Lie symmetry analysis, exact analytical solutions and dynamics of solitons for (2 + 1)-dimensional NNV equations
About: This article is published in Physica Scripta.The article was published on 2020-08-04. It has received 73 citations till now. The article focuses on the topics: Symmetry (physics).
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TL;DR: In this article, the Lie group of transformation method via one-dimensional optimal system is proposed to obtain some more exact solutions of the (4+1)-dimensional Fokas equation.
Abstract: In this article, the Lie group of transformation method via one-dimensional optimal system is proposed to obtain some more exact solutions of the (4+1)-dimensional Fokas equation. Lie infinitesimal generators, possible vector fields, and their commutative and adjoint relations are presented by employing the Lie symmetry method. An optimal system of the one-dimensional subalgebras is also constructed using Lie vectors. Meanwhile, based on the optimal system, Lie symmetry reductions of the Fokas equation is obtained. A repeated process of Lie symmetry reductions, using the single, double, triple, quadruple, and quintuple combinations between the considered vectors, transforms the Fokas equation into nonlinear ordinary differential equations which produce abundant group-invariant solutions. The same problem was studied by Sadat et al. (Chaos Solitons Fractals 140:110134, 2020) using the same Lie symmetry technique via commutative product approach but with the less number of vector fields and therefore could obtain only three exact solutions as compared to the number of analytic solutions in this paper. In order to provide rich localized structures, some solutions are supplemented via numerical simulation, which produces some breather-type solitons, oscillating multi-solitons on the parabolic-shaped surface, fractal dromions, lump-type solitons, and annihilation of different parabolic multi-solitons profiles. The dynamical behaviors of excitation-localized structures are demonstrated graphically via 3D plots for suitable values of the arbitrary free parameters and independent arbitrary functions.
84 citations
TL;DR: In this article, the variable coefficients fifth-order nonlinear Schrodinger equation (NLS), which can be used to describe the transmission of femtosecond pulse in the optical fiber, is studied.
Abstract: Optical fiber communication has developed rapidly because of the needs of the information age. Here, the variable coefficients fifth-order nonlinear Schrodinger equation (NLS), which can be used to describe the transmission of femtosecond pulse in the optical fiber, is studied. By virtue of the Hirota method, we get the one-soliton and two-soliton solutions. Interactions between solitons are presented, and the soliton stability is discussed through adjusting the values of dispersion and nonlinear effects. Results may potentially be useful for optical communications such as all-optical switches or the study of soliton control.
80 citations
66 citations
TL;DR: In this paper, the authors explore the new exact-soliton solutions to the higher-dimensional nonlinear Fokas equation and (2+1)-dimensional Breaking soliton equations via a generalized exponential rational function (GERF) method.
Abstract: The prime objective of this paper is to explore the new exact-soliton solutions to the higher-dimensional nonlinear Fokas equation and (2+1)-dimensional Breaking soliton equations via a generalized exponential rational function (GERF) method. Many different kinds of exact-soliton solutions are obtained, all of which are completely novel and have never been reported in the literature before. The dynamical behaviors of some obtained exact-soliton solutions are also exhibited via choosing the appropriate {values} of the free constants that aid in understanding the nonlinear complex phenomenon of such equations. These exact-soliton solutions are observed in the shapes of different dynamical structures of localized solitary wave solutions, singular form solitons, single-solitons, double-solitons, triple-solitons, bell-shaped solitons, combo singular solitons, breather-type solitons, elastic interaction between triple-solitons with kink-waves, elastic interaction between diverse solitons with kink-waves. With the help of less symbolic computation work and more constructed closed-form solutions, it is observed that the existing suggested technique is effective, robust, and straightforward. Moreover, several other these types of higher-dimensional NLEEs can be solved by utilizing the powerful GERF technique.
39 citations
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TL;DR: An exact solution for the Korteweg-de Vries equation for the case of multiple collisions of $N$ solitons with different amplitudes was obtained in this paper, which is the only known exact solution.
Abstract: An exact solution has been obtained for the Korteweg---de Vries equation for the case of multiple collisions of $N$ solitons with different amplitudes.
2,637 citations
TL;DR: In this article, a new method, called Exp-function method, is proposed to seek solitary solutions, periodic solutions and compacton-like solutions of nonlinear differential equations, and the modified KdV equation and Dodd-Bullough-Mikhailov equation are chosen to illustrate the effectiveness and convenience of the suggested method.
Abstract: In this paper, a new method, called Exp-function method, is proposed to seek solitary solutions, periodic solutions and compacton-like solutions of nonlinear differential equations. The modified KdV equation and Dodd–Bullough–Mikhailov equation are chosen to illustrate the effectiveness and convenience of the suggested method.
1,718 citations
24 Jun 2002
TL;DR: Backlund-Darboux transformations with their remarkable associated nonlinear superposition principles and importance in soliton theory have been explored in this article, where the authors also explore the extensive body of literature from the nineteenth and early twentieth centuries by such eminent geometers as Bianchi, Darboux, Backlund, and Eisenhart on transformations of privileged classes of surfaces which leave key geometric properties unchanged.
Abstract: This book describes the remarkable connections that exist between the classical differential geometry of surfaces and modern soliton theory. The authors also explore the extensive body of literature from the nineteenth and early twentieth centuries by such eminent geometers as Bianchi, Darboux, Backlund, and Eisenhart on transformations of privileged classes of surfaces which leave key geometric properties unchanged. Prominent amongst these are Backlund-Darboux transformations with their remarkable associated nonlinear superposition principles and importance in soliton theory. It is with these transformations and the links they afford between the classical differential geometry of surfaces and the nonlinear equations of soliton theory that the present text is concerned. In this geometric context, solitonic equations arise out of the Gaus-Mainardi-Codazzi equations for various types of surfaces that admit invariance under Backlund-Darboux transformations. This text is appropriate for use at a higher undergraduate or graduate level for applied mathematicians or mathematical physics.
835 citations
TL;DR: The Exp-function method is used to obtain generalized solitonary solutions and periodic solutions for nonlinear evolution equations arising in mathematical physics using symbolic computation as discussed by the authors, which is straightforward and concise, and its applications are promising.
Abstract: The Exp-function method is used to obtain generalized solitonary solutions and periodic solutions for nonlinear evolution equations arising in mathematical physics using symbolic computation. The method is straightforward and concise, and its applications are promising.
487 citations
TL;DR: In this article, an Exp-function method is used to find a unified solution of a nonlinear wave equation, and a generalized solitary solution with free parameters is obtained. But this method is not suitable for the case of non-linear wave equations.
Abstract: Exp-function method is used to find a unified solution of a nonlinear wave equation. Variant Boussinesq equations are selected to illustrate the effectiveness and simplicity of the method. A generalized solitary solution with free parameters is obtained.
221 citations