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We modified the truncated expansion method to construct the exact solutions for some nonlinear differential difference equations in mathematical physics via the general lattice equation, the discrete nonlinear Schrodinger with a saturable nonlinearity, the quintic discrete nonlinear Schrodinger equation, and the relativistic Toda lattice system. Also, we put a rational solitary wave function method to find the rational solitary wave solutions for some nonlinear differential difference equations. The proposed methods are more effective and powerful to obtain the exact solutions for nonlinear difference differential equations.

Nonlinear differential difference equations (NDDEs) play a crucial role in many branches of applied physical sciences such as condensed matter physics, biophysics, atomic chains, molecular crystals, and discretization in solid-state and quantum physics. They also play an important role in numerical simulation of soliton dynamics in high-energy physics because of their rich structures. Therefore, researchers have shown a wide interest in studying NDDEs since the original work of Fermi et al. [

The study of discrete nonlinear system governed by differential difference equations (DDEs) has drawn much attention in recent years particularly from the point of view of complete integrability. There is a vast body of work on nonlinear DDEs, including investigation of integrability criteria, the computation of densities, Backlund transformation, and recursion operator [

In the recent years, there have been a lot of papers devoted to obtain the solitary wave or periodic solutions for a variety of nonlinear differential difference equations by using the symbolic computations. Among these methods Liu [

In this paper, we use a modified truncated expansion method to construct the exact solutions of the following nonlinear difference differential equations in mathematical physics:

the general lattice equation [

the discrete nonlinear Schrodinger equation with a saturable nonlinearity [

the quintic discrete nonlinear Schrodinger equation [

the relativistic Toda lattice system [

In this section, we would like to outline the algorithm for using the modified truncated expansion method to solve the nonlinear DDEs. Consider a given system of

The main steps of the algorithm for the modified truncated expansion method to solve NDDEs are outlined as follows.

We seek the traveling wave transformation in the following form:

We suppose the following series expansion as a solution of (

Further, using the properties of expansion functions the iterative relations can be written in the following form:

Determine the degree

Substituting (

Solving the overdetermined system of nonlinear algebraic equations by using Mathematica or Maple, we end up with explicit expressions of,

Using the results obtained in above steps, we can finally obtain exact solutions of (

In this section, we apply the proposed modified truncated expansion method to construct the exact solutions for the nonlinear DDEs via the lattice equation, the discrete nonlinear Schrodinger equation with a saturable nonlinearity, the quintic discrete nonlinear Schrodinger equation, and the relativistic Toda lattice system which are very important in the mathematical physics and have been paid attention by many researchers.

In this subsection, we use the modified truncated expansion method to find the exact solutions of the general lattice equation. The traveling wave variable (

If we take the transformation

The transformation (

We substitute (

In this subsection, we study the quintic discrete nonlinear Schrodinger equation (

If we take the transformation

The transformation (

In this subsection, we use the modified truncated expansion method to study the relativistic Toda lattice system (

In this section, we would like to outline the algorithm for using the rational solitary wave functions method to solve nonlinear DDEs. Consider a given system of

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

We suppose the wave transformation in the following form:

We suppose the rational solitary wave series expansion solutions of (

Determine the degree

Substituting (

Solving the overdetermined system of nonlinear algebraic equations by using Maple or Mathematica software package, we end up with explicit expressions for

Substituting

In this section, we apply the proposed rational solitary wave functions method to construct the rational solitary wave solutions for some nonlinear DDEs via the discrete nonlinear Schrodinger equation with a saturable nonlinearity, the quintic discrete nonlinear Schrodinger equation, and the relativistic Toda lattice system, which are very important in the mathematical physics and modern physics.

We suppose that the solution of (

Consequently the rational hyperbolic solitary wave solution of (

In this subsection, we study the quintic discrete nonlinear Schrodinger equation (

We suppose that the solution of (

In this subsection, we study the relativistic Toda lattice system (

If we take the transformation

When we compare between the results which obtained in this paper and other exact solutions we get the following.

The solutions obtained in the modified truncated expansion functions method are equivalent to the solution obtained by the exp-functions method, but the modified truncated expansion is simple and allowed us to solve more complicated nonlinear difference differential equations such as the discrete nonlinear Schrodinger equation with a saturable nonlinearity, the quintic discrete nonlinear Schrodinger equation, and the relativistic Toda lattice system. For example, the solution (

In the special case when

These methods which are discussed in this paper allowed us to obtain some new rational solitary wave solutions for some complicated nonlinear differential difference equations.

These methods prefer to another methods to convert the complicated rational methods into a direct nonrational method.

In this paper, we use the modified truncated expansion method to obtain the exact solutions for some nonlinear differential difference equations in the mathematical physics. Also, we calculate the rational solitary wave solutions for the nonlinear differential difference equations. As a result, many new and more rational solitary wave solutions are obtained, from the hyperbolic function solutions and trigonometric function.

The authors wish to thank the referees for their suggestions and very useful comments.