# New Periodic Solutions for a Class of Zakharov Equations

Authors: Cong Sun; Shuguan Ji
Journal: Advances in Mathematical Physics
 (2016)

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## Abstract

Through applying the Jacobian elliptic function method, we obtain the periodic solution for a series of nonlinear Zakharov equations, which contain Klein-Gordon Zakharov equations, Zakharov equations, and Zakharov-Rubenchik equations.


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## Body

## 1. Introduction

For most of nonlinear evolution equations, we have many methods to obtain their exactly solutions, such as hyperbolic function expansion method [1], the transformation method [2], the trial function method [3], the automated method [4], and the extended tanh-function method [5]. But these methods can only obtain solitary wave solution and cannot be used to deduce periodic solutions. The Jacobian function method provides a way to find periodic solutions for some nonlinear evolution equations. In particular, in the research of plasma physics theory, quantum mechanics, fluid mechanics, and optical fiber communication, we frequently meet kinds of Zakharov equations.Hence, in this paper, inspired by Angulo Pava’s work [6], we are concerned to obtain exact periodic solutions of a series of Zakharov equations, (1)utt-uxx+u+nu=0,ntt-nxx=u2xx,utt-uxx-vxx=0,ivt+αvxx-uv=0,and Zakharov-Rubenchik equations, (2)iBt+ωBxx-ku-12λρ+qB2B=0,θρt+u-λρx=-kBx2,θut+βρ-λux=12kλBx2.


