Jacobi Elliptic Solutions for Nonlinear Differential Difference Equations in Mathematical Physics

Abstract

We put a direct new method to construct the rational Jacobi elliptic solutions for nonlinear differential difference equations which may be called the rational Jacobi elliptic functions method. We use the rational Jacobi elliptic function method to construct many new exact solutions for some nonlinear differential difference equations in mathematical physics via the lattice equation and the discrete nonlinear Schrodinger equation with a saturable nonlinearity. The proposed method is more effective and powerful to obtain the exact solutions for nonlinear differential difference equations.

1. Introduction

It is well known that the investigation of differential difference equations (DDEs) which describe many important phenomena and dynamical processes in many different fields, such as particle vibrations in lattices, currents in electrical networks, pulses in biological chains, and many others, has played an important role in the study of modern physics. Unlike difference equations which are fully discredited, DDEs are semidiscredited with some (or all) of their special variables discredited, while time is usually kept continuous. DDEs also play an important role in numerical simulations of nonlinear partial differential equations (NLPDEs), queuing problems, and discretization in solid state and quantum physics.

Since the work of Fermi et al. in the 1960s [1], DDEs have been the focus of many nonlinear studies. On the other hand, a considerable number of well-known analytic methods are successfully extended to nonlinear DDEs by researchers [2–17]. However, no method obeys the strength and the flexibility for finding all solutions to all types of nonlinear DDEs. Zhang et al. [18] and Aslan [19] used the (𝐺^/𝐺)-expansion method in some physically important nonlinear DDEs. Xu and Li [12] constructed the Jacobi elliptic solutions for nonlinear DDEs. Recently, S. Zhang and H.-Q. Zhang [20] and Gepreel [21] have used the Jacobi elliptic function method for constructing new and more general Jacobi elliptic function solutions of the integral discrete nonlinear Schrödinger equation. The main objective of this paper is to put a direct new method to construct the rational Jacobi elliptic solutions for nonlinear DDEs. We use this method to calculate the exact wave solutions for some nonlinear DDEs in mathematical physics via the lattice equation and the discrete nonlinear Schrodinger equation with a saturable nonlinearity.

2. Description of the Rational Jacobi Elliptic Functions Method

In this section, we would like to outline an algorithm for using the rational Jacobi elliptic functions method to solve nonlinear DDEs. For a given nonlinear DDEsΔ𝑣𝑢_𝑛+𝑝1(𝑥),…,𝑢_{𝑛+𝑝𝑘}(𝑥),𝑢^𝑛+𝑝₁(𝑥),…,𝑢^𝑛+𝑝_𝑘(𝑥),…,𝑢^{(𝑟)𝑛+𝑝}₁(𝑥),…,𝑢^{(𝑟)𝑛+𝑝}_𝑘(𝑥),_𝑛+𝑝1(𝑥),…,𝑣_{𝑛+𝑝𝑘}(𝑥),𝑣^𝑛+𝑝₁(𝑥),…,𝑣^𝑛+𝑝_𝑘(𝑥),…,𝑣^{(𝑟)𝑛+𝑝}₁(𝑥),…,𝑣^{(𝑟)𝑛+𝑝}_𝑘(𝑥),….=0,(2.1) where Δ=(Δ₁,…,Δ_𝑔), 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑚), 𝑛=(𝑛₁,…,𝑛_𝑄), and 𝑔,𝑚,𝑄,𝑝₁,…,𝑝_𝑘 are integers, 𝑢^𝑖(𝑟), 𝑣^𝑖(𝑟) denotes the set of all 𝑟th order derivatives of 𝑢_𝑖,𝑣_𝑖 with respect to 𝑥.

The main steps of the algorithm for the rational Jacobi elliptic functions method to solve nonlinear DDEs are outlined as follows.

Step 1. We seek the traveling wave solutions of the following form: 𝑢_𝑛(𝑥)=𝑈𝜉_𝑛,𝑣_𝑛(𝑥)=𝑉𝜉_𝑛,…,(2.2) where 𝜉_𝑛=^𝑄_𝑖=1𝑑_𝑖𝑛_𝑖+^𝑚_𝑗=1𝑐_𝑗𝑥_𝑗+𝜉₀,(2.3)𝑑_𝑖(𝑖=1,…,𝑄), 𝑐_𝑗, (𝑗=1,…,𝑚), and the phase 𝜉₀ are constants to be determined later. The transformations in (2.2) are reduced (2.1) to the following ordinary differential difference equations Ω𝑉𝑈𝜉𝜉_{𝑛+𝑝1𝑛+𝑝1},…,𝑈𝜉_{𝑛+𝑝𝑘},𝑈^𝜉_𝑛+𝑝1,…,𝑈^𝜉_{𝑛+𝑝𝑘},…,𝑈^(𝑟)𝜉_𝑛+𝑝1,…,𝑈^(𝑟)𝜉_{𝑛+𝑝𝑘},,…,𝑉𝜉_{𝑛+𝑝𝑘},𝑉^𝜉_𝑛+𝑝1,…,𝑉^𝜉_{𝑛+𝑝𝑘},…,𝑉^{(𝑟)𝑛+𝑝}₁𝜉_𝑛+𝑝1,…,𝑉^{(𝑟)𝑛+𝑝}_𝑘𝜉_{𝑛+𝑝𝑘},…=0,(2.4) where Ω=(Ω₁,…,Ω_𝑔). The transformations in (2.2) help in the calculation of the iteration relations between 𝑢_𝑛(𝑥), 𝑢_𝑛−1(𝑥), and 𝑢_𝑛+1(𝑥). For example, Langmuir chains equation 𝑑𝑢_𝑛(𝑡)/𝑑𝑡=𝑢_𝑛(𝑡)(𝑢_𝑛+1(𝑡)−𝑢_𝑛−1(𝑡)) under the wave transformation 𝑢_𝑛(𝑡)=𝑈(𝜉_𝑛), 𝜉_𝑛=𝑑𝑛+𝑐𝑡+𝜉₀ takes the form 𝑐𝑈^(𝜉_𝑛)=𝑈(𝜉_𝑛)(𝑈(𝜉_𝑛+𝑑)−𝑈(𝜉_𝑛−𝑑)).

