Rational Solitary Wave Solutions Research Articles

Rational solitary waves and optical solitons of the model having unrestricted dispersion and nonlinearity with polynomial form

Objective:The mathematical model having unrestricted dispersion and nonlinearity in the polynomial form of described by the nonlinear Schrödinger equation is considered. Rational solitary wave solutions and optical solitons of the nonlinear partial differential equation are looked for. Method:The Painlevé test is applied to analyze the integrability of the nonlinear partial differential equation. Two variants of the simplest equation method are used for finding rational solitary waves and optical solitons of the mathematical model. Results:It is shown that the mathematical model having unrestricted dispersion and nonlinearity in the form of polynomial is nonintegrable by the inverse scattering transform. It is proved the general solution of the nonlinear ordinary differential equation has the expansion of in the Laurent series with two arbitrary constants. Rational solitary wave solutions and optical solutions of the mathematical model are found.

Soliton solution of fractional Sharma-Tasso-Olever equation via an efficient (G′/G)-expansion method

This paper investigates the solutions of the conformable fractional Sharma-Tasso-Olever equation with the help of Atangana’s conformable differential operator. In this respect, an efficient (G′/G)-expansion method is used for obtaining the new solitary wave solutions. This work addresses the physical and dynamic behavior of some new exact trigonometric, hyperbolic, and rational solitary wave solutions in the form of 3D-plots, contour plots, and 2D- plots using different measures of parameters. The graphical representation of these results is very helpful for understanding the real physical importance of the studied model equation. The obtained results are new and more general, and show the efficiency of the proposed method for the analytical treatment of nonlinear problems in mathematical physics and engineering and helpful in better understanding the propagating wave dynamics in diverse situations.

Explicit and traveling wave solutions of the non-linear couple Drinfeld-Sokolov-Wilson dynamical system arising in shallow water waves

We efficaciously extended the ansatz in the Riccati equation rational expansion technique to a new broad form and is used to attain some new exact wave solutions for the couple non-linear Drinfeld-Sokolov-Wilson (DSW) system. This system arises in water waves, quantum mechanics, fluid mechanics, biology, chemistry etc. Several of the attained solutions which is in the shape of rational solitary wave solutions and periodic wave solutions are entirely new and could be vital for researchers to appreciate. Certain selected solutions are graphically verified by precise values allocated to constant parameters. This technique with the current ansatz can be implemented with a broader range of applicability to handle various other forms of non-linear evolution systems (NLES).

Computational wave solutions of generalized higher-order nonlinear Boussinesq dynamical wave equation

A Riccati equation rational expansion (RERE) method is implemented to attain solutions of nonlinear partial differential equations (NLPDEs). Subsequently, the technique is used to attain families or sets of nonlinear solutions for the third extended fifth-order nonlinear equation and the generalized (2 + 1)-dimensional Boussinesq equation. By the use of the RERE method, exciting and new forms of solution such as rational solitary wave solutions, solitary wave solutions and periodic solitary wave solutions are obtained. Some of the solutions are demonstrated graphically with specific values assigned to the parameters. It is worth noting that the application of this method is very effective and reliable.

Exact Solutions for Nonlinear Differential Difference Equations in Mathematical Physics

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.

Construction of new exact rational form non-travelling wave solutions to the (2 + 1)-dimensional generalized Broer–Kaup system

With the aid of symbolic computation Maple, several new families of rational form variable separation solutions with three arbitrary functions to the (2 + 1)-dimensional generalized Broer–Kaup system are derived by using an improved mapping approach and a variable separation approach. These solutions include rational solitary wave solutions, periodic wave solutions and rational wave solutions. The properties of the novel localized excitation are revealed by some figures.

New Families of Rational Form Variable Separation Solutions to (2+1)-Dimensional Dispersive Long Wave Equations

With the aid of symbolic computation system Maple, some families of new rational variable separation solutions of the (2+1)-dimensional dispersive long wave equations are constructed by means of a function transformation, improved mapping approach, and variable separation approach, among which there are rational solitary wave solutions, periodic wave solutions and rational wave solutions.

A generalized extended rational expansion method and its application to (1 + 1)-dimensional dispersive long wave equation

In this Letter, a generalized extended rational expansion method is used to construct exact solutions of the (1 + 1)-dimensional dispersive long wave equation. As a result, many new and more general exact solutions are obtained, the solutions obtained in this Letter include rational triangular periodic wave solutions, rational solitary wave solutions.

An Improved F-Expansion Method and Its Application to Coupled Drinfel'd–Sokolov–Wilson Equation

With the aid of computerized symbolic computation, an improved F-expansion method is presented to uniformly construct more new exact doubly periodic solutions in terms of rational formal Jacobi elliptic function of nonlinear partial differential equations (NPDEs). The coupled Drinfel'd–Sokolov–Wilson equation is chosen to illustrate the method. As a result, we can successfully obtain abundant new doubly periodic solutions without calculating various Jacobi elliptic functions. In the limit cases, the rational solitary wave solutions and trigonometric function solutions are obtained as well.