## 2. The Periodic Solution for Klein-Gordon Zakharov Equations

The Klein-Gordon Zakharov equations are used to show the interaction between langmuir wave and ion wave in the plasma, which has the following form:(3)utt-uxx+u+nu=0,ntt-nxx=u2xx,where u(x,t) denotes the biggest moment scale component produced by electron in electric field. n(x,t) denotes the speed of deviations between the ions at any position and that at equilibrium position.Now, we suppose that it possesses solitary wave solutions of the following form:(4)ux,t=eiθφξ,nx,t=nξ,ξ=x-ct,where c is a traveling wave speed and c2<1 is a constant.By substituting (4) into (3), we can obtain(5)c2φ′′-φ′′+φ+φn=0,(6)c2-1n′′=φ2′′.By (6), we have(7)c2-1n-φ2=c1ξ+m,where c1, m are integration constants.In what follows, we are concerned with the periodic solution of (3); thus we need to require c1=0.Therefore, (7) implies(8)n=φ2+mc2-1.Moreover, through (8) and (5), we have that(9)c2-1φ′′+φ+φ3+φmc2-1=0.Multiplying (9) by φ′ and integrating once, we obtain(10)c2-12φ′2+φ22+φ4+2mφ2c2-114=h,where h is a nonzero integration constant. Furthermore, h, m, and c satisfy the following condition: (11)1-c2-m2-4h1-c2≥0,1-c2-m-1-c2-m2-41-c2h≥0,0<h<1-c2-m241-c2,so that(12)φ′2=121c2-12η12-φ2φ2-η22,where -η1, η1, -η2, and η2 are the zeros of the polynomial F(λ)=-(c2-1+m/(c2-1)2)λ2-(1/2)(λ4/(c2-1)2)+2h/(c2-1). Without losing generality, we assume that η1>η2>0. Therefore, we can deduce that η1≤φ≤η2; η1 and η2 satisfy (13)η12+η22=21-c2-2m,η12η22=4h1-c2.Letϕ=φ/η1, k2=(η12-η22)/η12. Hence, (12) can be written as (14)ϕ′2=121c2-12η121-ϕ2ϕ2-1+k2.Moreover, we define a new variableψ, which satisfies ψ′≥0, ψ(0)=0, by the relation ϕ2=1-k2sin2ψ; through a tedious computation, we obtain that (15)ψ′2=121c2-12η121-k2sin2⁡ψ.Then we obtain(16)∫0ψξdτ1-k2sin2⁡τ=2211-c2η1ξ.According to the definition of the Jacobian elliptic functiony=sn(u;k), we can obtain that sin⁡ψ=sn(η1βξ,k). Here, β=2/2(1-c2). So (17)ϕξ=1-k2sn2η12211-c2ξ,k=dnη12211-c2ξ,k.By returning to initial variable, we obtain that(18)φξ=η1dnη12211-c2ξ,kis a dnoidal solution of (10).Furthermore,dn has fundamental period 2K(k); that is, dn(u+2k;k)=dn(u;k), and K(k) is the complete elliptic integral of first kind. So the dnoidal wave solution φ has fundamental period, T, given by (19)T=22Kk1-c2η1.So, by applying the method of the Jacobian elliptic function and inspired by Angulo Pava’s ideas, we obtain that (3) has the periodic traveling solution of the following form:(20)ux,t=eiθφξ=eiθη1dnη12211-c2ξ,k,nx,t=η1dnη12/21/1-c2ξ,k2+mc2-1.Moreover,T and k2 can be also rewritten as the following form:(21)Tη2=22Kkη21-c221-c2-2m-η22,k2=21-c2-2m-2η2221-c2-2m-η22.Furthermore, ifη2→0 then k(η2)→1. Hence K(k(η2))→+∞, so that T(η2)→+∞. If η2→1-c2-m, then k(η2)→0. Hence K(k(η2))→π/2, so that T(η2)→2π(1-c2)/1-c2-m.Next we will show that, for an arbitrary but fixedL, 1-(L2+LL2-8πm)/4π<|c0|<1+(L2+LL2-8πm)/4π. We can obtain that there is a unique η2,0=η2,0(c0)∈(0,1-c02-m) such that T(η2)=T(η2(c))=L is a fundamental period for the dnoidal wave solution (18).Theorem 1.
LetL>0 be arbitrary but fixed. Consider 1-(L2+LL2-8πm)/4π<|c0|<1+(L2+LL2-8πm)/4π and unique η2,0=η2,0(c0)∈(0,1-c02-m), such that T(η2)=L. Then, there exist an interval U(c0), an interval I(η2,0), and a unique smooth function F:U(c0)→I(η2,0) such that F(c)=η2 and(22)L=22Kkη21-c221-c2-2m-η22,where c∈U(c0), η2∈U(η2,0).Proof.
Based on the ideas establish in Angulo Pava’s work [6], we will give a brief proof. Now, we consider the open set(23)Ω=η2,c:η2∈0,1-c02-m,c∈-1+L2+LL2-8πm4π,1+L2+LL2-8πm4π∪-∞,-1-L2+LL2-8πm4π∪1-L2+LL2-8πm4π,+∞∈R2.
We defineΦ:Ω→R by(24)Φη2,c=22Kkη21-c221-c2-2m-η22.Here, k2=(2(1-c2)-2m-2η22)/(2(1-c2)-2m-η22). Hypothesise Φ(η2,0,c0)=L. In what follows, we will prove that ∂Φ/∂η2<0.
From (24), we can obtain that(25)∂Φ∂η2=22η2Kkη21-c221-c2-2m-η223/2+221-c221-c2-2m-η22dKdk∂k∂η2.
Fromk2, we can deduce k(η2,c) is a strictly decreasing function of η2.
According to Jacobian elliptic function theory [7], we have(26)dKdk=E-1-k2Kk1-k2,(27)dEdk=E-Kk.Here, E is the complete elliptic integral of second kind.
Next, we adopt the reduction to absurdity to prove∂Φ/∂η2<0. Now, we assume that ∂Φ/∂η2≥0. So, we have the following inequality: (28)k21-c2-m-η22Kkη2≥21-c2-mdKdk.
Indeed, substituting (26) into (25) and using the method of enlarging and reducing, we obtain that(29)1-k241-c2-m-2η22Kk≥21-c2-mEk.
Fromk2, we can easily deduce that (30)1-k2=η2221-c2-2m-η22,2-k2=21-c2-2m21-c2-2m-η22.
Hence,(31)21-k2Kk≥2-k2Ek.
Letγ2(η2,c)=1-k2, γ′=-k(∂k/∂η2)(1/γ)>0. So, γ is an increasing function of η2∈(0,1-c2-m). Moreover, γ(0)=1, γ(1-c2-m)=1.
Define(32)fγ=1+γ2E1-γ2-2γ2K1-γ2.Due to ∂Φ/∂η2≥0, f(γ)≤0.
However, by (27) and a simple computation, (33)f′γ=3γE1-γ2-K1-γ2<0,so f(γ) is a decreasing function. Furthermore, f(1)=0, so for γ∈(0,1), f(γ)>0.
It is in conflict with our assumption. So we obtain our affirmation that∂Φ/∂η2<0. Hence, by the implicit function theorem, there exists a unique smooth function F, defined in a neighborhood, U(c0) of c0, so that Φ(η2(c),c)=L for c∈U(c0). Hence, we obtain (22).


