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

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.

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 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.

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 𝐹^′(𝜉)=𝐴₀+𝐴₂𝐹²(𝜉)+𝐴₄𝐹⁴(𝜉)+𝐴₆𝐹⁶(𝜉),^′≡𝑑𝑑𝜉, (2.1) where 𝜉≡𝜉(𝑥,𝑡) and 𝐴₀,𝐴₂,𝐴₄,𝐴₆ are constants.

Case 1. If 𝐴₀=1,𝐴₂=−(1+𝑘²₁+𝑘²₂),𝐴₄=𝑘²₁+𝑘²₂+𝑘²₁𝑘²₂ and 𝐴₆=−𝑘²₁𝑘²₂, then (2.1) has a solution 𝑠(𝜉,𝑘₁,𝑘₂).

Case 2. If 𝐴₀=1−𝑘²₁−𝑘²₂+𝑘²₁𝑘²₂,𝐴₂=2𝑘²₁+2𝑘²₂−3𝑘²₁𝑘²₂−1, 𝐴₄=3𝑘²₁𝑘²₂−𝑘²₁−𝑘²₂ and 𝐴₆=−𝑘²₁𝑘²₂, then (2.1) has a solution 𝑐(𝜉,𝑘₁,𝑘₂).

Case 3. If 𝐴₀=𝑘²₁−1−𝑘²₂+𝑘²₂𝑘₁⁻², 𝐴₂=2𝑘²₂+2−𝑘²₁−3𝑘²₂𝑘₁⁻², 𝐴₄=3𝑘²₂𝑘₁⁻²−𝑘²₂−1 and 𝐴₆=−𝑘²₂𝑘₁⁻², then (2.1) has a solution 𝑑₁(𝜉,𝑘₁,𝑘₂).

Case 4. If 𝐴₀=𝑘²₂−1−𝑘²₁+𝑘²₁𝑘₂⁻², 𝐴₂=2𝑘²₁+2−𝑘²₂−3𝑘²₁𝑘₂⁻², 𝐴₄=3𝑘²₁𝑘₂⁻²−𝑘²₁−1 and 𝐴₆=−𝑘²₁𝑘₂⁻², 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 variables 𝑥, 𝑡, 𝑃𝑢,𝑢_𝑡,𝑢_𝑥,𝑢_𝑥𝑥,…=0, (2.2) 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: 𝑢(𝑥,𝑡)=𝑈(𝜉),𝜉=𝑘(𝑥−𝜔𝑡), (2.3) 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 𝑢(𝜉)=𝑎₀+^𝑛_𝑖=1𝑎_𝑖𝐹^𝑖(𝜉). (2.4) 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 𝑎₀,𝑎₁,…,𝑎_𝑛, 𝑘 and 𝜔.

