# Generalized Jacobi Elliptic Function Solution to a Class of Nonlinear Schrödinger-Type Equations

Authors: Zeid I. A. Al-Muhiameed; Emad A.-B. Abdel-Salam
Journal: Mathematical Problems in Engineering
 (2011)

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## Abstract

With the help of the generalized Jacobi elliptic function, an improved Jacobi elliptic function method is used to construct exact traveling wave solutions of the nonlinear partial differential equations in a unified way. A class of nonlinear Schrödinger-type equations including the generalized Zakharov system, the Rangwala-Rao equation, and the Chen-Lee-Lin equation are investigated, and the exact solutions are derived with the aid of the homogenous balance principle.


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## Body

## 1. Introduction

Nonlinear phenomena appear in a wide variety of scientific fields, such as applied mathematics, physics and engineering problems. However, solving nonlinear partial differential equations (NLPDEs) corresponding to the nonlinear problems is often complicate. Especially, obtaining their explicit solutions is even more difficult. Up to now, a lot of new methods for solving NLPDEs are developed, for example, Bäckland transformation method, inverse scattering method, Darboux transformation method, Hirota’s bilinear method, homogeneous balance method, Jacobi elliptic function method, tanh-function method, variational iteration method, the sine-cosine method, F-expansion method, Lucas Riccati method, and so on [1–15]. But, generally speaking, all of the above methods have their own advantages and shortcomings, respectively.Nowadays, many exact solutions of NLPDEs can be written as a polynomial in several elementary or special functions which satisfy first-order nonlinear ordinary differential equation (NLODE) with a sixth-degree nonlinear term. The aim of this paper, motivated by [13, 15], is to perform a first-order NLODE with sixth-degree nonlinear term which is, in nature, an extension of a type of elliptic equation, into a new algebraic or new auxiliary equation method to seek exact solutions to a class of nonlinear Schrödinger-type equations.The rest of this paper is organized as follows. In Section2, we give the description of the generalized improved Jacobi elliptic function method. In Section 3, we apply this method to the generalized Zakharov system, the Rangwala-Rao equation, and the Chen-Lee-Lin equation. Finally, we conclude the paper and give some futures and comments.


## 2. Description of the Improved Jacobi Elliptic Function Method

The main idea of this method is to take full advantage of the elliptic equation that the generalized Jacobi elliptic functions (GJEFs) satisfy [13, 16–18]. The desired elliptic equation read
(2.1)F′(ξ)=A0+A2F2(ξ)+A4F4(ξ)+A6F6(ξ),≡′ddξ,
where ξ≡ξ(x,t) and A0,A2,A4,A6 are constants.Case 1.
IfA0=1,A2=-(1+k12+k22),A4=k12+k22+k12k22 and A6=-k12k22, then (2.1) has a solution s(ξ,k1,k2).Case 2.
IfA0=1-k12-k22+k12k22,A2=2k12+2k22-3k12k22-1, A4=3k12k22-k12-k22 and A6=-k12k22, then (2.1) has a solution c(ξ,k1,k2).Case 3.
IfA0=k12-1-k22+k22k1-2, A2=2k22+2-k12-3k22k1-2, A4=3k22k1-2-k22-1 and A6=-k22k1-2, then (2.1) has a solution d1(ξ,k1,k2).Case 4.
IfA0=k22-1-k12+k12k2-2, A2=2k12+2-k22-3k12k2-2, A4=3k12k2-2-k12-1 and A6=-k12k2-2, then (2.1) has a solution d2(ξ,k1,k2).s(ξ,k1,k2) is the generalized Jacobi elliptic sine function, ξ is an independent variable, k1,k2(0≤k2≤k1≤1) are two modulus of the GJEFs, c(ξ,k1,k2) is the generalized Jacobi elliptic cosine function, d1(ξ,k1,k2) is the generalized Jacobi elliptic function of the third kind, and d2(ξ,k1,k2) is the generalized Jacobi elliptic function of the forth kind [13, 16–18]. The definitions and properties of the GJEFs are given in the appendix.For a given NLPDEs involving the two independent variablesx, t,
(2.2)P(u,ut,ux,uxx,…)=0,
where P is in general a polynomial function of its argument and the subscripts denote the partial derivatives, by using the traveling wave transformation, Equation (2.2) possesses the following ansätz:
(2.3)u(x,t)=U(ξ),ξ=k(x-ωt),
where k, ω are constants to be determined later. Substituting (2.3) into (2.2) yields an ordinary differential equation (ODE):O(u(ξ),u(ξ)ξ,u(ξ)ξξ,…)=0. Then, u(ξ) is expanded into a polynomial of F(ξ) in the form
(2.4)u(ξ)=a0+∑i=1naiFi(ξ).
The processes take the following steps.Step 1.
Determinen in (2.4) by balancing the linear term(s) of the highest order with the nonlinear term(s) in (2.2).Step 2.
Substituting (2.4) with (2.1) into (2.2), then the left-hand side of (2.2) can be converted into a polynomial in F(ξ). Setting each coefficient of the polynomial to zero yields system of algebraic equations for a0,a1,…,an, k and ω.Step 3.
Solving this system obtained in Step2, then a0,a1,…,an, k and ω can be expressed by A0, A2, A4, A6. Substituting these into (2.4), then general form of traveling wave solution of (2.2) can be obtained. In the following section, we apply this method to class of nonlinear Schrödinger-type equations to obtain new quasidoubly periodic solution.