## 3. The Periodic Solution for Zakharov Equations

Now, we consider the following Zakharov equations:(34)utt-uxx-vxx2=0,ivt-αvxx-uv=0,which describe the high frequency moment of plasma, where u(x,t) denotes the ion number density variation, v(x,t) denotes the electric field intensity of slowly varying amplitude, and α∈R.We seek the solitary solutions of (34) in the form(35)u=uξ,v=φξeiξ,ξ=x-ct,where u and φ are real functions, c is a traveling speed, and c2≠1.Substituting (35) into (34), we can obtain that(36)c2-1u′′-φ2′′=0,(37)-αφ′′-c+2αiφ′+c+αφ-uφ=0.By (36), we can deduce(38)c2-1u-φ2=c2ξ+r.Here, c2, r are integration constants.Next, we consider periodic solutions of (34). So, we need to require c2=0. Therefore, by (38)(39)u=φ2+rc2-1.Furthermore, by (37), we can obtain that(40)c=-2α.So, by (39) and (40), (37) can be rewritten:(41)φ′′+φ1+2rc-c3+2φ3c-c3=0.Multiplying (41) by φ′ and integrating once, we obtain(42)12φ′2+12φ21+2rc-c3+12φ4=h,where h is a nonzero integration constant.Hence,(43)φ′2=a2+φ2b2-φ2.Here, -ai, ai, -b, and b are polynomial roots of F(t)=-t4-t2(a2-b2)+2h, b>0, and(44)a2-b2=1+2rc-c3,a2b2=2h.Letχ=φ/b (suppose χ(0)=0) and k2=b2/(a2+b2). Hence, (43) becomes (45)χ′2=b21-χ2a2b2+χ2.Now, we defineχ2=1-sin2⁡ψ, so we get that (46)ψ′2=a2+b21-k2sin2⁡ψ.Letτ=a2+b2. According to the definition of the Jacobian elliptic function sn, we can obtain sin⁡ψ(ξ)=sn(τξ,k). So that, χ2=1-sn2(τξ,k).So, we obtain the solution of (41):(47)φξ=bcnτξ,k.According to Jacobian elliptic functions theory,cn has period 4K(k), and we can obtain that the cnoidal wave solution has period T, which is given by(48)T=4Kkτ.So, we can obtain the periodic solutions of (34):(49)ux,t=uξ=bcnτξ,k2+rc2-1,vx,t=vξ=beiξcnτξ,k.Furthermore, it follows that fork2=b2/(a2+b2), we have k2∈(0,1/2).Ifτ→0, then k→0, and K(k)→π/2, T(τ)→+∞. Furthermore, if τ→+∞, T(τ)→0.Next, we will prove that for an arbitrary but fixedL>0, there exists a unique α=α(c)∈(0,+∞) such that T(α(c))=L is a fundamental period of the cnoidal wave solution (47). So, we have the following theorem.Theorem 2.
LetL>0 but fixed. Consider c0>0 and the unique τ0=τ0(c0)∈(0,+∞) such that T(τ0)=L. Then, there exists an internal A(c0) around c0, an internal B(b0) around b0, and a unique smooth function Υ such that Υ(c0)=α0 and (50)L=4Kkτ,where c∈A(c0), τ=Υ(c).Proof.
The proof is similar to that of Theorem1. For details see Theorem 1 (see also Angulo Pava [6, 8, 9]).