Step 2. We suppose the rational series expansion solutions of (2.4) in the following form: 𝑈𝜉_𝑛=^𝐾_𝑖=0𝛼_𝑖𝐹^𝜉_𝑛𝐹𝜉_𝑛^𝑖,𝑉𝜉_𝑛=^𝐿_𝑖=0𝛽_𝑖𝐹^𝜉_𝑛𝐹𝜉_𝑛^𝑖,…,(2.5) where 𝛼_𝑖(𝑖=0,1,…,𝐾), and 𝛽_𝑖(𝑖=0,1,…,𝐿) are constants to be determined later, and 𝐹(𝜉_𝑛) satisfies a discrete Jacobi elliptic differential equation 𝐹′²𝜉_𝑛=𝑒₀+𝑒₁𝐹²𝜉_𝑛+𝑒₂𝐹⁴𝜉_𝑛,(2.6) where 𝑒₀, 𝑒₁, and 𝑒₂ are arbitrary constants.

Step 3. Since the general solution of the proposed (2.6) is difficult to obtain and so the iteration relations corresponding to the general exact solutions. So that we discuss the solutions of the proposed discrete Jacobi elliptic differential equation (2.6) at some special cases to 𝑒₀, 𝑒₁ and 𝑒₂ to cover all the Jacobi elliptic functions as follows:

Type 1. if 𝑒₀=1, 𝑒₁=−(1+𝑚²), 𝑒₂=𝑚². In this case (2.6) has the solution 𝐹(𝜉_𝑛)=𝑠𝑛(𝜉_𝑛,𝑚), where 𝑠𝑛(𝜉_𝑛,𝑚) is the Jacobi elliptic sine function, and 𝑚 is the modulus.
The Jacobi elliptic functions satisfy the following properties: 𝑑𝑛𝑠𝑛𝜉_𝑛𝜉_𝑛,𝑚^=𝑐𝑛𝜉_𝑛,𝑚𝑑𝑛𝜉_𝑛,𝑚,𝑐𝑛𝜉_𝑛,𝑚^=−𝑠𝑛𝜉_𝑛,𝑚𝑑𝑛𝜉_𝑛,𝑚,𝑠𝑑𝜉_𝑛,𝑚^=−𝑚²𝑠𝑛𝜉_𝑛,𝑚𝑐𝑛𝜉_𝑛,𝑚,𝑐𝑠𝜉_𝑛,𝑚^=−𝑛𝑠𝜉_𝑛,𝑚𝑑𝑠𝜉_𝑛,𝑚,,𝑚^=−𝑛𝑑𝜉_𝑛,𝑚𝑑𝑠𝜉_𝑛,𝑚,𝑑𝑐𝜉_𝑛,𝑚^=1−𝑚²𝑛𝑐𝜉_𝑛,𝑚𝑠𝑐𝜉_𝑛,𝑚,(2.7) where 𝑐𝑛(𝜉_𝑛,𝑚), and 𝑑𝑛(𝜉_𝑛,𝑚) are the Jacobi elliptic cosine function, and the Jacobi elliptic function of the third kind. The other Jacobi elliptic functions can be generated by 𝑠𝑛(𝜉_𝑛,𝑚), 𝑐𝑛(𝜉_𝑛,𝑚), and 𝑑𝑛(𝜉_𝑛,𝑚) as follows: 𝑐𝑑𝜉_𝑛,𝑚=𝑐𝑛𝜉^𝑛,𝑚𝑑𝑛𝜉_𝑛,𝑚,𝑑𝑐𝜉^𝑛,𝑚=𝑑𝑛𝜉^𝑛,𝑚𝑐𝑛𝜉_𝑛,𝑚,𝑛𝑐𝜉^𝑛,𝑚=1𝑐𝑛𝜉_𝑛,𝑚,𝑛𝑑𝜉^𝑛,𝑚=1𝑐𝑠𝜉_𝑛𝑑𝑛𝜉_𝑛,𝑚,,𝑚=𝑐𝑛𝜉^𝑛,𝑚𝑠𝑛𝜉_𝑛,𝑚,𝑠𝑐𝜉^𝑛,𝑚=𝑠𝑛𝜉^𝑛,𝑚𝑐𝑛𝜉_𝑛,𝑚,𝑠𝑑(𝜉)=𝑠𝑛𝜉^𝑛,𝑚𝑑𝑛𝜉_𝑛,𝑚,𝑑𝑠𝜉^𝑛,𝑚=𝑑𝑛𝜉^𝑛,𝑚𝑠𝑛𝜉₁𝑠𝑛𝜉_𝑛,𝑚,(2.8)±𝜉₂,𝑚=sn𝜉₁,𝑚cn𝜉₂,𝑚dn𝜉₂,𝑚±sn𝜉₂,𝑚cn𝜉₁,𝑚dn𝜉₁,𝑚𝑐𝑛𝜉₁1−𝑚²sn²𝜉₁,𝑚sn²𝜉₂,𝑚,±𝜉₂,𝑚=𝑐𝑛𝜉¹,𝑚𝑐𝑛𝜉₂,𝑚∓𝑠𝑛𝜉₁,𝑚𝑠𝑛𝜉₂,𝑚𝑑𝑛𝜉₁,𝑚𝑑𝑛𝜉₂,𝑚𝑑𝑛𝜉₁1−𝑚²sn²𝜉₁,𝑚sn²𝜉₂,𝑚,±𝜉₂,𝑚=𝑑𝑛𝜉¹,𝑚𝑑𝑛𝜉₂,𝑚∓sn𝜉₁,𝑚𝑠𝑛𝜉₂,𝑚𝑐𝑛𝜉₁,𝑚𝑐𝑛𝜉₂,𝑚1−𝑚²sn²𝜉₁,𝑚sn²𝜉₂,𝑚.(2.9) In this case from using the properties of Jacobi elliptic functions, the series expansion solutions (2.5) take the following form 𝑈𝜉_𝑛=^𝐾_𝑖=0𝛼_𝑖𝑐𝑛𝜉^𝑛,𝑚𝑑𝑛𝜉_𝑛,𝑚𝑠𝑛𝜉_𝑛,𝑚^𝑖𝑉𝜉_𝑛,=^𝐿_𝑖=0𝛽_𝑖𝑐𝑛𝜉^𝑛,𝑚𝑑𝑛𝜉_𝑛,𝑚𝑠𝑛𝜉_𝑛,𝑚^𝑖,….(2.10) Further by using the properties of Jacobi elliptic functions, the iterative relations can be written in the following form: 𝑈𝜉_𝑛±𝑝=^𝐾_𝑖=0𝛼_𝑖𝐹^𝜉_𝑛±𝑝𝐹𝜉_𝑛±𝑝^𝑖𝑉𝜉_𝑛±𝑝,=^𝐿_𝑖=0𝛽_𝑖𝐹^𝜉_𝑛±𝑝𝐹𝜉_𝑛±𝑝^𝑖,…,(2.11) where 𝐹^𝜉_𝑛±𝑑𝐹𝜉_𝑛±𝑑=𝑀1¹∓2±𝑚𝑐𝑛₂(𝑑,𝑚)𝑐𝑛𝜉_𝑛𝑑𝑛𝜉_𝑛,𝑚𝑑𝑛(𝑑,𝑚)±𝑚²𝑠𝑛(𝑑,𝑚)𝑠𝑛𝜉_𝑛,𝑚+𝑠𝑛(𝑑𝑠,𝑛𝑚(𝑑),𝑠𝑚𝑛)𝜉𝑠_𝑛𝑛³𝜉_𝑛,𝑚∓2𝑚²𝑠𝑛³(𝑑,𝑚)𝑠𝑛𝜉_𝑛,𝑚±𝑚²𝑠𝑛³(𝑑,𝑚)𝑠𝑛³𝜉_𝑛,𝑚∓𝑚₂,𝑚±𝑚⁴𝑠𝑛³(𝑑,𝑚)𝑠𝑛³𝜉_𝑛,𝑚𝑀𝑠𝑛²(𝑑,𝑚)𝑠𝑛²𝜉_𝑛,𝑚𝑑𝑛𝜉_𝑛,𝑚𝑑𝑛(𝑑,𝑚)𝑐𝑛(𝑑,𝑚)𝑐𝑛𝜉_𝑛,𝑚,(2.12)₁=×−𝑠𝑐𝑛𝑛₂(𝜙,𝑚)𝑑𝑛(𝜙,𝑚)𝑠𝑛𝜉_𝑛,𝑚∓𝑠𝑛(𝜙,𝑚)𝑑𝑛𝜉_𝑛,𝑚𝑐𝑛𝜉_𝑛,𝑚+𝑚²𝑠𝑛³𝜉_𝑛,𝑚(𝜙,𝑚)𝑐𝑛(𝜙,𝑚)𝑑𝑛(𝜙,𝑚)±𝑚²𝑠𝑛²𝜉_𝑛,𝑚𝑠𝑛³(𝜙,𝑚)𝑐𝑛𝜉_𝑛,𝑚𝑑𝑛𝜉_𝑛,𝑚,(2.13)𝑑=𝑝_𝑠1𝑑₁+𝑝_𝑠2𝑑₂+⋯+𝑝_𝑠𝑄𝑑_𝑄, 𝑝_𝑠𝑗 is the 𝑗th component of shift vector 𝑝_𝑠.