Step 3. Solving this system obtained in Step 2, then 𝑎₀,𝑎₁,…,𝑎_𝑛, 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] 𝑢_𝑡𝑡−𝑐²_𝑠𝑢_𝑥𝑥=𝛽|𝐸|²_𝑥𝑥,𝑖𝐸_𝑡+𝛼𝐸_𝑥𝑥−𝛿₁𝑢𝐸+𝛿₂|𝐸|²𝐸+𝛿₃|𝐸|⁴𝐸=0. (3.1) 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 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 𝐸(𝑥,𝑡)=𝐻(𝜉)𝑒^{𝑖(𝑘𝑥−𝜔𝑡)},𝑢(𝑥,𝑡)=𝑢(𝜉),𝜉=𝑥−𝑐𝑡, (3.2) where 𝑘, 𝜔, and 𝑐 are constants and 𝐻(𝜉) is real function. Therefore, system (3.1) reduces to 𝑐²−𝑐²_𝑠𝑢^′′=𝛽𝐻²_𝜉𝜉, (3.3)𝛼𝐻^′′+𝑖(2𝛼𝑘−𝑐)𝐻^′+𝜔−𝛼𝑘²𝐻−𝛿₁𝑢𝐻+𝛿₂𝐻³+𝛿₃𝐻⁵=0. (3.4) Integrating (3.3) with respect to 𝜉 and taking the integration constants to zero yield 𝛽𝑢=𝑐²−𝑐²_𝑠𝐻²,𝑐²−𝑐²_𝑠≠0. (3.5) Substituting (3.5) into (3.4) results in 𝐻^′′+1𝛼𝜔−𝛼𝑘²𝐻+𝛿₂−𝛽𝛿₁𝑐²−𝑐²_𝑠𝐻³+𝛿₃𝐻⁵=0,𝑐=2𝛼𝑘,𝛼≠0. (3.6) According to Step 3, we assume that (3.6) possesses the solutions in the form 𝐻(𝜉)=𝑎₀+𝑎₁𝐹(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: 𝑎₀=0,𝑐=2𝛼𝑘,𝜔=𝛼𝑘²−𝐴₂,𝑎₁=±3𝐴₆2𝛿₃𝐴₄𝛿₂−𝛽𝛿₁4𝛼²𝑘²−𝑐²_𝑠. (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) 𝑢(𝑥,𝑡)=3𝐴₆𝛽2𝛿₃𝐴₄4𝛼²𝑘²−𝑐²_𝑠𝛿₂−𝛽𝛿₁4𝛼²𝑘²−𝑐²_𝑠𝐹²(),𝐸()𝑥−2𝛼𝑘𝑡𝑥,𝑡=±3𝐴₆2𝛿₃𝐴₄𝛿₂−𝛽𝛿₁4𝛼²𝑘²−𝑐²_𝑠𝐹(𝑥−2𝛼𝑘𝑡)𝑒^{𝑖(𝑘𝑥−𝜔𝑡)}. (3.9) 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): 𝑢₁(𝑥,𝑡)=−3𝑘²₁𝑘²₂𝛽2𝛿₃𝑘²₁+𝑘²₂+𝑘²₁𝑘²₂4𝛼²𝑘²−𝑐²_𝑠𝛿₂−𝛽𝛿₁4𝛼²𝑘²−𝑐²_𝑠𝑠²𝑥−2𝛼𝑘𝑡,𝑘₁,𝑘₂,𝐸₁(𝑥,𝑡)=±−3𝑘²₁𝑘²₂2𝛿₃𝑘²₁+𝑘²₂+𝑘²₁𝑘²₂𝛿₂−𝛽𝛿₁4𝛼²𝑘²−𝑐²_𝑠𝑠𝑥−2𝛼𝑘𝑡,𝑘₁,𝑘₂𝑒^{𝑖(𝑘𝑥−𝜔𝑡)},𝑢₂(𝑥,𝑡)=−3𝑘²₁𝑘²₂𝛽2𝛿₃3𝑘²₁𝑘²₂−𝑘²₁−𝑘²₂4𝛼²𝑘²−𝑐²_𝑠𝛿₂−𝛽𝛿₁4𝛼²𝑘²−𝑐²_𝑠𝑐²𝑥−2𝛼𝑘𝑡,𝑘₁,𝑘₂,𝐸₂(𝑥,𝑡)=±−3𝑘²₁𝑘²₂2𝛿₃3𝑘²₁𝑘²₂−𝑘²₁−𝑘²₂𝛿₂−𝛽𝛿₁4𝛼²𝑘²−𝑐²_𝑠𝑐𝑥−2𝛼𝑘𝑡,𝑘₁,𝑘₂𝑒^{𝑖(𝑘𝑥−𝜔𝑡)}. (3.10) We omitted the reminder solutions for simplicity.

