Dimensional Dispersive Long Wave Equation Research Articles

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 two-variable (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 produce 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 differential equations (NLFPDEs) is efficient, and simple to find further and new-fangled solutions in the arena of mathematical physics and coastal engineering.

Variational Principles for Two Kinds of Coupled Nonlinear Equations in Shallow Water

It is a very important but difficult task to seek explicit variational formulations for nonlinear and complex models because variational principles are theoretical bases for many methods to solve or analyze the nonlinear problem. By designing skillfully the trial-Lagrange functional, different groups of variational principles are successfully constructed for two kinds of coupled nonlinear equations in shallow water, i.e., the Broer-Kaup equations and the (2+1)-dimensional dispersive long-wave equations, respectively. Both of them contain many kinds of soliton solutions, which are always symmetric or anti-symmetric in space. Subsequently, the obtained variational principles are proved to be correct by minimizing the functionals with the calculus of variations. The established variational principles are firstly discovered, which can help to study the symmetries and find conserved quantities for the equations considered, and might find lots of applications in numerical simulation.

Oblique closed form solutions of some important fractional evolution equations via the modified Kudryashov method arising in physical problems

The paper deals with the obliquely propagating wave solutions of fractional nonlinear evolution equations (NLEEs) arising in science and engineering. The conformable time fractional (2 + 1)-dimensional extended Zakharov-Kuzetsov equation (EZKE), coupled space-time fractional (2 + 1)-dimensional dispersive long wave equation (DLWE) and space-time fractional (2 + 1)-dimensional Ablowitz-Kaup-Newell-Segur (AKNS) equation are considered to investigate such physical phenomena. The modified Kudryashov method along with the properties of conformable and modified Riemann-Liouville derivatives is employed to construct the oblique wave solutions of the considered equations. The obtained results may be useful for better understanding the nature of internal oblique propagating wave dynamics in ocean engineering.

Nonlocal Symmetries and Consistent Riccati Expansions of the (2+1)-Dimensional Dispersive Long Wave Equation

Abstract In this article, the (2+1)-dimensional dispersive long wave equation (DLWE) is investigated, which is derived in the context of a water wave propagating in narrow infinitely long channels of finite constant depth. By using of the truncated Painlevé expansion, we construct its nonlocal symmetry and Bäcklund transformation. After implanting the equation into an enlarged one, then the residual symmetry is localised. Meanwhile, the symmetry group transformation can be computed from the prolonged system. Furthermore, the equation is verified to be consistent Riccati expansion (CRE) solvable. Outing from the CRE, the soliton-cnoidal wave interaction solution in terms of Jacobi elliptic functions and the third type of incomplete elliptic integral are studied, respectively.

Exact soliton solutions for \((2+1)\)-dimensional dispersive long wave equation

In this paper, the first integral method is used to construct exact traveling wave solutions of $(2+1)-$ dimensional dispersive long wave equation. The first integral method is an efficient method for obtaining exact solutions some of nonlinear partial differential equations. This method can be applied to nonintegrable equations as well as to integrable ones.

Construction of new infinite sequence complexion soliton-like solutions of nonlinear evolution equations

The G'(ξ)/G(ξ) expansion method is further studied for constructing new infinite sequence complexion soliton-like solutions of nonlinear evolution equations. First, to solve a linear ordinary differential equation with constant coefficients of second order is changed into the solving of one unknown quadratic equation and Riccati equation by a function transformation. Then a nonlinear superposition formula of the solutions to Riccati equation is presented to seek new infinite sequence complexion solutions of a second order linear ordinary differential equation with constant coefficients. Based on this, the new infinite sequence complexion soliton-like solutions to (2+1)-dimensional modified dispersive water wave system and (2+1)-dimensional dispersive long-wave equation are obtained with the help of symbolic computation system Mathematica.

Fission and Fusion Interaction Phenomena of the (2+1)-Dimensional Dispersive Long Wave Equations

With the aid of symbolic computation, the (2+1)-dimensional dispersive long-wave equations are investigated via Backlund transformation and the Hirota bilinear method, two types of new multiple soliton solutions with 2n + 1 arbitrary functions for that system are derived. Based on the derived solutions, abundant coherent structures like dromions, compactons and peakons are derived, and moreover, the fission and fusion interactions among different types of localized structures are discussed graphically, which might be helpful for understanding the travel of the shallow water wave.

Kac–Moody–Virasoro Symmetry Algebra of (2+1)-Dimensional Dispersive Long-Wave Equation with Arbitrary Order Invariant

By Lie symmetry method, the Lie point symmetries and its Kac–Moody–Virasoro (KMV) symmetry algebra of (2+1)-dimensional dispersive long-wave equation (DLWE) are obtained, and the finite transformation of DLWE is given by symmetry group direct method, which can recover Lie point symmetries. Then KMV symmetry algebra of DLWE with arbitrary order invariant is also obtained. On basis of this algebra the group invariant solutions and similarity reductions are also derived.

Symmetry Groups and New Exact Solutions of (2+1)-Dimensional Dispersive Long-Wave Equations

Using the modified CK's direct method, we derive a symmetry group theorem of (2+1)-dimensional dispersive long-wave equations. Based upon the theorem, Lie point symmetry groups and new exact solutions of (2+1)-dimensional dispersive long-wave equations are obtained.

