Exact travelling wave solutions for （1＋ 1）-dimensional dispersive long wave equation

A complete discrimination system for the fourth order polynomial is given. As an application, we have reduced a （1＋1）-dimensional dispersive long wave equation with general coefficients to an elementary integral form and obtained its all possible exact travelling wave solutions including rational function type solutions, solitary wave solutions, triangle function type periodic solutions and Jacobian elliptic functions double periodic solutions. This method can be also applied to many other similar problems.

Rational Form Solitary Wave Solutions and Doubly Periodic Wave Solutions to （1＋1）-Dimensional Dispersive Long Wave Equation

In this work we devise an algebraic method to uniformly construct rational form solitary wave solutions and Jacobi and Weierstrass doubly periodic wave solutions of physical interest for nonlinear evolution equations. With the aid of symbolic computation, we apply the proposed method to solving the (1+1)-dimensional dispersive long wave equation and explicitly construct a series of exact solutions which include the rational form solitary wave solutions and elliptic doubly periodic wave solutions as special cases.

Exact Solutions to the Generalized Dispersive Long Wave Equation with Variable Coefficients

By using the homogeneous balance principle（HBP）, we derive a Backtund transformation（BT） to the generalized dispersive long wave equation with variable coefficients.Based on the BT, we give many kinds of the exact solutions of the equation, such as, singlesolitary solutions, multi-soliton solutions and generalized exact solutions.

New Complexiton Solutions of （1＋1）-Dimensional Dispersive Long Wave Equation

By means of two different Riccati equations with different parameters as subequation in the components of finite rational expansion method, new complexiton solutions for the （1＋1）-dimensional dispersive long wave equation are successfully constructed, which include various combination of trigonometric periodic and hyperbolic function solutions, various combination of trigonometric periodic and rational function solutions, and various combination of hyperbolic and rational function solutions.

A Generalized Extended F-Expansion Method and Its Application in （2＋1）-Dimensional Dispersive Long Wave Equation

A new generalized extended F-expansion method is presented for finding periodic wave solutions of nonlinear evolution equations in mathematical physics. As an application of this method, we study the （2＋1）-dimensional dispersive long wave equation. With the aid of computerized symbolic computation, a number of doubly periodic wave solutions expressed by various Jacobi elliptic functions are obtained. In the limit cases, the solitary wave solutions are derived as well.

Construction of Recursion Formulas for Lax Pairs of (2+1)-Dimensional Equation: Modified Generalized Dispersive Long Wave Equation

In this article, we study the Lax pairs of -dimensional equation: the modified generalized dispersive long wave (MGDLW) equation. Based on the well-known binary Darboux transformation, we dig out the recursion formulas of the first part of the Lax pairs. Then by further discussion and doing some revisional work, we make the recursion formulas fit for the second part of Lax pairs. At last, some solutions to the MGDLW equation are worked out by using the recursion formula.

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.

Symbolic Computation Study of （2＋1）-Dimensional Dispersive Long Wave Equations

Seeking exact analytical solutions of nonlinear evolution equations is of fundamental importance in mathematlcal physics. In this paper, based on a constructive algorithm and symbolic computation, abundant new exact solutions of the （2＋1）-dimensional dispersive long wave equations are obtained, among which there are soliton-like solutions, mult-soliton-like solutions and formal periodic solutions, etc. Certain special solutions are considered and some interesting localized structures are revealed.

Bifurcation of travelling wave solutions for (2+1)-dimension nonlinear dispersive long wave equation

In this paper,the bifurcation of solitary,kink,anti-kink,and periodic waves for （2＋1）-dimension nonlinear dispersive long wave equation is studied by using the bifurcation theory of planar dynamical systems.Bifurcation parameter sets are shown,and under various parameter conditions,all exact explicit formulas of solitary travelling wave solutions and kink travelling wave solutions and periodic travelling wave solutions are listed.

New Exact Travelling Wave Solutions to Hirota Equation and (1+1)-Dimensional Dispersive Long Wave Equation

Based on the computerized symbolic Maple, we study two important nonlinear evolution equations, i.e., the Hirota equation and the (1+1)-dimensional dispersive long wave equation by use of a direct and unified algebraic method named the general projective Riccati equation method to find more exact solutions to nonlinear differential equations. The method is more powerful than most of the existing tanh method. New and more general form solutions are obtained. The properties of the new formal solitary wave solutions are shown by some figures.