On the rational solitary wave solutions for the nonlinear Hirota–Satsuma coupled KdV system

The objective of this article is to investigate an algebraic method for constructing new rational exact wave soliton solutions in terms of hyperbolic and triangular functions for the generalized nonlinear Hirota–Satsuma coupled KdV systems of partial differential equations using symbolic software like Mathematica or Maple. These studies reveal that the generalized nonlinear Hirota–Satsuma coupled KdV system has a rich variety of solutions.

New rational solitary wave solutions of (2+1) -dimensional Burgers equation

Based on the computer symbolic system Maple and the rational solitary wave solutions and rational periodic solutions obtained by others previously, we present a new direct ansätz in terms of the hyperbolic functions and periodic function to solve the nonlinear evolution equations, that is to say, we assume the equations have the solution in the following form u i ( ξ ) = a i 0 + ∑ j = 1 n i g j − 1 ( a i j f + b i j g + c i j h + d i j r ) , where f = tanh ( ξ ) sech ( ξ ) A + B tanh ( ξ ) + C sech 2 ( ξ ) , g = sech ( ξ ) A + B tanh ( ξ ) + C sech 2 ( ξ ) , h = sech 2 ( ξ ) A + B tanh ( ξ ) + C sech 2 ( ξ ) , r = tanh ( ξ ) A + B tanh ( ξ ) + C sech 2 ( ξ ) and A 2 + B 2 + C 2 ≠ 0 . Compared with the solutions obtained by the existing tanh methods, Riccati equation method and all kinds of their improved methods, the solutions obtained by the proposed direct method not only include some obtained solutions, but also include some new and general rational solutions. And we choose the (2+1)-dimensional Burgers equation to illustrate the key step. As a result, we obtain abundant families of rational solitary wave solutions and rational periodic solutions. Furthermore, the direct method is proved to be efficient.

Multiple solution profiles to the higher-dimensional Kadomtsev–Petviashvilli equations via Wronskian determinant

Abstract Many types of new exact solutions of (3 + 1)-dimensional KP equation are obtained via an unified Wronskian determinant and three linear partial differential equations, which involve many types of multiple solitary wave solutions, rational solutions, and rational-solitary wave solutions. It is shown that the collisions of the obtained multiple solitary wave solutions are elastic, which implies that (3 + 1)-dimensional KP equation admits multisoliton solutions. Moreover we also give the Wronskian formal solutions of ( n + 1)-dimensional KP equations.

New Explicit Rational Solitary WaveSolutions for Discretized mKdV LatticeEquation**The project supported by National Key BasicResearch Project of China under Grant No. 2004CB318000

In this letter, we study discretized mKdV lattice equation byusing a new generalized ansätz. As a result, many explicit rationalexact solutions, including some new solitary wave solutions,are obtained by symbolic computation code Maple.

A new Riccati equation rational expansion method and its application to (2+1)-dimensional Burgers equation

In this paper, we present a new Riccati equation rational expansion method to uniformly construct a series of exact solutions for nonlinear evolution equations. Compared with most existing tanh methods and other sophisticated methods, the proposed method not only recover some known solutions, but also find some new and general solutions. The solutions obtained in this paper include rational triangular periodic wave solutions, rational solitary wave solutions and rational wave solutions. The efficiency of the method can be demonstrated on (2 + 1)-dimensional Burgers equation.

Fractional transformation and new solutions to mKdV equation

A fractional transformation is introduced to solve modified KdV (mKdV for short) equation, where this fractional transformation is used to map the solutions of an elliptic equation to another elliptic equation. Thus, more new kinds of solutions are obtained, such as rational periodic wave solutions, rational solitary wave solutions and so on. It is shown that this method is more powerful to give more kinds of solutions.

Something hidden in the Fourier series and its partial sum: Solitary wave solutions to polynomial nonlinear differential equations

Transformation proposed in the author's recent work is utilized to obtain the polynomial and rational solitary wave solutions of polynomial nonlinear 1 + 1-dimension evolution equations, and thus some special solutions of polynomial nonlinear ODEs. In particular, several theorems are established to provide the necessary and sufficient conditions for the existence of these solutions. With the new transformation, polynomial solitary waves can be hidden in the form of a partial sum of the Fourier series of an intermediate variable (a curvilinear coordinate system), and therefore the balance between instabilities (nonlinear) and dispersive, dissipative, and other effects is well represented in a given polynomial nonlinear differential equation. A direct generalization of the method yields a solitary wave-like solution to the cylindrical KdV equation. In addition, we prove that the main transformation in this paper is equivalent to a simplified sine-Gordon equation. Finally, some new (complex) traveling wave solutions of the KdV equation, the Burgers equation and the modified KdV equation are obtained via our method. Among those new solutions, one of the KdV equation is shown taking similar form to the solitary wave solution of the nonlinear Schrödinger equation, which once again exposes the relationship between the two well-known nonlinear wave equations.