Dispersive Long Wave Equation Research Articles

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.

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.

Numerical investigation of the dispersive long wave equation using an adaptive moving mesh method and its stability

This paper investigates the exact and numerical solutions of the dispersive long wave equations (DLWE) using the extended Jacobian elliptic function expansion method (EJEFET) and the adaptive moving mesh method (AMMT). The uniform mesh method (UMM) is also applied to obtain some numerical results. The analysis of the stability of these solutions is also presented. The achieved solutions are stable. We compare both numerical and exact solutions and obtained that the numerical results are accurate. Some 3D figures are plotted to reinforce and ensure the integrity of the qualitative and quantitative results obtained in this work. Finally, L2 and L∞ errors are analyzed to discover the proper performance of the considered techniques in solving DLWE.

Conservation laws, soliton-like and stability analysis for the time fractional dispersive long-wave equation

In this manuscript we investigate the time fractional dispersive long wave equation (DLWE) and its corresponding integer order DLWE. The symmetry properties and reductions are derived. We construct the conservation laws (Cls) with Riemann–Liouville (RL) for the time fractional DLWE via a new conservation theorem. The conformable derivative is employed to establish soliton-like solutions for the governing equation by using the generalized projective method (GPM). Moreover, the Cls via the multiplier technique and the stability analysis via the concept of linear stability analysis for the integer order DLWE are established. Some graphical features are presented to explain the physical mechanism of the solutions.

Solitary Wave Solutions of Some Coupled Nonlinear Evolution Equations

In this article, the modified simple equation (MSE) method has been executed to find the traveling wave solutions of the coupled (1+1)-dimensional Broer-Kaup (BK) equations and the dispersive long wave (DLW) equations. The efficiency of the method for finding exact solutions has been demonstrated. It has been shown that the method is direct, effective and can be used for many other nonlinear evolution equations (NLEEs) in mathematical physics. Moreover, this procedure reduces the large volume of calculations. Keywords: MSE method; NLEE; BK equations; DLW equations; Solitary wave solutions. © 2014 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. doi: http://dx.doi.org/10.3329/jsr.v6i2.16671 J. Sci. Res. 6 (2), 273-284 (2014)

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.

A Few Expanding Integrable Models, Hamiltonian Structures and Constrained Flows

Two kinds of higher-dimensional Lie algebras and their loop algebras are introduced, for which a few expanding integrable models including the coupling integrable couplings of the Broer—Kaup (BK) hierarchy and the dispersive long wave (DLW) hierarchy as well as the TB hierarchy are obtained. From the reductions of the coupling integrable couplings, the corresponding coupled integrable couplings of the BK equation, the DLW equation, and the TB equation are obtained, respectively. Especially, the coupling integrable coupling of the TB equation reduces to a few integrable couplings of the well-known mKdV equation. The Hamiltonian structures of the coupling integrable couplings of the three kinds of soliton hierarchies are worked out, respectively, by employing the variational identity. Finally, we decompose the BK hierarchy of evolution equations into x-constrained flows and tn-constrained flows whose adjoint representations and the Lax pairs are given.

Chaotic and fractal patterns for interacting nonlinear waves

Using an appropriate reduction method, a quite general new integrable system of equations 2 + 1 dimensions can be derived from the dispersive long-wave equation. Various soliton and dromion solutions are obtaining by selecting some types of solutions appropriately. The interaction between the localized solutions is completely elastic, because they pass through each other and preserve their shapes and velocities, the only change being a phase shift. The arbitrariness of the functions included in the general solution implies that approximate lower dimensional chaotic patterns such as chaotic–chaotic patterns, periodic–chaotic patterns, chaotic line soliton patterns and chaotic dromion patterns can appear in the solution. In a similar way, fractal dromion patterns and stochastic fractal excitations also exist for appropriate choices of the boundary conditions and/or initial conditions.

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.

The multi-component dispersive long wave equation hierarchy, its integrable couplings and their Hamiltonian structures

An isospectral problem is established based on loop algebra X ∼ , by making use of the generalized Tu scheme the multi-component Liouville integrable dispersive long wave (DLW) equation hierarchy is obtained. Then, two expanding loop algebra F ∼ M and Y ∼ are presented, which devoted to working out two integrable couplings of the multi-component DLW equation hierarchy. Finally, the Hamiltonian structures of the multi-component DLW equation hierarchy and the second integrable couplings are obtained by employing the trace variational identity.

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.

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.

Finite symmetry transformation groups and exact solutions of Lax integrable systems

Abstract In this paper, a method is developed to directly find finite symmetry transformation groups and then symmetries of Lax integrable nonlinear physical systems. Some symmetry groups, symmetry algebras and exact solutions for the Kadomtsev–Petviashvili equation, the dispersive long-wave equation and the extended self-dual Yang–Mills equations are explicitly given.

Nonpropagating Solitary Waves in (2+1)-Dimensional Generalized Dispersive Long Wave Systems

An extended mapping approach is used to obtain a new type of variable separation excitation, with three arbitrary functions, of the 2+1-dimensional generalized dispersive long wave equation (DLWE). By selecting appropriate functions, the richness of nonpropagating solitons, such as nonpropagating dromion, nonpropagating ring, nonpropagating lump, and nonpropagating foldon, etc., is displayed for the ($2+1$)-dimensional generalized dispersive long wave equation (DLWE) in this paper. Meanwhile, we conclude that the solution v1 and v2 are essentially equivalent to the “universal” formula.

Application of an extended homogeneous balance method to new exact solutions of nonlinear evolution equations

The multiple soliton solutions of the approximate equations for long water waves and soliton-like solutions for the dispersive long-wave equations in 2+1 dimensions are constructed by using an extended homogeneous balance method. Solitary wave solutions are shown to be a special case of the present results. This method is simple and has a wide-ranging practicability, and can solve a lot of nonlinear partial differential equations.