Type 2. if 𝑒₀=1−𝑚², 𝑒₁=2𝑚²−1, 𝑒₂=−𝑚². In this case, (2.6) has the solution 𝐹(𝜉_𝑛)=𝑐𝑛(𝜉_𝑛,𝑚). From using the properties of Jacobi elliptic functions, the series expansion solutions (2.5) take the following form𝑈𝜉_𝑛=^𝐾_𝑖=0𝛼_𝑖−𝑠𝑛𝜉^𝑛,𝑚𝑑𝑛𝜉_𝑛,𝑚𝑐𝑛𝜉_𝑛,𝑚^𝑖𝑉𝜉_𝑛,=^𝐿_𝑖=0𝛽_𝑖−𝑠𝑛𝜉^𝑛,𝑚𝑑𝑛𝜉_𝑛,𝑚𝑐𝑛𝜉_𝑛,𝑚^𝑖,….(2.14)

Type 3. if 𝑒₀=𝑚²−1, 𝑒₁=2−𝑚², 𝑒₂=−1. In this case, (2.6) has the solution 𝐹(𝜉_𝑛)=𝑑𝑛(𝜉_𝑛,𝑚). From using the properties of Jacobi elliptic functions the series expansion solutions (2.5) take the following form 𝑈𝜉_𝑛=^𝐾_𝑖=0𝛼_𝑖−𝑚²𝑠𝑛𝜉_𝑛,𝑚𝑐𝑛𝜉_𝑛,𝑚𝑑𝑛𝜉_𝑛,𝑚^𝑖𝑉𝜉_𝑛,=^𝐿_𝑖=0𝛽_𝑖−𝑚²𝑠𝑛𝜉_𝑛,𝑚𝑐𝑛𝜉_𝑛,𝑚𝑑𝑛𝜉_𝑛,𝑚^𝑖,….(2.15)

Type 4. if 𝑒₀=1−𝑚², 𝑒₁=2−𝑚², 𝑒₂=1. In this case, (2.6) has the solution 𝐹(𝜉_𝑛)=𝑐𝑠(𝜉_𝑛,𝑚), then the series expansion solutions (2.5) take the following form 𝑈𝜉_𝑛=^𝐾_𝑖=0𝛼_𝑖−𝑛𝑠𝜉^𝑛,𝑚𝑑𝑠𝜉_𝑛,𝑚𝑐𝑠𝜉_𝑛,𝑚^𝑖,𝑉𝜉_𝑛=^𝐿_𝑖=0𝛽_𝑖−𝑛𝑠𝜉^𝑛,𝑚𝑑𝑠𝜉_𝑛,𝑚𝑐𝑠𝜉_𝑛,𝑚^𝑖,….(2.16) Equation (2.16) can be written in the following form: 𝑈𝜉_𝑛=^𝐾_𝑖=0𝛼_𝑖−𝑑𝑛𝜉^𝑛,𝑚𝑠𝑛𝜉_𝑛,𝑚𝑐𝑛𝜉_𝑛,𝑚^𝑖,𝑉𝜉_𝑛=^𝐿_𝑖=0𝛽_𝑖−𝑑𝑛𝜉^𝑛,𝑚𝑠𝑛𝜉_𝑛,𝑚𝑐𝑛𝜉_𝑛,𝑚^𝑖,….(2.17)