## 3. Applications

In the following, we use the  improved Jacobi elliptic function method to seek exact traveling wave solutions of class of nonlinear Schrödinger-type equations which are of interest in plasma physics, wave propagation in nonlinear optical fibers, Ginzburg-Landau theory of superconductivity, and so forth.


### 3.1. Generalized Zakharov’s System

In the interaction of laser-plasma the system of Zakharov’s equation plays an important role. This system has wide interest and attention for many scientists.Let us consider the generalized Zakharov system [19]
(3.1)utt-cs2uxx=β(|E|2)xx,iEt+αExx-δ1uE+δ2|E|2E+δ3|E|4E=0.
When δ2=δ3=0, the generalized Zakharov system reduces to the famous Zakharov system which describe the propagation Langmuir waves in plasmas. The real unknown function u(x,t) is the fluctuation in the ion density about its equilibrium value, and the complex unknown function E(x,t) is the slowly varying envelope of highly oscillatory electron field. The parameters α, β, δ1, δ2, δ3, and cs are real numbers, where cs is proportional to the ion acoustic speed (or electron sound speed). Here, we seek its traveling wave solution in the forms
(3.2)E(x,t)=H(ξ)ei(kx-ωt),u(x,t)=u(ξ),ξ=x-ct,
where k, ω, and c are constants and H(ξ) is real function. Therefore, system (3.1) reduces to
(3.3)(c2-cs2)u′′=β(H2)ξξ,(3.4)αH′′+i(2αk-c)H′+(ω-αk2)H-δ1uH+δ2H3+δ3H5=0.
Integrating (3.3) with respect to ξ and taking the integration constants to zero yield
(3.5)u=βc2-cs2H2,c2-cs2≠0.
Substituting (3.5) into (3.4) results in
(3.6)H′′+1α[(ω-αk2)H+(δ2-βδ1c2-cs2)H3+δ3H5]=0,c=2αk,α≠0.
According to Step 3, we assume that (3.6) possesses the solutions in the form
(3.7)H(ξ)=a0+a1F(ξ).
Substituting (3.7) with (2.1) into (3.6) and equating each of the coefficients of Fi(ξ), i=0,1,…,5 to zero, we obtain system of algebraic equations. To avoid tediousness, we omit the overdetermined algebraic equations. From the output of Maple, we obtain the following solution:
(3.8)a0=0,c=2αk,ω=α(k2-A2),a1=±3A62δ3A4(δ2-βδ14α2k2-cs2).
Now, based on the solutions of (2.1), one can obtain new types of quasiperiodic wave solution of the generalized Zakharov system. We obtain the general formulae of the solution of system (3.1)
(3.9)u(x,t)=3A6β2δ3A4(4α2k2-cs2)(δ2-βδ14α2k2-cs2)F2(x-2αkt),E(x,t)=±3A62δ3A4(δ2-βδ14α2k2-cs2)F(x-2αkt)ei(kx-ωt).
By selecting the special values of the A0, A2, A4, A6 and the corresponding function F(ξ), we have the following solutions of the generalized Zakharov system (3.1):
(3.10)u1(x,t)=-3k12k22β2δ3(k12+k22+k12k22)(4α2k2-cs2)(δ2-βδ14α2k2-cs2)s2(x-2αkt,k1,k2),E1(x,t)=±-3k12k222δ3(k12+k22+k12k22)(δ2-βδ14α2k2-cs2)s(x-2αkt,k1,k2)ei(kx-ωt),u2(x,t)=-3k12k22β2δ3(3k12k22-k12-k22)(4α2k2-cs2)(δ2-βδ14α2k2-cs2)c2(x-2αkt,k1,k2),E2(x,t)=±-3k12k222δ3(3k12k22-k12-k22)(δ2-βδ14α2k2-cs2)c(x-2αkt,k1,k2)ei(kx-ωt).
We omitted the reminder solutions for simplicity.


