics ofn(x,t) in (15) (α=-2, β=1, δ=1, cs=1, γ=-4, k1=2, k2=5, r=2, and c2=5).


## 3. Conclusion

To summarize, by using the Exp-function method, we have obtained the periodic wave solution, bisoliton solution, and various breather solutions of the Zakharov equations. Firstly, by using the appropriate transformation, the Zakharov equations are transformed into the(1+1)-dimensional nonlinear Schrödinger equation. Then, based on the Exp-function method, a unified solution formula of the (1+1)-dimensional nonlinear Schrödinger equation is given. Under different parameter conditions, the exact solutions of the Zakharov equations which contain the periodic wave solution, bisoliton solution, Akhmediev breather solution, Ma breather solution, and Peregrine breather solution are obtained. The results greatly enrich the diversity of exact solutions of the Zakharov equations.Secondly, based on the numerical simulations of these solutions, we obtain some physics behind our results. Figures3 and 4 show the bisolitons which move to each other; they meet and have an elastic collision. When the elastic collision occurs, the amplitude is obviously increased. After separation, they do the reverse motion. Figures 5 and 6 show Akhmediev breather; it yields the breather which affects spatial direction. Figures 7 and 8 show Ma breather; it yields the breather which affects time direction. Figures 9 and 10 show Peregrine breather which is a high amplitude wave. With the increase of time, the wavelength becomes larger, and the amplitude decreases gradually. Finally, it becomes a plane wave.Finally, whenα=1/2 and γ=1, based on the work of Akhmediev et al. [19, 20], we can obtain the following higher-order rational rogue wave solutions of (3):(16)Ex,t=1γ1-G+iHDeit,γ>0,where G=4x2+12αx2+72α2t2+48αx2t2+80α2t4-3α2/16α2, H=t(4x2+8αt2+16αx2t2+16α2t4-15α2-12αx2)/8α2, and D=8x6+α(12+48t2)x4+6α2(4t2-3)2x2+α3(9+64t6+432t4+396t2)/192α3. Then we have the following higher-order rational rogue wave solutions of (1):(17)Ex,t=1γ1-G+iHDeit,nx,t=βk12k22-cs2k12Ex,t2.



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