3.2. Rangwala-Rao Equation

The Rangwala-Rao equation [20] is 𝑢_𝑥𝑡−𝛽₁𝑢_𝑥𝑥+𝑢+𝑖𝑇𝛽₂|𝑢|²𝑢_𝑥=0,𝑇=±1, (3.11) 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 𝑢(𝑥,𝑡)=𝑒^{−𝑖𝜔𝑡}𝑒^{𝑖𝜓(𝑥−𝑐𝑡)}𝐻(𝑥−𝑐𝑡), (3.12) where 𝜔,𝑐 are constants determined later and 𝜓, 𝐻 are undetermined functions with one variable only. Set the relation of 𝜓, 𝐻 as 𝜓^′(𝜉)=𝜔2𝑐+𝛽₁+𝑇𝛽₂4𝑐+𝛽₁𝐻²(𝜉),^′=𝑑𝑑𝜉,𝜉=𝑥−𝑐𝑡. (3.13) Substituting (3.12) with (3.13) into (3.11) simultaneously yields 𝐻^′′−4𝑐+𝛽₁−𝜔²4𝑐+𝛽₁²𝐻−𝑇𝛽₂𝜔2𝑐+𝛽₁²𝐻³+3𝑇²𝛽²₂16𝑐+𝛽₁²𝐻⁵=0. (3.14) According to the homogeneous balance principle, we suppose that the exact solutions of (3.14) take the form 𝐻(𝜉)=𝑎₀+𝑎₁𝐹(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: 𝑎₀=0,𝜔=2𝑐+𝛽₁1−𝑐+𝛽₁𝐴₂,𝑎₁=±−8𝐴₆𝜔3𝑇𝐴₄𝛽₂. (3.16) The general formulae of the solutions of Rangwala-Rao equation 𝑢(𝑥,𝑡)=±−8𝐴₆𝜔3𝑇𝐴₄𝛽₂𝐹(𝑥−𝑐𝑡)𝑒^{−𝑖𝜔𝑡}𝑒^{𝑖𝜓(𝑥−𝑐𝑡)}, (3.17) 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 the 𝐴₀,𝐴₂,𝐴₄,𝐴₆ and the corresponding function 𝐹(𝜉), we have the following intensities of the solutions of the Rangwala-Rao equation.

When 𝐴₀=1, 𝐴₂=−(1+𝑘²₁+𝑘²₂), 𝐴₄=𝑘²₁+𝑘²₂+𝑘²₁𝑘²₂ and 𝐴₆=−𝑘²₁𝑘²₂, we have 𝑢₁²=−16𝑘²₁𝑘²₂𝑐+𝛽₁1+1+𝑘²₁+𝑘²₂𝑐+𝛽₁3𝑇𝛽₂𝑘²₁+𝑘²₂+𝑘²₁𝑘²₂𝑠²𝑥−𝑐𝑡,𝑘₁,𝑘₂, (3.18) and when 𝐴₀=1−𝑘²₁−𝑘²₂+𝑘²₁𝑘²₂, 𝐴₂=2𝑘²₁+2𝑘²₂−3𝑘²₁𝑘²₂−1, 𝐴₄=3𝑘²₁𝑘²₂−𝑘²₁−𝑘²₂, and 𝐴₆=−𝑘²₁𝑘²₂, we have 𝑢₂²=−16𝑘²₁𝑘²₂𝑐+𝛽₁1−2𝑘²₁+2𝑘²₂−3𝑘²₁𝑘²₂𝑐+𝛽₁3𝑇𝛽₂3𝑘²₁𝑘²₂−𝑘²₁−𝑘²₂𝑐²𝑥−𝑐𝑡,𝑘₁,𝑘₂. (3.19) We omitted the reminder intensities for simplicity.