## 4. The Periodic Solution for Zakharov-Rubenchik Equations

Now, we consider the Zakharov-Rubenchik equations:(51)iBt+ωBxx-lu-12λρ+qB2B=0,θρt+u-λρx=-lBx2,θut+βρ-λux=12lλBx2,which describe the dynamics of small amplitude Alfvén waves propagating in a plasma. Here, B(x,t) denotes the magnetic field, u(x,t) the fluid speed, and ρ(x,t) the density of mass. Moreover, ω, l, λ, θ, β, q are real constants.Motivated by Oliveira [10], we look for the solitary waves of (51), as follows:(52)Bx,t=eiξAξ,ux,t=aA2ξ,ρx,t=bA2ξ,ξ=x-ct,where a, b, and A are real functions and c denotes traveling speed.Putting (52) into (51), from the second and third equations of (51), we can deduce (53)a=ac=lλ/2λ+cθ-ββ-cθ+λ2,b=bc=l-cθ-λ/2β-cθ+λ2.Moreover, from the first equation of (51), we can obtain that(54)c-ωA+i2ω-cA′+ωA′′-la-12λb+qA3=0.So, by (54), it implies (55)c=2ω,(56)A′′+A-2la-λ/2b+qcA3=0.Letb1=-l(a-(λ/2)b+q)/c>0 and multiplying (56) by A′ and integrating once, we obtain(57)A′2=b1A2+a12b22-A2,where -a1i, a1i, -b2, b2 are the polynomial roots of F(r)=-b1r4-r2+h. Moreover,(58)b22-a12=-1b1,b22a12=hb1.Letχ=A/b2 (suppose χ(0)=0) and k2=b22/(a12+b22). Hence, (57) becomes (59)χ′2=b1b221-χ2a12b22+χ2.Now, we defineχ2=1-sin2⁡ψ, so we get that (60)ψ′2=b1a12+b221-k2sin2⁡ψ.Letα=b1(a12+b22). According to the definition of the Jacobin elliptic function sn, we can obtain sin⁡ψ(ξ)=sn(αξ,k). So, χ2=1-sn2(αξ,k).So, we obtain the solution of (56):(61)Aξ=b2cnαξ,k.Sincecn has fundamental period 4K(k), we can obtain that solution (61) has fundamental period, T, which is (62)T=4Kkb1a12+b22.Hence, (51) have the periodic solutions of the following form:(63)Bξ=eiξb2cnαξ,k,uξ=ab22cn2αξ,k,ρξ=bb22cn2αξ,k.Moreover, from (58) and the definition of k2, it follows that (64)k2=a12-1/b12a12-1/b1,Ta1=4Kk2a12b1-1.Moreover, ifa1→0, k2→1, T(a1)→+∞. If a1→+∞, k2→1/2, T(a1)→0.Next, we will show that for an arbitrary but fixedL>0, there exists a1=a1(c)∈(0,+∞) such that T(a1(c))=L is a fundamental period of the cnoidal wave solution (61). Motivated by Angulo Pava’s result [6], we have the following theorem.Theorem 3.
ForL>0 arbitrary and fixed, consider c0>0 and the unique a1=a1(c0)∈(0,+∞). Then, there exist an interval C(c0) around c0, an interval D(a1) around a1, and a unique smooth function H:C(c0)→D(a1) such that H(c0)=a1 and (65)L=4Kk2a12b1-1,where c∈C(c0), a1=H(c).Proof.
The idea and method are similar to those of Theorem1. For details, please see Theorem 1 (see also Angulo Pava [6, 8, 9]).


## 5. Conclusion

Inspired by Angulo Pava’s idea, by applying Jacobian elliptic function method, we have obtained new period wave solution for Klein-Gordon Zakharov equations, Zakharov equations, and Zakharov-Rubenchik equations. In particular, the solutions of (18), (47), and (58) were not found in the previous work. The method can help to look for periodic solution for a class of nonlinear equations.



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*Source: 6219251-2016-09-29.xml*