Type 5. if 𝑒₀=1, 𝑒₁=2𝑚²−1, and 𝑒₂=𝑚²(𝑚²−1). In this case, (2.6) has the solution 𝐹(𝜉_𝑛)=𝑠𝑑(𝜉_𝑛,𝑚), then the series expansion solutions (2.5) take the following form 𝑈𝜉_𝑛=^𝐾_𝑖=0𝛼_𝑖𝑛𝑑𝜉^𝑛,𝑚𝑐𝑑𝜉_𝑛,𝑚𝑠𝑑𝜉_𝑛,𝑚^𝑖,𝑉𝜉_𝑛=^𝐿_𝑖=0𝛽_𝑖𝑛𝑑𝜉^𝑛,𝑚𝑐𝑑𝜉_𝑛,𝑚𝑠𝑑𝜉_𝑛,𝑚^𝑖,…(2.18) Equation (2.18) can be written in the following form: 𝑈𝜉_𝑛=^𝐾_𝑖=0𝛼_𝑖𝑐𝑛𝜉^𝑛,𝑚𝑠𝑛𝜉_𝑛,𝑚𝑑𝑛𝜉_𝑛,𝑚^𝑖,𝑉𝜉_𝑛=^𝐿_𝑖=0𝛽_𝑖𝑐𝑛𝜉^𝑛,𝑚𝑠𝑛𝜉_𝑛,𝑚𝑑𝑛𝜉_𝑛,𝑚^𝑖,…(2.19)

Type 6. if 𝑒₀=𝑚², 𝑒₁=−(𝑚²+1), and 𝑒₂=1. In this case, (2.6) has the solution 𝐹(𝜉_𝑛)=𝑑𝑐(𝜉_𝑛,𝑚), then the series expansion solutions (2.5) take the following form 𝑈𝜉_𝑛=^𝐾_𝑖=0𝛼_𝑖1−𝑚²𝑛𝑐𝜉_𝑛,𝑚𝑠𝑐𝜉_𝑛,𝑚𝑑𝑐𝜉_𝑛,𝑚^𝑖𝑉𝜉_𝑛,=^𝐿_𝑖=0𝛽_𝑖1−𝑚²𝑛𝑐𝜉_𝑛,𝑚𝑠𝑐𝜉_𝑛,𝑚𝑑𝑐𝜉_𝑛,𝑚^𝑖,….(2.20) Equation (2.20) can be written in the following form: 𝑈𝜉_𝑛=^𝐾_𝑖=0𝛼_𝑖1−𝑚²𝑠𝑛𝜉_𝑛,𝑚𝑐𝑛𝜉_𝑛,𝑚𝑑𝑛𝜉_𝑛,𝑚^𝑖,𝑉_𝑛𝜉_𝑛=^𝐿_𝑖=0𝛽_𝑖1−𝑚²𝑠𝑛𝜉_𝑛,𝑚𝑐𝑛𝜉_𝑛,𝑚𝑑𝑛𝜉_𝑛,𝑚^𝑖,….(2.21) From the properties of the Jacobi elliptic functions, we can deduce the iterative relation to the above kind of solutions from Types 2−6 as we show in Type 1.
Equations (2.10)–(2.21) lead to getting all formulas of solutions from Types 1–6 as different. Consequently, we will discuss all solutions from Types 1–6.

Step 4. Determine the degree 𝐾,𝐿,… of (2.5) by balancing the nonlinear term(s) and the highest-order derivatives of 𝑈(𝜉_𝑛), 𝑉(𝜉_𝑛),…   in (2.4). It should be noted that the leading terms 𝑈(𝜉_𝑛±𝑝), 𝑉(𝜉_𝑛±𝑝), …, 𝑝≠0will not affect the balance because we are interested in balancing the terms of 𝐹^(𝜉_𝑛)/𝐹(𝜉_𝑛).

Step 5. Substituting 𝑈(𝜉_𝑛), 𝑉(𝜉_𝑛), and … in each type form 1–6 and the given values of 𝐾, 𝐿, and … into (2.4). Cleaning the denominator and collecting all terms with the same degree of 𝑠𝑛(𝜉_𝑛,𝑚), 𝑑𝑛(𝜉_𝑛,𝑚), and 𝑐𝑛(𝜉_𝑛,𝑚) together, the left hand side of (2.4) is converted into a polynomial in 𝑠𝑛(𝜉_𝑛,𝑚), 𝑑𝑛(𝜉_𝑛,𝑚), and 𝑐𝑛(𝜉_𝑛,𝑚). Setting each coefficient of this polynomial to zero, we derive a set of algebraic equations for 𝛼_𝑖, 𝛽_𝑖, 𝑑_𝑖, and 𝑐_𝑖.

Step 6. Solving the over determined system of nonlinear algebraic equations by using Maple or Mathematica. We end up with explicit expressions for 𝛼_𝑖, 𝛽_𝑖, 𝑑_𝑖, and 𝑐_𝑗.

Step 7. Substituting 𝛼_𝑖, 𝛽_𝑖, 𝑑_𝑖, and 𝑐_𝑖 into 𝑈(𝜉_𝑛), 𝑉(𝜉_𝑛), and … in the corresponding type from 1–6, we can finally obtain the exact solutions for (2.1).

3. Applications

In this section, we apply the proposed rational Jacobi elliptic functions method to construct the traveling wave solutions for some nonlinear DDEs via the lattice equation and the discrete nonlinear Schrodinger equation with a saturable nonlinearity which are very important in the mathematical physics and have been paid attention to by many researchers.

3.1. Example  1. The Lattice Equation

In this section, we study the lattice equation which takes the following form [22–25]𝑑𝑢_𝑛(𝑡)𝑑𝑡=𝛼+𝛽𝑢^𝑛+𝛾𝑢^2𝑛𝑢_𝑛−1−𝑢_𝑛+1,(3.1) where 𝛼, 𝛽, and 𝛾 are nonzero constants. The equation contains hybrid lattice equation, mKdV lattice equation, modified Volterra lattice equation, and Langmuir chain equation:(i)(1+1) dimensional Hybrid lattice equation [25]: 𝑑𝑢_𝑛(𝑡)𝑑𝑡=1+𝛽𝑢^𝑛+𝛾𝑢^2𝑛𝑢_𝑛−1−𝑢_𝑛+1;(3.2)(ii)mKdV lattice equation [25]: 𝑑𝑢_𝑛(𝑡)𝑑𝑡=𝛼−𝑢^2𝑛𝑢_𝑛−1−𝑢_𝑛+1;(3.3)(iii)modified Volterra equation [24]: 𝑑𝑢_𝑛(𝑡)𝑑𝑡=𝑢^2𝑛𝑢_𝑛−1−𝑢_𝑛+1;(3.4)(iv)Langmuir chain equation [25]: 𝑑𝑢_𝑛(𝑡)𝑑𝑡=𝑢^𝑛𝑢_𝑛+1−𝑢_𝑛−1.(3.5)