3.3. Chen-Lee-Lin Equation

The Chen-Lee-Lin equation [20] is 𝑖𝑢_𝑡+𝑢_𝑥𝑥+𝑖𝛿|𝑢|²𝑢_𝑥=0, (3.20) 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 𝜓^′(𝜉)=𝑐2−𝛿4𝐻²(𝜉),^′=𝑑𝑑𝜉,𝜉=𝑥−𝑐𝑡. (3.22) Substituting (3.21) with (3.22) into (3.20) simultaneously yields 𝐻^′′+𝑐𝜔+²4𝐻−𝑐𝛿2𝐻³+3𝛿²𝐻16⁵=0. (3.23) According to the homogeneous balance principle, we suppose that the exact solutions of (3.23) take the form 𝐻(𝜉)=𝑎₀+𝑎₁𝐹(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: 𝑎₀=0,𝜔=𝐴₂−𝑐²4,𝑎₁=±2−𝑐𝐴₆𝛿𝐴₄. (3.25) The general formulae of the solution of Chen-Lee-Lin equation 𝑢(𝑥,𝑡)=±2−𝑐𝐴₆𝛿𝐴₄𝐹(𝑥−𝑐𝑡)𝑒^{−𝑖𝜔𝑡}𝑒^{𝑖𝜓(𝑥−𝑐𝑡)}, (3.26) 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.

When 𝐴₀=1, 𝐴₂=−(1+𝑘²₁+𝑘²₂), 𝐴₄=𝑘²₁+𝑘²₂+𝑘²₁𝑘²₂ and 𝐴₆=−𝑘²₁𝑘²₂, we have 𝑢₁²=−4𝑐𝑘²₁𝑘²₂𝛿𝑘²₁+𝑘²₂+𝑘²₁𝑘²₂𝑠²𝑥−𝑐𝑡,𝑘₁,𝑘₂, (3.27) and when 𝐴₀=1−𝑘²₁−𝑘²₂+𝑘²₁𝑘²₂, 𝐴₂=2𝑘²₁+2𝑘²₂−3𝑘²₁𝑘²₂−1, 𝐴₄=3𝑘²₁𝑘²₂−𝑘²₁−𝑘²₂ and 𝐴₆=−𝑘²₁𝑘²₂, we have 𝑢₂²=−4𝑐𝑘²₁𝑘²₂𝛿3𝑘²₁𝑘²₂−𝑘²₁−𝑘²₂𝑐²𝑥−𝑐𝑡,𝑘₁,𝑘₂. (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 𝑘₁ 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 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 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.