### 3.2. Rangwala-Rao Equation

The Rangwala-Rao equation [20] is
(3.11)uxt-β1uxx+u+iTβ2|u|2ux=0,T=±1,
where β1, β2 are real constants. Rangwala and Rao introduced Equation (3.11) as the integrability condition when they studied the mixed, derivative, nonlinear Schrödinger equations and looked for the Bäcklund transformation and solitary wave solutions.Suppose the exact solutions of (3.11) is of the form
(3.12)u(x,t)=e-iωteiψ(x-ct)H(x-ct),
where ω,c are constants determined later and ψ, H are undetermined functions with one variable only.  Set the relation of  ψ, H as
(3.13)ψ′(ξ)=ω2(c+β1)+Tβ24(c+β1)H2(ξ),=′ddξ,ξ=x-ct.
Substituting (3.12) with (3.13) into (3.11) simultaneously yields
(3.14)H′′-4(c+β1)-ω24(c+β1)2H-Tβ2ω2(c+β1)2H3+3T2β2216(c+β1)2H5=0.
According to the homogeneous balance principle, we suppose that the exact solutions of (3.14) take the form
(3.15)H(ξ)=a0+a1F(ξ).
Substituting (3.15) with (2.1) into (3.14) and equating each of the coefficients of Fi(ξ), i=0,1,…,5 to zero, we obtain system of algebraic equations. Solving this system with the aid of Maple, we obtain the following solution:
(3.16)a0=0,ω=2(c+β1)[1-(c+β1)A2],a1=±-8A6ω3TA4β2.
The general formulae of the solutions of Rangwala-Rao equation
(3.17)u(x,t)=±-8A6ω3TA4β2F(x-ct)e-iωteiψ(x-ct),
with ψ(ξ)=ω/6A4(c+β1)∫[3A4-4A6F2(ξ)]dξ, ω=2(c+β1)[1-(c+β1)A2].By selecting the special values of theA0,A2,A4,A6 and the corresponding function F(ξ), we have the following intensities of the solutions of the Rangwala-Rao equation.WhenA0=1, A2=-(1+k12+k22), A4=k12+k22+k12k22 and A6=-k12k22, we have
(3.18)|u1|2=-16k12k22(c+β1)[1+(1+k12+k22)(c+β1)]3Tβ2(k12+k22+k12k22)s2(x-ct,k1,k2),
and when A0=1-k12-k22+k12k22, A2=2k12+2k22-3k12k22-1, A4=3k12k22-k12-k22, and A6=-k12k22, we have
(3.19)|u2|2=-16k12k22(c+β1)[1-(2k12+2k22-3k12k22)(c+β1)]3Tβ2(3k12k22-k12-k22)c2(x-ct,k1,k2).
We omitted the reminder intensities for simplicity.