Further Extended Jacobi Elliptic Function Rational Expansion Method and New Families of Jacobi Elliptic Function Solutions to （2＋1）-Dimensional Dispersive Long Wave Equation

In this paper, a further extended Jacobi elliptic function rationM expansion method is proposed for constructing new forms of exact solutions to nonlinear partial differential equations by making a more general transformation. For illustration, we apply the method to （2＋1）-dimensionM dispersive long wave equation and successfully obtain many new doubly periodic solutions. When the modulus m→1, these sohitions degenerate as soliton solutions. The method can be also applied to other nonlinear partial differential equations.

Non-completely elastic interactions in a(2+1)-dimensional dispersive long wave equation

With the help of a modified mapping method, we obtain two kinds of variable separation solutions with two arbitrary functions for the （24-1）-dimensional dispersive long wave equation. When selecting appropriate multi-valued functions in the variable separation solution, we investigate the interactions among special multi-dromions, dromion-like multi-peakons, and dromion-like multi-semifoldons, which all demonstrate non-completely elastic properties.

（1＋1）—Dimensional Turbulent and Chaotic Systems Reduced from（2＋1）—Dimensional Lax Integrable Dispersive Long Wave Equation

After generalizing the Clarkson-Kruskal direct similarity reduction ansatz, one can obtain various newtypes of reduction equations. Especially, some lower-dimensional turbulent systems or chaotic systems may be obtainedfrom the general form of the similarity reductions of a higher-dimensional Lax integrable model. Furthermore, anarbitrary three-order quasi-linear equation, which includes the Korteweg de-Vries Burgers equation and the generalLorenz equation as two special cases, has been obtained from the reductions of the (2+1)-dimensional dispersive longwave equation system. Some types of periodic and chaotic solutions of the system are also discussed.

On similarity solutions to (2+1)-dispersive long-wave equations

This work is devoted to get a new family of analytical solutions of the(2+1)-coupled dispersive long wave equations propagating in an infinitely long channel with constant depth,and can be observed in an open sea or in wide channels.The solutions are obtained by using the invariance property of the similarity transformations method via one-parameter Lie group theory.The repeated use of the similarity transformations method can transform the system of PDEs into system of ODEs.Under adequate restrictions,the reduced system of ODEs is solved.Numerical simulation is performed to describe the solutions in a physically meaningful way.The profiles of the solutions are simulated by taking an appropriate choice of functions and constants involved therein.In each animation,a frame for dominated behavior is captured.They exhibit elastic multisolitons,single soliton,doubly solitons,stationary,kink and parabolic nature.The results are significant since these have confirmed some of the established results of S.Kumar et al.(2020)and K.Sharma et al.(2020).Some of their solutions can be deduced from the results derived in this work.Other results in the existing literature are different from those in this work.

ABUNDANT LOCALIZED COHERENT STRUCTURES FOR THE (2+1)-DIMENSIONAL LONG DISPERSIVE WAVE SYSTEM

In the present paper, a simple and direct method was proposed to solve the (2+ 1)-dimensional long dispersive wave equations. A variable-dependent transformation was intorducedto convert the equations into the simpler forms, which are coupled and linear partial differentialequations, then obtain its general solution. Some special types of the localized excitations, suchas oscillating dromion, multi-solitoff, multi-dromion, multi-lump and multi-ring soliton solutionsare derived by selecting the arbitrary functions appropriately.

New Extended Jacobi Elliptic Function Rational Expansion Method and Its Application

In this paper, an extended Jacobi elliptic function rational expansion method is proposed for constructing new forms of exact Jacobi elliptic function solutions to nonlinear partial differential equations by means of making a more general transformation. For illustration, we apply the method to the （2＋1）-dimensional dispersive long wave equation and successfully obtain many new doubly periodic solutions, which degenerate as soliton solutions when the modulus m approximates 1. The method can also be applied to other nonlinear partial differential equations.

Investigation of adequate closed form travelling wave solution to the space-time fractional non-linear evolution equations

This work aims to construct exact solutions for the space-time fractional(2+1)-dimensional dispersive longwave(DLW)equation and approximate long water wave equation(ALW)utilizing the twovariable(G′/G,1/G)-expansion method and the modified Riemann-Liouville fractional derivative.The recommended equations play a significant role to describe the travel of the shallow water wave.The fractional complex transform is used to convert fractional differential equations into ordinary differential equations.Several wave solutions have been successfully achieved using the proposed approach and the symbolic computer Maple package.The Maple package program was used to set up and validate all of the computations in this investigation.By choosing particular values of the embedded parameters,we pro-duce multiple periodic solutions,periodic wave solutions,single soliton solutions,kink wave solutions,and more forms of soliton solutions.The achieved solutions might be useful to comprehend nonlinear phenomena.It is worth noting that the implemented method for solving nonlinear fractional partial dif-ferential equations(NLFPDEs)is efficient,and simple to find further and new-fangled solutions in the arena of mathematical physics and coastal engineering.