Appendix

In this appendix, we review the GJEFs and study some properties of these functions [13, 16–18]. We consider the (pseudo-) hyperelliptic integral 𝑦𝑥,𝑘₁,𝑘₂=^𝑥₀𝑑𝑡1−𝑡²1−𝑘²₁𝑡²1−𝑘²₂𝑡². (A.1) We define the generalized Jacobi elliptic sine function as the inverse function 𝑥=𝑠(𝑦,𝑘₁,𝑘₂), where 𝑦 is an independent variable and 𝑘₁,𝑘₂(0≤𝑘₂≤𝑘₁≤1) are two modulus of the GJEFs. Similarly, √1−𝑥², 1−𝑘²₁𝑥², and 1−𝑘²₂𝑥² are defined as the generalized Jacobi elliptic cosine function, the generalized Jacobi elliptic function of the third kind, and the generalized Jacobi elliptic function of the forth kind. They are expressed as √1−𝑥²=𝑐𝑦,𝑘₁,𝑘₂,1−𝑘²₁𝑥²=𝑑₁𝑦,𝑘₁,𝑘₂,1−𝑘²₂𝑥²=𝑑₂𝑦,𝑘₁,𝑘₂. (A.2) The GJEFs possess the following properties of the triangular functions (we use the abbreviated notations 𝑠(𝑦)≡𝑠(𝑦,𝑘₁,𝑘₂), 𝑐(𝑦)≡(𝑦,𝑘₁,𝑘₂),…, and so forth): 𝑐²𝑦=1−𝑠²𝑦,𝑑²₁𝑦=1−𝑘²₁𝑠²𝑦,𝑑²₂𝑦=1−𝑘²₂𝑠²𝑦,𝑘²₁𝑑²₂𝑦−𝑘²₂𝑑²₁𝑦=𝑘²₁−𝑘²₂,𝑑²_𝑖𝑦−𝑘²_𝑖𝑐²𝑦=1−𝑘²_𝑖,(𝑖=1,2). (A.3) The first derivatives of these functions are given by 𝑠^′𝑦=𝑐𝑦𝑑₁𝑦𝑑₂𝑦,𝑐^′𝑦=−𝑠𝑦𝑑₁𝑦𝑑₂𝑦,𝑑^′₁𝑦=−𝑘²₁𝑠𝑦𝑐𝑦𝑑₂𝑦,𝑑^′₂𝑦=−𝑘²₂𝑠𝑦𝑐𝑦𝑑₁𝑦. (A.4) Moreover, in the limiting case 𝑘₂→0, the GJEF reduced to the usual JEFs 𝑠𝑦,𝑘₁,0⟶sn𝑦,𝑘₁,𝑐𝑦,𝑘₁,0⟶cn𝑦,𝑘₁,𝑑₁𝑦,𝑘₁,0,𝑑₂𝑦,𝑘₁,0⟶dn𝑦,𝑘₁. (A.5) When 𝑘₁→1, 𝑘₂→0, we have 𝑠𝑦,1,0⟶tanh𝑦,𝑐𝑦,1,0,𝑑₁𝑦,1,0,𝑑₂𝑦,1,0⟶sech𝑦. (A.6) Also, in the limiting case 𝑘₁→0, 𝑘₂→0, we have 𝑠𝑦,0,0⟶sin𝑦,𝑐𝑦,0,0⟶cos𝑦,𝑑₁𝑦,0,0,𝑑₂𝑦,0,0⟶1. (A.7) The GJEFs can be expressed in terms of the standard Jacobi elliptic functions 𝑠𝑦,𝑘₁,𝑘₂=sn𝑘^′₂𝑦,𝑘1−𝑘²₂+𝑘²₂sn²𝑘^′₂𝑦,𝑘,𝑐𝑦,𝑘₁,𝑘₂=𝑘^′₂cn𝑘^′₂𝑦,𝑘1−𝑘²₂cn²𝑘^′₂,𝑑𝑦,𝑘₁𝑦,𝑘₁,𝑘₂=𝑘²₁−𝑘²₂dn𝑘^′₂𝑦,𝑘𝑘²₁−𝑘²₂dn²𝑘^′₂𝑦,𝑘,𝑑₂𝑦,𝑘₁,𝑘₂=𝑘²₁−𝑘²₂𝑘²₁−𝑘²₂dn²𝑘^′₂,𝑦,𝑘 (A.8) with 𝑘^′₂=1−𝑘²₂, 𝑘=(𝑘²₁−𝑘²₂)/(1−𝑘²₂), and 0≤𝑘₂≤𝑘₁≤1. From the double periodic properties of the Jacobi elliptic functions, one can see that the GJEFs are quasidouble periodic 𝑠(𝑘)𝑦+4𝐊𝑘^′₂𝑘=𝑠𝑦+2𝑖𝐊^′𝑘^′₂𝑦,𝑐(𝑘)=±𝑠𝑦+4𝐊𝑘^′₂(𝑘)𝑘=𝑐𝑦+2𝐊+2𝑖𝐊^′𝑘^′₂𝑦,𝑑=±𝑐₁(𝑘)𝑦+2𝐊𝑘^′₂=𝑑₁𝑘𝑦+4𝑖𝐊^′𝑘^′₂=±𝑑₁𝑦,𝑑₂(𝑘)𝑦+2𝐊𝑘^′₂=𝑑₂𝑘𝑦+2𝑖𝐊^′𝑘^′₂=±𝑑₂𝑦, (A.9) where 𝐊(𝑘) is the complete elliptic integral of the first kind and 𝑘′=1−𝑘² [13, 16–18].

Acknowledgment

This work has been supported by the Deanship of Scientific Research at Qassim University.