### 3.3. Chen-Lee-Lin Equation

The Chen-Lee-Lin equation [20] is
(3.20)iut+uxx+iδ|u|2ux=0,
where δ is a real constant. Similarly as before, we suppose the exact solution of (3.20) is of the form
(3.21)u(x,t)=e-iωteiψ(x-ct)H(x-ct).
Set the relation of ψ, H as
(3.22)ψ′(ξ)=c2-δ4H2(ξ),=′ddξ,ξ=x-ct.
Substituting (3.21) with (3.22) into (3.20) simultaneously yields
(3.23)H′′+(ω+c24)H-cδ2H3+3δ216H5=0.
According to the homogeneous balance principle, we suppose that the exact solutions of (3.23) take the form
(3.24)H(ξ)=a0+a1F(ξ).
Substituting (3.24) with (2.1) into (3.23) and equating each of the coefficients of Fi(ξ),i=0,1,…,5 to zero, we obtain system of algebraic equations. Solving this system with the aid of Maple, we obtain the following solution:
(3.25)a0=0,ω=A2-c24,a1=±2-cA6δA4.
The general formulae of the solution of Chen-Lee-Lin equation
(3.26)u(x,t)=±2-cA6δA4F(x-ct)e-iωteiψ(x-ct),
with ψ(ξ)=(c/2A4)∫[A4-2A6F2(ξ)]dξ and ω=A2-c2/4. By selecting the special values of the A0, A2, A4, A6 and the corresponding function F(ξ), we have the following intensities of the solutions of the Chen-Lee-Lin equation.WhenA0=1, A2=-(1+k12+k22), A4=k12+k22+k12k22 and A6=-k12k22, we have
(3.27)|u1|2=-4ck12k22δ(k12+k22+k12k22)s2(x-ct,k1,k2),
and when A0=1-k12-k22+k12k22, A2=2k12+2k22-3k12k22-1, A4=3k12k22-k12-k22 and A6=-k12k22, we have
(3.28)|u2|2=-4ck12k22δ(3k12k22-k12-k22)c2(x-ct,k1,k2).
We omitted the reminder intensities for simplicity.Besides the solutions obtained above, the ODE Equation (2.1), albeit with different parameters, has been studied in the different context [21–24]. It has been shown that this equation possesses abundant solutions, Including Weierstrass function solutions, kink solutions, periodic solutions, and so forth. To the best of our knowledge, some of our explicit solutions are new.Notice that the GJEFs are generalization of the Jacobi elliptic, hyperbolic, and trigonometric functions as stated in the appendix. Also, the two modulus parametersk1 and k2 describe the degree of the wave energy localization in the obtained solutions.