According to the above steps, to seek traveling wave solutions of (3.1), we construct the transformation𝑢_𝑛(𝑡)=𝑈𝜉_𝑛,𝜉_𝑛=𝑑𝑛+𝑐₁𝑡+𝜉₀,(3.6) where 𝑑, 𝑐₁, and 𝜉₀ are constants. The transformation in (3.6) permits us to convert (3.1) into the following form:𝑐₁𝑈^𝜉_𝑛=𝛼+𝛽𝑈𝜉_𝑛+𝛾𝑈²𝜉_𝑛𝑈𝜉_𝑛−𝑑−𝑈𝜉_𝑛+𝑑,(3.7) where ^=𝑑/𝑑𝜉_𝑛. Considering the homogeneous balance between the highest-order derivative and the nonlinear term in (3.7), we get 𝐾=1. Thus, the solution of (3.7) has the following form:𝑈𝜉_𝑛=𝛼₁𝐹^𝜉_𝑛𝐹𝜉_𝑛+𝛼⁰,(3.8) where 𝛼₀, and 𝛼₁ are constants to be determined later, and 𝐹(𝜉_𝑛) satisfies a discrete Jacobi elliptic ordinary differential (2.6). When, we discuss the solutions of the Jacobi elliptic differential difference (2.6), we get the following types.

Type 1. If 𝑒₀=1, 𝑒₁=−(1+𝑚²), and 𝑒₂=𝑚². In this case, the series expansion solution of (3.7) has the form: 𝑈𝜉_𝑛=𝛼₀+𝛼¹𝑐𝑛𝜉_𝑛,𝑚𝑑𝑛𝜉_𝑛,𝑚𝑠𝑛𝜉_𝑛,𝑚.(3.9) With help of Maple, we substitute (3.9) and (2.12) into (3.7), cleaning the denominator and collecting all terms with the same degree of 𝑠𝑛(𝜉_𝑛,𝑚), 𝑑𝑛(𝜉_𝑛,𝑚), and 𝑐𝑛(𝜉_𝑛,𝑚) together, the left hand side of (3.7) is converted into polynomial in 𝑠𝑛(𝜉_𝑛,𝑚), 𝑑𝑛(𝜉_𝑛,𝑚), and 𝑐𝑛(𝜉_𝑛,𝑚). Setting each coefficient of this polynomial to zero, we derive a set of algebraic equations for 𝛼₀, 𝛼₁, 𝑑, and 𝑐₁. Solving the set of algebraic equations by using Maple or Mathematica, we have 𝛼₀=−𝛽2𝛾,𝛼¹=√𝛽²−4𝛼𝛾𝑠𝑛(𝑑,𝑚)2𝛾𝑐𝑛(𝑑,𝑚)𝑑𝑛(𝑑,𝑚),𝑐¹=−4𝛼𝛾−𝛽²𝑠𝑛(𝑑,𝑚)2𝛾𝑐𝑛(𝑑,𝑚)𝑑𝑛(𝑑,𝑚).(3.10) From (3.9) and (3.10), the solution of (3.7) takes the following form: 𝑈𝜉_𝑛=√𝛽²−4𝛼𝛾𝑠𝑛(𝑑,𝑚)𝑐𝑛𝜉_𝑛,𝑚𝑑𝑛𝜉_𝑛,𝑚2𝛾𝑐𝑛(𝑑,𝑚)𝑑𝑛(𝑑,𝑚)𝑠𝑛𝜉_𝑛,𝑚−2𝛽𝛾,(3.11) where 𝜉_𝑛=𝑑𝑛−((4𝛼𝛾−𝛽²)𝑠𝑛(𝑑,𝑚)/[2𝛾𝑐𝑛(𝑑,𝑚)𝑑𝑛(𝑑,𝑚)])𝑡+𝜉₀.

Type 2. If 𝑒₀=1−𝑚², 𝑒₁=2𝑚²−1, and 𝑒₂=−𝑚². In this case, the series expansion solution of (3.7) has the form: 𝑈𝜉_𝑛=𝛼₀−𝛼₁𝑠𝑛𝜉_𝑛𝑑𝑛𝜉_𝑛𝑐𝑛𝜉_𝑛,𝑚.(3.12) With the help of Maple, we substitute (3.12) into (3.7), cleaning the denominator and collecting all terms with the same degree of 𝑠𝑛(𝜉_𝑛,𝑚),𝑑𝑛(𝜉_𝑛,𝑚), and 𝑐𝑛(𝜉_𝑛,𝑚) together, the left hand side of (3.7) is converted into polynomial in 𝑠𝑛(𝜉_𝑛,𝑚), 𝑑𝑛(𝜉_𝑛,𝑚), and 𝑐𝑛(𝜉_𝑛,𝑚). Setting each coefficient of this polynomial to zero, we derive a set of algebraic equations for 𝛼₀, 𝛼₁, 𝑑, and 𝑐₁. Solving the set of algebraic equations by using Maple or Mathematica, we get 𝛼₀=−𝛽2𝛾,𝛼¹=√𝛽²−4𝛼𝛾𝑠𝑛(𝑑,𝑚)𝑑𝑛(𝑑,𝑚)2𝛾𝑐𝑛(𝑑,𝑚),𝑐¹=−4𝛼𝛾−𝛽²𝑑𝑛(𝑑,𝑚)𝑠𝑛(𝑑,𝑚)2𝛾𝑐𝑛(𝑑,𝑚).(3.13) In this case the solution of (3.7) takes the following form: 𝑈𝜉_𝑛=−𝛽2𝛾−√𝛽²−4𝛼𝛾𝑠𝑛(𝑑,𝑚)𝑑𝑛(𝑑,𝑚)𝑠𝑛𝜉_𝑛,𝑚𝑑𝑛𝜉_𝑛,𝑚2𝛾𝑐𝑛(𝑑,𝑚)𝑐𝑛𝜉_𝑛,𝑚,(3.14) where 𝜉_𝑛=𝑑𝑛−((4𝛼𝛾−𝛽²)𝑑𝑛(𝑑,𝑚)𝑠𝑛(𝑑,𝑚)/[2𝛾𝑐𝑛(𝑑,𝑚)])𝑡+𝜉₀.

