Bilinear form, solitons, breathers, lumps and hybrid solutions for a (3+1)-dimensional Date-Jimbo-Kashiwara-Miwa equation
TL;DR: In this paper, a (3+1)-dimensional Date-Jimbo-Kashiwara-Miwa equation for the nonlinear dispersive waves in an inhomogeneous medium is studied.
Abstract: In this work, we study a (3+1)-dimensional Date-Jimbo-Kashiwara-Miwa equation for the nonlinear dispersive waves in an inhomogeneous medium. Bilinear form and N-soliton solutions are derived, where N is a positive integer. The higher-order breather and lump solutions are constructed based on the N-soliton solutions. Hybrid solutions comprising the solitons and breathers, breathers and lumps, as well as solitons and lumps are worked out. Amplitudes and velocities of the one solitons as well as periods of the first-order breathers are investigated. Amplitudes of the first-order lumps reach the maximum and minimum values at certain points given in the paper. Interactions between any two of those waves are discussed graphically.
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TL;DR: In this paper, a generalized (2 + 1 )-dimensional dispersive long-wave system was investigated and four sets of similarity reductions were obtained, each of which leads to a known ordinary differential equation.
Abstract: 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.
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TL;DR: In this article, a generalized (2+1)-dimensional dispersive long-wave system with scaling transformations, including Bell polynomials and symbolic computation, has been studied.
Abstract: Oceanic water-wave studies are attractive. Hereby, with a view to modelling 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 inquire into a generalized (2+1)-dimensional dispersive long-wave system, by virtue of the scaling transformations we hereby obtain, Bell polynomials and symbolic computation. As for the horizontal velocity and wave elevation above the undisturbed water surface, we find two branches of the hetero-Backlund transformations, two branches of the bilinear forms and two branches of the N -soliton solutions, with N as a positive integer. What we have obtained relies on the coefficients in the system.
56 citations
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01 Jan 2004TL;DR: In this paper, Bilinearization of soliton equations is discussed and the Backlund transformation is used to transform the soliton equation into a linear combination of determinants and pfaffians.
Abstract: Preface Foreword 1. Bilinearization of soliton equations 2. Determinants and pfaffians 3. Structure of soliton equations 4. Backlund transformations Afterword References Index.
2,132 citations
TL;DR: In this paper, a new approach to soliton equations, based on τ functions (or Hirota's dependent variables), vertex operators and the Clifford algebra of free fermions, is applied to study a new hierarchy of Kadomtsev-Petviashvili type equations (the BKP hierarchy).
582 citations
TL;DR: In this article, two-dimensional lump solutions which decay to a uniform state in all directions are obtained for the Kadomtsev-Petviashvili and Schrodinger type equation.
Abstract: Two‐dimensional lump solutions which decay to a uniform state in all directions are obtained for the Kadomtsev–Petviashvili and a two‐dimensional nonlinear Schrodinger type equation. The amplitude of these solutions is rational in its independent variables. These solutions are constructed by taking a ’’long wave’’ limit of the corresponding N‐soliton solutions obtained by direct methods. The solutions describing multiple collisions of lumps are also presented.
548 citations
TL;DR: In this paper, the Korteweg-de Vries equation with the isospectral property is considered and rational solutions are obtained by performing an appropriate limiting procedure on the soliton solutions obtained by direct methods.
Abstract: Rational solutions of certain nonlinear evolution equations are obtained by performing an appropriate limiting procedure on the soliton solutions obtained by direct methods. In this note specific attention is directed at the Korteweg–de Vries equation. However, the methods used are quite general and apply to most nonlinear evolution equations with the isospectral property, including certain multidimensional equations. In the latter case, nonsingular, algebraically decaying, soliton solutions can be constructed.
348 citations
01 Sep 1978
310 citations