## 3.1. Generalized Zakharov’s System

In the interaction of laser-plasma the system of Zakharov’s equation plays an important role. This system has wide interest and attention for many scientists.Let us consider the generalized Zakharov system [19]
(3.1)utt-cs2uxx=β(|E|2)xx,iEt+αExx-δ1uE+δ2|E|2E+δ3|E|4E=0.
When δ2=δ3=0, the generalized Zakharov system reduces to the famous Zakharov system which describe the propagation Langmuir waves in plasmas. The real unknown function u(x,t) is the fluctuation in the ion density about its equilibrium value, and the complex unknown function E(x,t) is the slowly varying envelope of highly oscillatory electron field. The parameters α, β, δ1, δ2, δ3, and cs are real numbers, where cs is proportional to the ion acoustic speed (or electron sound speed). Here, we seek its traveling wave solution in the forms
(3.2)E(x,t)=H(ξ)ei(kx-ωt),u(x,t)=u(ξ),ξ=x-ct,
where k, ω, and c are constants and H(ξ) is real function. Therefore, system (3.1) reduces to
(3.3)(c2-cs2)u′′=β(H2)ξξ,(3.4)αH′′+i(2αk-c)H′+(ω-αk2)H-δ1uH+δ2H3+δ3H5=0.
Integrating (3.3) with respect to ξ and taking the integration constants to zero yield
(3.5)u=βc2-cs2H2,c2-cs2≠0.
Substituting (3.5) into (3.4) results in
(3.6)H′′+1α[(ω-αk2)H+(δ2-βδ1c2-cs2)H3+δ3H5]=0,c=2αk,α≠0.
According to Step 3, we assume that (3.6) possesses the solutions in the form
(3.7)H(ξ)=a0+a1F(ξ).
Substituting (3.7) with (2.1) into (3.6) and equating each of the coefficients of Fi(ξ), i=0,1,…,5 to zero, we obtain system of algebraic equations. To avoid tediousness, we omit the overdetermined algebraic equations. From the output of Maple, we obtain the following solution:
(3.8)a0=0,c=2αk,ω=α(k2-A2),a1=±3A62δ3A4(δ2-βδ14α2k2-cs2).
Now, based on the solutions of (2.1), one can obtain new types of quasiperiodic wave solution of the generalized Zakharov system. We obtain the general formulae of the solution of system (3.1)
(3.9)u(x,t)=3A6β2δ3A4(4α2k2-cs2)(δ2-βδ14α2k2-cs2)F2(x-2αkt),E(x,t)=±3A62δ3A4(δ2-βδ14α2k2-cs2)F(x-2αkt)ei(kx-ωt).
By selecting the special values of the A0, A2, A4, A6 and the corresponding function F(ξ), we have the following solutions of the generalized Zakharov system (3.1):
(3.10)u1(x,t)=-3k12k22β2δ3(k12+k22+k12k22)(4α2k2-cs2)(δ2-βδ14α2k2-cs2)s2(x-2αkt,k1,k2),E1(x,t)=±-3k12k222δ3(k12+k22+k12k22)(δ2-βδ14α2k2-cs2)s(x-2αkt,k1,k2)ei(kx-ωt),u2(x,t)=-3k12k22β2δ3(3k12k22-k12-k22)(4α2k2-cs2)(δ2-βδ14α2k2-cs2)c2(x-2αkt,k1,k2),E2(x,t)=±-3k12k222δ3(3k12k22-k12-k22)(δ2-βδ14α2k2-cs2)c(x-2αkt,k1,k2)ei(kx-ωt).
We omitted the reminder solutions for simplicity.


## 3.2. Rangwala-Rao Equation

The Rangwala-Rao equation [20] is
(3.11)uxt-β1uxx+u+iTβ2|u|2ux=0,T=±1,
where β1, β2 are real constants. Rangwala and Rao introduced Equation (3.11) as the integrability condition when they studied the mixed, derivative, nonlinear Schrödinger equations and looked for the Bäcklund transformation and solitary wave solutions.Suppose the exact solutions of (3.11) is of the form
(3.12)u(x,t)=e-iωteiψ(x-ct)H(x-ct),
where ω,c are constants determined later and ψ, H are undetermined functions with one variable only.  Set the relation of  ψ, H as
(3.13)ψ′(ξ)=ω2(c+β1)+Tβ24(c+β1)H2(ξ),=′ddξ,ξ=x-ct.
Substituting (3.12) with (3.13) into (3.11) simultaneously yields
(3.14)H′′-4(c+β1)-ω24(c+β1)2H-Tβ2ω2(c+β1)2H3+3T2β2216(c+β1)2H5=0.
According to the homogeneous balance principle, we suppose that the exact solutions of (3.14) take the form
(3.15)H(ξ)=a0+a1F(ξ).
Substituting (3.15) with (2.1) into (3.14) and equating each of the coefficients of Fi(ξ), i=0,1,…,5 to zero, we obtain system of algebraic equations. Solving this system with the aid of Maple, we obtain the following solution:
(3.16)a0=0,ω=2(c+β1)[1-(c+β1)A2],a1=±-8A6ω3TA4β2.
The general formulae of the solutions of Rangwala-Rao equation
(3.17)u(x,t)=±-8A6ω3TA4β2F(x-ct)e-iωteiψ(x-ct),
with ψ(ξ)=ω/6A4(c+β1)∫[3A4-4A6F2(ξ)]dξ, ω=2(c+β1)[1-(c+β1)A2].By selecting the special values of theA0,A2,A4,A6 and the corresponding function F(ξ), we have the following intensities of the solutions of the Rangwala-Rao equation.WhenA0=1, A2=-(1+k12+k22), A4=k12+k22+k12k22 and A6=-k12k22, we have
(3.18)|u1|2=-16k12k22(c+β1)[1+(1+k12+k22)(c+β1)]3Tβ2(k12+k22+k12k22)s2(x-ct,k1,k2),
and when A0=1-k12-k22+k12k22, A2=2k12+2k22-3k12k22-1, A4=3k12k22-k12-k22, and A6=-k12k22, we have
(3.19)|u2|2=-16k12k22(c+β1)[1-(2k12+2k22-3k12k22)(c+β1)]3Tβ2(3k12k22-k12-k22)c2(x-ct,k1,k2).
We omitted the reminder intensities for simplicity.


