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

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 derivedwith the aid of the homogenous balance principle.


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, Mathematical Problems in Engineering 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 Schr ödinger-type equations.
The rest of this paper is organized as follows.In Section 2, 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.

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 where ξ ≡ ξ x, t and A 0 , A 2 , A 4 , A 6 are constants.
s ξ, k 1 , k 2 is the generalized Jacobi elliptic sine function, ξ is an independent variable, k 1 , k 2 0 ≤ k 2 ≤ k 1 ≤ 1 are two modulus of the GJEFs, c ξ, k 1 , k 2 is the generalized Jacobi elliptic cosine function, d 1 ξ, k 1 , k 2 is the generalized Jacobi elliptic function of the third kind, and d 2 ξ, k 1 , k 2 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 variables x, t, P u, u t , u x , u xx , . . .0, 2.
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: 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 The processes take the following steps.
Step 1. Determine n 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 a 0 , a 1 , . . ., a n , k and ω.
Step 3. Solving this system obtained in Step 2, then a 0 , a 1 , . . ., a n , k and ω can be expressed by A 0 , A 2 , A 4 , A 6 .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 Schr ödinger-type equations to obtain new quasidoubly periodic solution.

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

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
When δ 2 δ 3 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, δ 1 , δ 2 , δ 3 , and c s are real numbers, where c s is proportional to the ion acoustic speed or electron sound speed .Here, we seek its traveling wave solution in the forms where k, ω, and c are constants and H ξ is real function.Therefore, system 3.1 reduces to Integrating 3.3 with respect to ξ and taking the integration constants to zero yield According to Step 3, we assume that 3.6 possesses the solutions in the form H ξ a 0 a 1 F ξ .

3.7
Substituting 3.7 with 2.1 into 3.6 and equating each of the coefficients of F i 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
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
By selecting the special values of the A 0 , A 2 , A 4 , A 6 and the corresponding function F ξ , we have the following solutions of the generalized Zakharov system 3.1 :

3.10
We omitted the reminder solutions for simplicity.

3.14
According to the homogeneous balance principle, we suppose that the exact solutions of 3.14 take the form H ξ a 0 a 1 F ξ .

3.15
Substituting 3.15 with 2.1 into 3.14 and equating each of the coefficients of F i 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: The general formulae of the solutions of Rangwala-Rao equation By selecting the special values of the A 0 , A 2 , A 4 , A 6 and the corresponding function F ξ , we have the following intensities of the solutions of the Rangwala-Rao equation. When

3.19
We omitted the reminder intensities for simplicity.

Chen-Lee-Lin Equation
The Chen-Lee-Lin equation 20 is where δ is a real constant.Similarly as before, we suppose the exact solution of 3.20 is of the form u x, t e −iωt e iψ x−ct H x − ct .

3.21
Set the relation of ψ, H as Substituting 3.21 with 3.22 into 3.20 simultaneously yields According to the homogeneous balance principle, we suppose that the exact solutions of 3.23 take the form H ξ a 0 a 1 F ξ .

3.24
Substituting 3.24 with 2.1 into 3.23 and equating each of the coefficients of F i 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
The general formulae of the solution of Chen-Lee-Lin equation . By selecting the special values of the A 0 , A 2 , A 4 , A 6 and the corresponding function F ξ , we have the following intensities of the solutions of the Chen-Lee-Lin equation. When

3.28
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 parameters k 1 and k 2 describe the degree of the wave energy localization in the obtained solutions.

Conclusion
There is no systematic way for solving 2.1 .Nevertheless, this ansätz with four arbitrary parameters A 0 , A 2 , A 4 , A 6 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 a i 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 A 0 , A 2 , A 4 , A 6 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.
The GJEFs possess the following properties of the triangular functions we use the abbreviated notations s y ≡ s y, k 1 , k 2 , c y ≡ y, k 1 , k 2 , . .., and so forth : The first derivatives of these functions are given by The GJEFs can be expressed in terms of the standard Jacobi elliptic functions s y, k