Type 3. if 𝑒₀=𝑚²−1, 𝑒₁=2−𝑚², and 𝑒₂=−1. In this case, the series expansion solution of (3.7) has the form: 𝑈𝜉_𝑛=𝛼₀−𝑚²𝛼₁𝑠𝑛𝜉_𝑛𝑐𝑛𝜉_𝑛𝑑𝑛𝜉_𝑛.(3.15) Consequently, by using Maple or Mathematica, we obtain the following results: 𝛼₀=−𝛽2𝛾,𝛼¹=√𝛽²−4𝛼𝛾𝑠𝑛(𝑑,𝑚)𝑐𝑛(𝑑,𝑚)2𝛾𝑑𝑛(𝑑,𝑚),𝑐¹=−4𝛼𝛾−𝛽²𝑐𝑛(𝑑,𝑚)𝑠𝑛(𝑑,𝑚)2𝛾𝑑𝑛(𝑑,𝑚).(3.16) In this case, the solution takes the following form: 𝑈𝜉_𝑛=−𝛽2𝛾−√𝛽²−4𝛼𝛾𝑚²𝑠𝑛(𝑑,𝑚)𝑐𝑛(𝑑,𝑚)𝑠𝑛𝜉_𝑛,𝑚𝑐𝑛𝜉_𝑛,𝑚2𝛾𝑑𝑛(𝑑,𝑚)𝑑𝑛𝜉_𝑛,𝑚,(3.17) where 𝜉_𝑛=𝑑𝑛−((4𝛼𝛾−𝛽²)𝑐𝑛(𝑑,𝑚)𝑠𝑛(𝑑,𝑚)/[2𝛾𝑑𝑛(𝑑,𝑚)])𝑡+𝜉₀.

Type 4. if 𝑒₀=1−𝑚², 𝑒₁=2−𝑚², and 𝑒₂=1. In this case, the series expansion solution of (3.7) has the form: 𝑈_𝑛𝜉_𝑛=𝛼₀−𝛼¹𝑛𝑠𝜉_𝑛𝑑𝑠𝜉_𝑛𝑐𝑠𝜉_𝑛.(3.18) Consequently, using the Maple or Mathematica we get the following results: 𝛼₀=−𝛽2𝛾,𝛼¹=√𝛽²−4𝛼𝛾𝑠𝑛(𝑑,𝑚)𝑐𝑛(𝑑,𝑚)2𝛾𝑑𝑛(𝑑,𝑚),𝑐¹=−4𝛼𝛾−𝛽²𝑐𝑛(𝑑,𝑚)𝑠𝑛(𝑑,𝑚)2𝛾𝑑𝑛(𝑑,𝑚).(3.19) In this case, the solution of (3.7) takes the following form: 𝑈_𝑛𝜉_𝑛=−𝛽2𝛾−√𝛽²−4𝛼𝛾𝑠𝑛(𝑑,𝑚)𝑐𝑛(𝑑,𝑚)𝑛𝑠𝜉_𝑛,𝑚𝑑𝑠𝜉_𝑛,𝑚2𝛾𝑑𝑛(𝑑,𝑚)𝑐𝑠𝜉_𝑛,𝑚,(3.20) where 𝜉_𝑛=𝑑𝑛−((4𝛼𝛾−𝛽²)𝑐𝑛(𝑑,𝑚)𝑠𝑛(𝑑,𝑚)/[2𝛾𝑑𝑛(𝑑,𝑚)])𝑡+𝜉₀.

Type 5. if 𝑒₀=1, 𝑒₁=2𝑚²−1, and 𝑒₂=𝑚²(𝑚²−1). In this case, the series expansion solution of (3.7) has the form: 𝑈𝜉_𝑛=𝛼₀+𝛼¹𝑛𝑑𝜉_𝑛𝑐𝑑𝜉_𝑛𝑠𝑑𝜉_𝑛.(3.21) Consequently, by using Maple or Mathematica, we get the following results: 𝛼₀=−𝛽2𝛾,𝛼¹=√𝛽²−4𝛼𝛾𝑠𝑛(𝑑,𝑚)𝑑𝑛(𝑑,𝑚)2𝛾𝑐𝑛(𝑑,𝑚),𝑐¹=𝛽²−4𝛼𝛾𝑠𝑛(𝑑,𝑚)𝑑𝑛(𝑑,𝑚)2𝛾𝑐𝑛(𝑑,𝑚).(3.22) In this case, the solution takes of (3.7) the following form: 𝑈𝜉_𝑛=−𝛽2𝛾+√𝛽²−4𝛼𝛾𝑠𝑛(𝑑,𝑚)𝑑𝑛(𝑑,𝑚)𝑛𝑑𝜉_𝑛,𝑚𝑐𝑑𝜉_𝑛,𝑚2𝛾𝑐𝑛(𝑑,𝑚)𝑠𝑑𝜉_𝑛,𝑚,(3.23) where 𝜉_𝑛=𝑑𝑛+((𝛽²−4𝛼𝛾)𝑠𝑛(𝑑,𝑚)𝑑𝑛(𝑑,𝑚)/[2𝛾𝑐𝑛(𝑑,𝑚)])𝑡+𝜉₀.

Type 6. if 𝑒₀=𝑚², 𝑒₁=−(𝑚²+1), and 𝑒₂=1. In this case, the series expansion solution of (3.7) has the form: 𝑈𝜉_𝑛=𝛼₀+1−𝑚²𝛼₁𝑛𝑐𝜉_𝑛𝑠𝑐𝜉_𝑛𝑑𝑐𝜉_𝑛.(3.24) Consequently, by using Maple or Mathematica, we get the following results: 𝛼₀=−𝛽2𝛾,𝛼¹=√𝛽²−4𝛼𝛾𝑠𝑛(𝑑,𝑚)2𝛾𝑐𝑛(𝑑,𝑚)𝑑𝑛(𝑑,𝑚),𝑐¹=𝛽²−4𝛼𝛾𝑠𝑛(𝑑,𝑚)2𝛾𝑐𝑛(𝑑,𝑚)𝑑𝑛(𝑑,𝑚).(3.25) In this case, the solution of (3.7) takes the following form: 𝑈𝜉_𝑛=−𝛽2𝛾+√𝛽²−4𝛼𝛾1−𝑚²𝑠𝑛(𝑑,𝑚)𝑛𝑐𝜉_𝑛,𝑚𝑠𝑐𝜉_𝑛,𝑚2𝛾𝑑𝑛(𝑑,𝑚)𝑐𝑛(𝑑,𝑚)𝑑𝑐𝜉_𝑛,𝑚,(3.26) where 𝜉_𝑛=𝑑𝑛+((𝛽²−4𝛼𝛾)𝑠𝑛(𝑑,𝑚)/[2𝛾𝑐𝑛(𝑑,𝑚)𝑑𝑛(𝑑,𝑚)])𝑡+𝜉₀.