Numerical analysis of higher-dimensional dispersive long-wave equations

In this paper, numerical analysis of the (2+1)-dimensional dispersive long-wave equation (DLWE) is studied by using the homotopy perturbation method (HPM). For this purpose, the available analytical solutions obtained by multiple traveling-wave solution will be compared to show the validity and accuracy of the presented numerical algorithm. The obtained results prove the convergence and accuracy of the HPM for the numerically analyzed (2+1)-dimensional DLWE system.

A New Rational Algebraic Approach to Find Exact AnalyticalSolutions to a (2+1)-Dimensional System

In this paper, we present a new rational algebraicapproach to uniformly construct a series of exactanalytical solutions for nonlinear partial differentialequations. Compared with most existing tanhmethods and other sophisticated methods, the proposed method not onlyrecovers some known solutions, but also finds some new andgeneral solutions. The solutions obtained in this paperinclude rational form triangular periodic wave solutions,solitary wave solutions, and elliptic doubly periodic wave solutions.The efficiency of the method can be demonstrated on(2+1)-dimensional dispersive long-wave equation.

Oscillating Solitons for (2+1)-Dimensional NonlinearModels

Using extended homogeneous balance method and variableseparation hypothesis, we found new variable separation solutionswith three arbitrary functions of the (2+1)-dimensional dispersivelong-wave equations. Based on derived solutions, we revealedabundant oscillating solitons such as dromion, multi-dromion,solitoff, solitary waves, and so on, by selecting appropriate functions.

Solitons and Waves in (2+1)-Dimensional DispersiveLong-Wave Equation

For a higher-dimensional integrable nonlinear dynamical system,there are abundant coherent soliton excitations. With the aid of an improvedprojective Riccati equation approach, the paper obtains several types ofexact solutions to the (2+1)-dimensional dispersive long-wave equation,including multiple-soliton solutions, periodic soliton solutions, andWeierstrass function solutions. From these solutions, apart from severalmultisoliton excitations, we derive some novel features of wave structures byintroducing some types of lower-dimensional patterns.

A Generalized Extended F-ExpansionMethod and Its Application in (2+1)-Dimensional DispersiveLong Wave Equation

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

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

By means of two different Riccatiequations with different parameters as subequation in thecomponents of finite rational expansion method, new complexitonsolutions for the (1+1)-dimensional dispersive long wave equationare successfully constructed, which include various combination oftrigonometric periodic and hyperbolic function solutions, variouscombination of trigonometric periodic and rational functionsolutions, and various combination of hyperbolic and rationalfunction solutions.

Further Extended Jacobi Elliptic Function Rational Expansion Method andNew Families of Jacobi Elliptic Function Solutions to (2+1)-Dimensional DispersiveLong Wave Equation

In this paper, a further extended Jacobi elliptic function rational expansionmethod is proposed for constructing new forms of exact solutionsto nonlinear partial differential equations by making a moregeneral transformation. For illustration, we apply the method to(2+1)-dimensional dispersive long wave equation andsuccessfully obtain many new doubly periodic solutions. When themodulus m→1, these solutions degenerate as solitonsolutions. The method can be also applied to other nonlinearpartial differential equations.

New Exact Solutions to Dispersive Long-Wave Equations in(2+1)-Dimensional Space

New exact solutions expressed by the Jacobi ellipticfunctions are obtained to the (2+1)-dimensional dispersive long-waveequations by using the modified F-expansion method. In the limit case, newsolitary wave solutions and triangular periodic wave solutions are obtainedas well.

Symbolic computation and new families of exact solutions to the (2+1)-dimensional dispersive long-wave equations

Abstract In this Letter, We present a further generalized algebraic method to the (2 + 1)-dimensional dispersive long-wave equations (DLWS), As a result, we can obtain abundant new formal exact solutions of the equation. The method can also be applied to solve more (2 + 1)-dimensional (or (3 + 1)-dimensional) nonlinear partial differential equations (NPDEs).

A NEW GENERAL ALGEBRAIC METHOD WITH SYMBOLIC COMPUTATION TO CONSTRUCT NEW TRAVELING SOLUTION FOR THE (1 +1)-DIMENSIONAL DISPERSIVE LONG WAVE EQUATION

A new algebraic method, named Riccati equation rational expansion (RERE) method, is devised for constructing multiple traveling wave solutions for nonlinear evolution equations (NEEs). With the aid of symbolic computation, we choose (1 +1)-dimensional dispersive long wave equation (DLWE) to illustrate our method. As a result, we obtain many types of solutions including rational form solitary wave solutions, triangular periodic wave solutions and rational wave solutions.

New multi-soliton solutions and travelling wave solutions of the dispersive long-wave equations

Using the extended homogeneous balance method,the (1+1)-dimensional dispersive long-wave equations have been solved.Starting from the homogeneous balance method,we have obtained a nonlinear transformation for simplifying a dispersive long-wave equation into a linear partial differential equation.Usually,we can obtain only a type of soliton-like solution.In this paper,we have further found some new multi-soliton solutions and exact travelling solutions of the dispersive long-wave equations from the linear partial equation.