## 3.3. Chen-Lee-Lin Equation

The Chen-Lee-Lin equation [20] is
(3.20)iut+uxx+iδ|u|2ux=0,
where δ is a real constant. Similarly as before, we suppose the exact solution of (3.20) is of the form
(3.21)u(x,t)=e-iωteiψ(x-ct)H(x-ct).
Set the relation of ψ, H as
(3.22)ψ′(ξ)=c2-δ4H2(ξ),=′ddξ,ξ=x-ct.
Substituting (3.21) with (3.22) into (3.20) simultaneously yields
(3.23)H′′+(ω+c24)H-cδ2H3+3δ216H5=0.
According to the homogeneous balance principle, we suppose that the exact solutions of (3.23) take the form
(3.24)H(ξ)=a0+a1F(ξ).
Substituting (3.24) with (2.1) into (3.23) and equating each of the coefficients of Fi(ξ),i=0,1,…,5 to zero, we obtain system of algebraic equations. Solving this system with the aid of Maple, we obtain the following solution:
(3.25)a0=0,ω=A2-c24,a1=±2-cA6δA4.
The general formulae of the solution of Chen-Lee-Lin equation
(3.26)u(x,t)=±2-cA6δA4F(x-ct)e-iωteiψ(x-ct),
with ψ(ξ)=(c/2A4)∫[A4-2A6F2(ξ)]dξ and ω=A2-c2/4. By selecting the special values of the A0, A2, A4, A6 and the corresponding function F(ξ), we have the following intensities of the solutions of the Chen-Lee-Lin equation.WhenA0=1, A2=-(1+k12+k22), A4=k12+k22+k12k22 and A6=-k12k22, we have
(3.27)|u1|2=-4ck12k22δ(k12+k22+k12k22)s2(x-ct,k1,k2),
and when A0=1-k12-k22+k12k22, A2=2k12+2k22-3k12k22-1, A4=3k12k22-k12-k22 and A6=-k12k22, we have
(3.28)|u2|2=-4ck12k22δ(3k12k22-k12-k22)c2(x-ct,k1,k2).
We omitted the reminder intensities for simplicity.Besides the solutions obtained above, the ODE Equation (2.1), albeit with different parameters, has been studied in the different context [21–24]. It has been shown that this equation possesses abundant solutions, Including Weierstrass function solutions, kink solutions, periodic solutions, and so forth. To the best of our knowledge, some of our explicit solutions are new.Notice that the GJEFs are generalization of the Jacobi elliptic, hyperbolic, and trigonometric functions as stated in the appendix. Also, the two modulus parametersk1 and k2 describe the degree of the wave energy localization in the obtained solutions.


## 4. Conclusion

There is no systematic way for solving (2.1). Nevertheless, this ansätz with four arbitrary parameters A0, A2, A4, A6 is reasonable since its solution can be expressed in terms of functions, such as generalized Jacobi elliptic functions, that appear only in the nonlinear problems. In addition, these functions go back, in some limiting cases, to sn, cn, dn, tanh, sech, sin, and cos functions that describe the double periodic, periodic, solitary, and shock wave propagation. The values of the constants ai(i=0,1,…,n) in (2.4) depend crucially on the nature of differential equations whereas different types of their solutions can be classified in terms of A0, A2, A4, A6 as shown in Cases 1–4. In this work, we obtain the exact solutions of the generalized Zakharov system, the Rangwala-Rao equation, and the Chen-Lee-Lin equation by using GJEFs. We believe one can apply this method to many other nonlinear partial differential equations in mathematical physics.



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