3.2. Example  2. The Discrete Nonlinear Schrodinger Equation

The discrete nonlinear Schrodinger equation (DNSE) is one of the most fundamental nonlinear lattice models [8]. It arises in nonlinear optics as a model of infinite wave guide arrays [26] and has been recently implemented to describe Bose-Einstein condensates in optical lattices. The class of DNSE model with saturable nonlinearity is also of particular interest in their own right, due to a feature first unveiled in [27]. In this section, we study the DNSE with a saturable nonlinearity [28, 29] having the form𝑖𝜕𝜓^𝑛𝜕𝑡+𝜓^𝑛+1+𝜓_𝑛−1−2𝜓_𝑛+𝜈||𝜓^𝑛||²1+𝜇||𝜓_𝑛||²𝜓_𝑛=0,(3.27) which describes optical pulse propagations in various doped fibers, 𝜓_𝑛 is a complex valued wave function at sites 𝑛 while 𝜈 and 𝜇. We make the transformation𝜓_𝑛=𝜙𝜉_𝑛𝑒^{−𝑖(𝜎𝑡+𝜌)},𝜉_𝑛=𝛼𝑛+𝛽,(3.28) where 𝜎,𝜌, 𝛼, and 𝛽 are arbitrary real constants. The transformation (3.28) permits us converting (3.27) into the following nonlinear difference equation(𝜎−2)𝜙𝜉_𝑛+𝜙𝜉_𝑛+1+𝜙𝜉_𝑛−1+𝜈𝜙³𝜉_𝑛1+𝜇𝜙²𝜉_𝑛=0.(3.29) We assume that (3.29) has a solution of the form:𝜙𝜉_𝑛=𝑈𝜉_𝑛=𝛼₁𝐹^𝜉_𝑛𝐹𝜉_𝑛+𝛼⁰,(3.30) where 𝛼₁, and 𝛼₀ are constants to be determined later and 𝐹(𝜉_𝑛) satisfying a discrete Jacobi elliptic differential equation (2.6). When, we discuss the solutions of (2.6), we have the following types.

Type 1. If 𝑒₀=1, 𝑒₁=−(1+𝑚²), and 𝑒₂=𝑚². In this case, the series expansion solution of (3.29) has the form: 𝑈𝜉_𝑛=𝛼₁𝑐𝑛𝜉_𝑛,𝑚𝑑𝑛𝜉_𝑛,𝑚𝑠𝑛𝜉_𝑛,𝑚+𝛼⁰.(3.31) With the help of Maple, we substitute (3.31) and (2.12) into (3.29), cleaning the denominator and collecting all terms with the same order of 𝑐𝑛(𝜉_𝑛,𝑚), 𝑑𝑛(𝜉_𝑛,𝑚), and 𝑠𝑛(𝜉_𝑛,𝑚) together, the left hand side of (3.29) is converted into polynomial in 𝑐𝑛(𝜉_𝑛,𝑚), 𝑑𝑛(𝜉_𝑛,𝑚), and 𝑠𝑛(𝜉_𝑛,𝑚). Setting each coefficient of this polynomial to zero, we derive a set of algebraic equations for 𝛼₀, 𝛼₁, 𝜎, 𝜌, 𝛼, and 𝛽. Solving the set of algebraic equations by using Maple or Mathematica, we obtain 𝛼₀=0,𝛼₁=𝑠𝑛(𝛼,𝑚)√−𝜇𝑐𝑛(𝛼,𝑚)𝑑𝑛(𝛼,𝑚),𝜈=−2𝜇𝑚²𝑠𝑛⁴(𝛼,𝑚)−1𝜎=−2𝑠𝑛₂𝑐𝑛²(𝛼,𝑚)𝑑𝑛²(𝛼,𝑚),(𝛼,𝑚)𝑚²𝑐𝑛²(𝛼,𝑚)+𝑑𝑛²(𝛼,𝑚)𝑐𝑛²(𝛼,𝑚)𝑑𝑛²(𝛼,𝑚),𝜇<0.(3.32) In this case, the solution of (3.27) takes the following form: 𝜓_𝑛=𝑠𝑛(𝛼,𝑚)𝑐𝑛𝜉^𝑛,𝑚𝑑𝑛𝜉_𝑛,𝑚√−𝜇𝑐𝑛(𝛼,𝑚)𝑑𝑛(𝛼,𝑚)𝑠𝑛𝜉_𝑛,𝑚Exp−𝑖−2𝑡𝑠𝑛²(𝛼,𝑚)𝑚²𝑐𝑛²(𝛼,𝑚)+𝑑𝑛²(𝛼,𝑚)𝑐𝑛²(𝛼,𝑚)𝑑𝑛²(𝛼,𝑚)+𝜌,(3.33) where 𝜉_𝑛=𝛼𝑛+𝛽.

Type 2. If 𝑒₀=1−𝑚², 𝑒₁=2𝑚²−1, and 𝑒₂=−𝑚². In this case the solution of (3.29) has the form: 𝑈𝜉_𝑛=𝛼₀−𝛼₁𝑠𝑛𝜉_𝑛,𝑚𝑑𝑛𝜉_𝑛,𝑚𝑐𝑛𝜉_𝑛,𝑚,(3.34) Consequently, by using Maple or Mathematica, we get the following results: 𝛼₀=0,𝛼₁=𝑠𝑛(𝛼,𝑚)𝑑𝑛(𝛼,𝑚)√−𝜇𝑐𝑛(𝛼,𝑚),𝜈=2𝜇𝑚²𝑠𝑛⁴(𝛼,𝑚)−2𝑚²𝑠𝑛²(𝛼,𝑚)+1𝜎=−2𝑠𝑛₂𝑐𝑛²(𝛼,𝑚),(𝛼,𝑚)𝑚²𝑠𝑛²(𝛼,𝑚)+1−2𝑚²𝑐𝑛²(𝛼,𝑚),𝜇<0.(3.35) In this case, the solution takes the following form: 𝜓_𝑛=−𝑠𝑛(𝛼,𝑚)𝑑𝑛(𝛼,𝑚)𝑠𝑛𝜉^𝑛,𝑚𝑑𝑛𝜉_𝑛,𝑚×Exp−√𝑖−−𝜇2𝑐𝑡𝑛𝑠(𝑛𝛼₂,𝑚)𝑐𝑛𝜉_𝑛,𝑚(𝛼,𝑚)𝑚²𝑠𝑛²(𝛼,𝑚)+1−2𝑚²𝑐𝑛²(𝛼,𝑚)+𝜌.(3.36)

