Dimensional Dispersive Long Wave System Research Articles

Hetero-Bäcklund Transformation for a (2+1)-Dimensional Generalized Modified Dispersive Water-Wave System

This work is designed for a (2+1)-dimensional generalized modified dispersive water-wave system for the nonlinear and dispersive long gravity waves travelling along two horizontal directions in the shallow water of uniform depth, with our results as a hetero-B acklund transformation, from that system to a known generalized (2+1)-dimensional dispersive long-wave system. As for the height of the water surface and horizontal velocity of the water wave, our hetero-Backlund transformation depends on the shallow-water coefficients in that system.

Considering the Shallow Water of a Wide Channel or an Open Sea Through a Generalized (2+1)-dimensional Dispersive Long-wave System

Considering the Shallow Water of a Wide Channel or an Open Sea Through a Generalized (2+1)-dimensional Dispersive Long-wave System

Comment on “Shallow water in an open sea or a wide channel: Auto- and non-auto-Bäcklund transformations with solitons for a generalized (2+1)-dimensional dispersive long-wave system”

• We work on a generalized ( 2 + 1 )-dimensional dispersive long-wave system. • With the help of symbolic computation, we give rise to four sets of the similarity reductions, each of which leads to a known ordinary differential equation . • Our similarity reductions are in respect of the horizontal velocity and the wave elevation above the undisturbed water surface. • All of our results depend on the constant coefficients in the system. Active researches on the oceanic water waves have been done. As for the nonlinear and dispersive long gravity waves in two horizontal directions on the shallow water of an open sea or a wide channel of finite depth, the paper commented [i.e., Chaos Solitons Fract. 138, 109950 (2020)] has investigated a generalized ( 2 + 1 )-dimensional dispersive long-wave system. In respect of the horizontal velocity and the wave elevation above the undisturbed water surface, with the help of symbolic computation, we give rise to four sets of the similarity reductions, each of which leads to a known ordinary differential equation. All of our results depend on the constant coefficients in the original system.

Looking at an open sea via a generalized $$(2+1)$$-dimensional dispersive long-wave system for the shallow water: scaling transformations, hetero-Bäcklund transformations, bilinear forms and N solitons

Looking at an open sea via a generalized $$(2+1)$$-dimensional dispersive long-wave system for the shallow water: scaling transformations, hetero-Bäcklund transformations, bilinear forms and N solitons

Using the improved $$\exp ( - \Phi (\xi ))$$ expansion method to find the soliton solutions of the nonlinear evolution equation

With the help of the symbolic calculation software Mathematica, the nonlinear coupled Schrodinger system, (2 + 1)-dimensional dispersive long-wave system and (2 + 1)-dimensional Nizhnik–Novikov–Veselov system are solved by using the improved $$\exp ( - \Phi (\xi ))$$ expansion method. Diverse new soliton solutions are obtained, and the structure and properties of soliton solutions are studied. We derive seven sets of exact solutions under different conditions for the nonlinear coupled Schrodinger system and the (2 + 1)-dimensional dispersive long wave equation, respectively, and for the (2 + 1)-dimensional Nizhnik–Novikov–Veselov system, we find the inelastic interaction solitary waves and fussion soliton.

Shallow water in an open sea or a wide channel: Auto- and non-auto-Bäcklund transformations with solitons for a generalized (2+1)-dimensional dispersive long-wave system

Abstract Oceanic water waves are actively studied. Hereby, taking into account the nonlinear and dispersive long gravity waves in two horizontal directions on the shallow water of an open sea or a wide channel of finite depth, we investigate a generalized (2+1)-dimensional dispersive long-wave system. With symbolic computation while with respect to the horizontal velocity and wave elevation above the undisturbed water surface, we work out two non-auto-Backlund transformations and two auto-Backlund transformations with solitons. All of our results are dependent on the constant coefficients in the original system.

Lie symmetry analysis and generalized invariant solutions of (2+1)-dimensional dispersive long wave (DLW) equations

This present work applies the Lie group of point transformation method to construct the generalized invariant solutions for the (2+1)-dimensional dispersive long wave (DLW) equations under some constraints imposed on infinitesimal generators. In this connection, Lie point symmetries, vector fields and commutation relation for DLW system are well established and then the system is reduced into number of nonlinear ODEs through various symmetry reductions. An optimal system of one dimensional subalgebras of the Lie invariance algebra is formed. We exhaustively carry out symmetry reductions on the basis of these subalgebras. All the obtained solutions are more general in terms of arbitrary functions, and completely different from the previous work of the Sharma et al 2019, Phys. Scr. (Physica Scripta, 2019). Wherever possible, the relative comparison of our findings with the previous work is exhibited. Furthermore, we discuss the dynamic behavior of general solutions like annihilation of single soliton, nonlinear wave profile, curved shaped multisoliton and annihilation of doubly soliton through their evolutionary profiles.

