Multiwave interaction solutions for a (3+1)-dimensional nonlinear evolution equation

The residual symmetry of a (1 + 1)-dimensional nonlinear evolution equation (NLEE) ut+uxxx−6u2ux+6λux=0 is obtained through Painlevé expansion. By introducing a new dependent variable, the residual symmetry is localized into Lie point symmetry in an enlarged system, and the related symmetry reduction solutions are obtained using the standard Lie symmetry method. Furthermore, the (1 + 1)-dimensional NLEE equation is proved to be integrable in the sense of having a consistent Riccati expansion (CRE), and some new Bäcklund transformations (BTs) are given. In addition, some explicitly expressed solutions including interaction solutions between soliton and cnoidal waves are derived from these BTs.

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POSITON, NEGATON, SOLITON AND COMPLEXITON SOLUTIONS TO A FOUR-DIMENSIONAL NONLINEAR EVOLUTION EQUATION

Modern Physics Letters B ◽

10.1142/s0217984909021053 ◽

2009 ◽

Vol 23 (25) ◽

pp. 2971-2991 ◽

Cited By ~ 13

Author(s):

ZHAQILAO ◽

ZHI-BIN LI

Keyword(s):

Evolution Equation ◽

Exact Solutions ◽

Nonlinear Evolution Equation ◽

Nonlinear Evolution ◽

Direct Method ◽

Sufficient Conditions ◽

Comprehensive Approach ◽

Wronskian Determinant ◽

Interaction Solutions ◽

Complexiton Solutions

A generalized Wronskian formulation is presented for a four-dimensional nonlinear evolution equation. The representative systems are explicitly solved by selecting a broad set of sufficient conditions which make the Wronskian determinant a solution to the bilinearized four-dimensional nonlinear evolution equation. The obtained solution formulas provide us with a comprehensive approach to construct explicit exact solutions to the four-dimensional nonlinear evolution equation, by which positons, negatons, solitons and complexitons are computed for the four-dimensional nonlinear evolution equation. Applying the Hirota's direct method, multi-soliton, non-singular complexiton, and their interaction solutions of the four-dimensional nonlinear evolution equation are also obtained.

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Bäcklund Transformation and Exact Solutions to a Generalized (3 + 1)-Dimensional Nonlinear Evolution Equation

Discrete Dynamics in Nature and Society ◽

10.1155/2022/5598381 ◽

2022 ◽

Vol 2022 ◽

pp. 1-12

Author(s):

Yali Shen ◽

Ying Yang

Keyword(s):

Evolution Equation ◽

Nonlinear Evolution Equation ◽

Nonlinear Evolution ◽

Bäcklund Transformation ◽

Dynamic Properties ◽

Traveling Wave Solutions ◽

Mixed Solution ◽

Backlund Transformation ◽

Bilinear Method ◽

Interaction Solutions

In this article, a generalized (3 + 1)-dimensional nonlinear evolution equation (NLEE), which can be obtained by a multivariate polynomial, is investigated. Based on the Hirota bilinear method, the N-soliton solution and bilinear Bäcklund transformation (BBT) with explicit formulas are successfully constructed. By using BBT, two traveling wave solutions and a mixed solution of the generalized (3 + 1)-dimensional NLEE are obtained. Furthermore, the lump and the interaction solutions for the equation are constructed. Finally, the dynamic properties of the lump and the interaction solutions are described graphically.

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Lump Solutions and Interaction Solutions for the Dimensionally Reduced Nonlinear Evolution Equation

Complexity ◽

10.1155/2019/5765061 ◽

2019 ◽

Vol 2019 ◽

pp. 1-9 ◽

Cited By ~ 2

Author(s):

Baoyong Guo ◽

Huanhe Dong ◽

Yong Fang

Keyword(s):

Evolution Equation ◽

Nonlinear Evolution Equation ◽

Nonlinear Evolution ◽

Quadratic Functions ◽

Bilinear Method ◽

Lump Solutions ◽

Dynamical Characteristics ◽

Interaction Solutions ◽

Physical Quantities ◽

Hirota Bilinear

In this paper, by means of the Hirota bilinear method, a dimensionally reduced nonlinear evolution equation is investigated. Through its bilinear form, lump solutions are obtained. We construct interaction solutions between lump solutions and one soliton solution by choosing quadratic functions and exponential function. Interaction solutions with the combinations of exponential functions and sine function are also given. Meanwhile, the figures of these solutions are plotted. The dynamical characteristics and properties of obtained solutions are discussed, respectively. The results show that the corresponding physical quantities and properties of nonlinear waves are associated with the values of the parameters.

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Dynamics of localized waves in a (3+1)-dimensional nonlinear evolution equation

Modern Physics Letters B ◽

10.1142/s021798491950101x ◽

2019 ◽

Vol 33 (09) ◽

pp. 1950101 ◽

Cited By ~ 2

Author(s):

Yunfei Yue ◽

Yong Chen

Keyword(s):

Evolution Equation ◽

Soliton Solution ◽

Nonlinear Evolution Equation ◽

Nonlinear Evolution ◽

Rogue Wave ◽

Soliton Solutions ◽

Rogue Waves ◽

Dark Soliton ◽

Interaction Solutions ◽

Localized Waves

In this paper, a (3[Formula: see text]+[Formula: see text]1)-dimensional nonlinear evolution equation is studied via the Hirota method. Soliton, lump, breather and rogue wave, as four types of localized waves, are derived. The obtained N-soliton solutions are dark solitons with some constrained parameters. General breathers, line breathers, two-order breathers, interaction solutions between the dark soliton and general breather or line breather are constructed by choosing suitable parameters on the soliton solution. By the long wave limit method on the soliton solution, some new lump and rogue wave solutions are obtained. In particular, dark lumps, interaction solutions between dark soliton and dark lump, two dark lumps are exhibited. In addition, three types of solutions related with rogue waves are also exhibited including line rogue wave, two-order line rogue waves, interaction solutions between dark soliton and dark lump or line rogue wave.

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