Type 3. if 𝑒₀=𝑚²−1, 𝑒₁=2−𝑚², and 𝑒₂=−1. In this case, the series expansion solution of (3.29) has the form: 𝑈𝜉_𝑛=𝛼₀−𝑚²𝛼₁𝑠𝑛𝜉_𝑛𝑐𝑛𝜉_𝑛𝑑𝑛𝜉_𝑛,(3.37) Consequently, by using Maple or Mathematica, we get the following results: 𝛼₀=0,𝛼₁=𝑠𝑛(𝛼,𝑚)𝑐𝑛(𝛼,𝑚)√−𝜇𝑑𝑛(𝛼,𝑚),𝜈=2𝜇𝑚²𝑠𝑛⁴(𝛼,𝑚)−2𝑠𝑛²(𝛼,𝑚)+1𝜎=−2𝑠𝑛₂𝑑𝑛²(𝛼,𝑚),(𝛼,𝑚)𝑚²𝑠𝑛²(𝛼,𝑚)−2+𝑚²𝑑𝑛²(𝛼,𝑚),𝜇<0.(3.38) In this case, the solution takes the following form: 𝜓_𝑛=−𝑚²𝑠𝑛(𝛼,𝑚)𝑐𝑛(𝛼,𝑚)𝑠𝑛𝜉_𝑛,𝑚𝑐𝑛𝜉_𝑛,𝑚×Exp−𝑖√−−𝜇2𝑑𝑡𝑠𝑛𝑛(₂𝛼,𝑚)𝑑𝑛𝜉_𝑛,𝑚(𝛼,𝑚)𝑚²𝑠𝑛²(𝛼,𝑚)−2+𝑚²𝑑𝑛²(𝛼,𝑚)+𝜌.(3.39)

Type 4. if 𝑒₀=1−𝑚², 𝑒₁=2−𝑚², and 𝑒₂=1. In this case, the series expansion solution of (3.29) has the form: 𝑈𝜉_𝑛=𝛼₀−𝛼¹𝑛𝑠𝜉_𝑛𝑑𝑠𝜉_𝑛𝑐𝑠𝜉_𝑛.(3.40) After some calculation, the solution of (3.27) takes the following form: 𝜓_𝑛=−𝑠𝑛(𝛼,𝑚)𝑐𝑛(𝛼,𝑚)𝑛𝑠𝜉^𝑛,𝑚𝑑𝑠𝜉_𝑛,𝑚×Exp−√𝑖−𝜇−2𝑑𝑡𝑛𝑠(𝑛𝛼₂,𝑚)𝑐𝑛𝜉_𝑛,𝑚(𝛼,𝑚)𝑚²𝑠𝑛²(𝛼,𝑚)−2+𝑚²𝑑𝑛²(𝛼,𝑚)+𝜌,(3.41) where 𝜈=2𝜇(𝑚²𝑠𝑛⁴(𝛼,𝑚)−2𝑠𝑛²(𝛼,𝑚)+1)/𝑑𝑛²(𝛼,𝑚).

Type 5. if 𝑒₀=1, 𝑒₁=2𝑚²−1, and 𝑒₂=𝑚²(𝑚²−1). In this case, the series expansion solution of (3.29) has the form: 𝑈𝜉_𝑛=𝛼₀+𝛼¹𝑛𝑑𝜉_𝑛𝑐𝑑𝜉_𝑛𝑠𝑑𝜉_𝑛.(3.42) After some calculation, the solution of (3.27) takes the following form: 𝜓_𝑛=−𝑠𝑛(𝛼,𝑚)𝑑𝑛(𝛼,𝑚)𝑐𝑑𝜉^𝑛,𝑚𝑛𝑑𝜉_𝑛,𝑚×Exp−√𝑖−−𝜇2𝑐𝑡𝑛𝑠(𝑛𝛼₂,𝑚)𝑠𝑑𝜉_𝑛,𝑚(𝛼,𝑚)𝑚²𝑠𝑛²(𝛼,𝑚)+1−2𝑚²𝑐𝑛²(𝛼,𝑚)+𝜌,(3.43) where 𝜈=2𝜇(𝑚²𝑠𝑛⁴(𝛼,𝑚)−2𝑠𝑛²(𝛼,𝑚)+1)/𝑐𝑛²(𝛼,𝑚).

Type 6. if 𝑒₀=𝑚², 𝑒₁=−(𝑚²+1), and 𝑒₂=1. In this case, the series expansion solution of (3.29) has the form: 𝑈𝜉_𝑛=𝛼₀+1−𝑚²𝛼₁𝑛𝑐𝜉_𝑛𝑠𝑐𝜉_𝑛𝑑𝑐𝜉_𝑛.(3.44) After some calculation, the solution of (3.27) takes the following form: 𝜓_𝑛=1−𝑚²𝑠𝑛(𝛼,𝑚)𝑛𝑐𝜉_𝑛,𝑚𝑠𝑐𝜉_𝑛,𝑚×Ex√p−𝜇−𝑐𝑖𝑛(𝛼−,2𝑚𝑡𝑠)𝑛𝑑₂𝑛(𝛼,𝑚)𝑑𝑐𝜉_𝑛,𝑚(𝛼,𝑚)𝑚²𝑐𝑛²(𝛼,𝑚)+𝑑𝑛²(𝛼,𝑚)𝑐𝑛²(𝛼,𝑚)𝑑𝑛²(𝛼,𝑚)+𝜌,(3.45) where 𝜈=−2𝜇(𝑚²𝑠𝑛⁴(𝛼,𝑚)−1)/[𝑐𝑛²(𝛼,𝑚)𝑑𝑛²(𝛼,𝑚)].

Remark 3.1. (1) The formulas of the exact solutions from Types 1–6 are different, and consequently, we must discuss the exact solutions in all types from 1–6.
(2) The values of 𝛼_𝑖, 𝛽_𝑖, 𝑑_𝑖, and 𝑐_𝑖 in Examples  1 and 2 have a unique determination in all types of this method.

4. Conclusion

In this paper, we put a direct method to calculate the rational Jacobi elliptic solutions for the nonlinear difference differential equations via the lattice equation and the discrete nonlinear Schrodinger equation with a saturable nonlinearity. As a result, many new and more rational Jacobi elliptic solutions are obtained, from which hyperbolic function solutions and trigonometric function solutions are derived when the modulus 𝑚→1 and 𝑚→0.