Invariance analysis, exact solutions and conservation laws of (2+1)-dimensional dispersive long wave equations

In this article, some new exact explicit solutions of (2+1)-dimensional dispersive long wave (DLW) equations are obtained by using the similarity transformation method under some restrictions imposed on the infinitesimals. This method reduces the dimension of PDEs by one after applying once. By choosing the suitable values of arbitrary functions involved in the expressions of infinitesimals, the system of PDEs is converted into the system of ODEs with the help of similarity variables. Under the suitable choice of arbitrary constants, the graphical representation of the obtained solutions are shown in order to highlight the importance of the study. The adjoint table, conservation laws and optimal system of DLW system are also obtained.

Residual symmetry, interaction solutions, and conservation laws of the (2+1)-dimensional dispersive long-wave system**Project supported by the National Natural Science Foundation of China (Grant Nos. 11371293 and 11505090), the Natural Science Foundation of Shaanxi Province, China (Grant No. 2014JM2-1009), the Research Award Foundation for Outstanding Young Scientists of Shandong Province, China (Grant No. BS2015SF009), and the Science and Technology Innovation Foundation

We explore the (2+1)-dimensional dispersive long-wave (DLW) system. From the standard truncated Painlevé expansion, the Bäcklund transformation (BT) and residual symmetries of this system are derived. The introduction to an appropriate auxiliary dependent variable successfully localizes the residual symmetries to Lie point symmetries. In particular, it is verified that the (2+1)-dimensional DLW system is consistent Riccati expansion (CRE) solvable. If the special form of (CRE)-consistent tanh-function expansion (CTE) is taken, the soliton-cnoidal wave solutions and corresponding images can be explicitly given. Furthermore, the conservation laws of the DLW system are investigated with symmetries and Ibragimov theorem.

SOLITONS AND LOCALIZED EXCITATIONS FOR THE (2+1)-DIMENSIONAL DISPERSIVE LONG WAVE SYSTEM VIA SYMBOLIC COMPUTATION

For describing some nonlinear localized excitations, the (2+1)-dimensional dispersive long wave (DLW) system is investigated with symbolic computation in this paper. Based on two different dependent variable transformations obtained through the truncated Painlevé expansion, the (2+1)-dimensional DLW system can be bilinearized or linearized. Through the Hirota bilinear method, the analytic one-, two-, three-, and N-soliton solutions are derived. On the other hand, by means of the variable separation approach, localized excitations, such as the resonant dromion, resonant solitoff, lump and compacton excitations, are obtained. Figures are plotted to illustrate the structures of those solutions.

Periodic Structures Based on the Symmetrical Lucas Function of the (2+1)-Dimensional Dispersive Long-Wave System

By introducing the Lucas Riccati method and a linear variable separation method, new variable separation solutions with arbitrary functions are derived for a (2+1)-dimensional dispersive longwave system. The main idea of this method is to express the solutions of this system as polynomials in the solution of the Riccati equation that satisfies the symmetrical Lucas functions. From the variable separation solution and by selecting appropriate functions, some novel Jacobian elliptic wave structures and periodic wave evolutional behaviours are investigated.

Diversity of exact solutions to a restricted Boiti–Leon–Pempinelli dispersive long-wave system

A variety of exact solutions is explored for a restricted Boiti–Leon–Pempinelli (2+1)-dimensional dispersive long-wave system and thus also for the Boiti–Leon–Pempinelli (2+1)-dimensional dispersive long-wave system. The solutions presented involve a couple of arbitrary functions of space and time variables, contain solutions of dromion type only with peaks and with both peaks and pits, and exhibit various complex behaviors of solutions such as chaotic and fractal behaviors and behaviors with changing shape and exchanging energy during wave interaction.

The new extended Jacobian elliptic function expansion algorithm and its applications in nonlinear mathematical physics equations

More recently we have presented the extended Jacobian elliptic function expansion method and its algorithm to seek more types of doubly periodic solutions. Based on the idea of the method, by studying more relations among all twelve kinds of Jacobian elliptic functions. we further extend the method to be a more general method, which is still called the extended Jacobian elliptic function expansion method for convenience. The new method is more powerful to construct more new exact doubly periodic solutions of nonlinear equations. We choose the (2+1)-dimensional dispersive long-wave system to illustrate our algorithm. As a result, twenty-four families of new doubly periodic solutions are obtained. When the modulus m→1 or 0, these doubly periodic solutions degenerate as soliton solutions and trigonometric function solutions. This algorithm can be also applied to other nonlinear equations.

Localized solutions with chaotic and fractal behaviours in a (2+1)-dimensional dispersive long-wave system

It is known that in high dimensions there exist abundant localized excitations such as dromions, lumps, ring soliton solutions and so on. In this paper, the possible chaotic and fractal localized structures are revealed for the (2+1)-dimensional dispersive long-wave equation. The chaotic and fractal dromion and lump patterns of the model are constructed by some types of lower-dimensional chaotic and fractal patterns.