Multiple Lump Solutions of the (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mn>4</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:math>)-Dimensional Fokas Equation

Ma, Hongcai; Bai, Yunxiang; Deng, Aiping

doi:10.1155/2020/3407676

Cited by 13 publications

(8 citation statements)

References 33 publications

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“…Recently, Clarkson and Dowie solves the (1+1)-dimensional Boussinesq equation using the considered approach [44], also it is reported recently that the zeros of these polynomials have interesting patterns and the integral of the solutions representations are also well behaved [44,52]. The considered approach is further extended to higher dimensional soliton equation in (2+1)-D, (3+1)-D, (4+1)-D, and its nonlocal Alice Bob equivalent models [45,53,54,55]. In [55], it was predicted that the method is applicable for soliton equation whose bilinear form is free from mixed Hirota D-operators, but recently this approach is used for nonlinear models whose consist of mixed Hirota D-operators [56].…”

Section: Methodology To Construct Higher-order Rogue Wave Solutionsmentioning

confidence: 99%

Painlevé Analysis and Higher-Order Rogue Waves of a Generalized(3+1)-dimensional Shallow Water Wave Equation

Singh,

Sakkaravarthi,

Tamizhmani

et al. 2021

Preprint

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Considering the importance of ever-increasing interest in exploring localized waves, we investigate a generalized (3+1)-dimensional Hirota-Satsuma-Ito equation describing the unidirectional propagation of shallow-water waves and perform Painlevé analysis to understand its integrability nature. We construct the explicit form of higher-order rogue wave solutions by adopting Hirota's bilinearization and generalized polynomial functions. Further, we explore their dynamics in detail, depicting different pattern formation that reveal potential advantages with available arbitrary constants in their manipulation mechanism. Particularly, we demonstrate the existence of singly-localized line-rogue waves and doubly-localized rogue waves with multiple (single, triple, and sextuple) structures generating triangular and pentagon type geometrical patterns with controllable orientations that can be altered appropriately by tuning the parameters. The presented analysis will be an essential inclusion in the context of rogue waves in higher-dimensional systems.

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Section: Methodology To Construct Higher-order Rogue Wave Solutionsmentioning

confidence: 99%

Painlevé Analysis and Higher-Order Rogue Waves of a Generalized(3+1)-dimensional Shallow Water Wave Equation

Singh,

Sakkaravarthi,

Tamizhmani

et al. 2021

Preprint

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“…The explict solutions of nonlinear PDE play an important role in nonlinear science and engineering, and there are many kinds of effective methods that have been established to construct explict solutions for nonlinear equations, such as symmetry reductions [30,31], bilinear method [41][42][43], Darboux transformation [44], and Painlevé analysis [45], but most of them are more difficult to find interaction solutions which are an important and meaningful research topic [46][47][48]. Fortunately, the above consistent tanh expansion (CTE) method can be easily applied to investigate interaction solutions between a soliton and any other types of nonlinear excitations.…”

Section: Explict Solution To the Higher-order Broer-kaup Systemmentioning

confidence: 99%

Residual Symmetry, Bäcklund Transformation, and Soliton Solutions of the Higher-Order Broer-Kaup System

Xia

Yao

Xin

2021

Advances in Mathematical Physics

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Under investigation in this paper is the higher-order Broer-Kaup(HBK) system, which describes the bidirectional propagation of long waves in shallow water. Via the standard truncated Painlevé expansion method, the residual symmetry of this system is derived. By introducing an appropriate auxiliary-dependent variable, the residual symmetry is successfully localized to Lie point symmetries. Via solving the initial value problems, the finite symmetry transformations are presented. However, the solution which obtained from the residual symmetry is a special group invariant solutions. In order to find more general solution of HBK system, we further generalize the residual symmetry method to the consistent tanh expansion (CTE) method and prove that the HBK system is CTE solvable, then the resonant soliton solutions and interaction solutions among different nonlinear excitations are obtained by the CET method.

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“…where a 3 a 6 ̸ = 0. Therefore, by comparing equation ( 6) and limiting conditions (10), we can write a type of quadratic function solutions of equation ( 5) Then, substituting quadratic function (11) into transformation (4), we can obtain…”

Section: Lump Solution Of Equationmentioning

confidence: 99%

Lump Solutions, Multi Lump Solutions and More Soliton Solutions of a Novel (2+1)-dimensional Nonlinear Evolution Equation

Yue

Gao

et al. 2021

Preprint

Self Cite

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Exact solutions of a new (2+1)-dimensional nonlinear evolution equation are studied. Through the Hirota bilinear method, the test function method and the improved tanh-coth and tah-cot method, with the assisstance of symbolic operations, one can obtain the lump solutions, multi lump solutions and more soliton solutions. Finally, by determining different parameters, we draw the three-dimensional plots and density plots at different times.

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Multiple Lump Solutions of the (4+1)-Dimensional Fokas Equation

Cited by 13 publications

References 33 publications

Painlevé Analysis and Higher-Order Rogue Waves of a Generalized(3+1)-dimensional Shallow Water Wave Equation

Painlevé Analysis and Higher-Order Rogue Waves of a Generalized(3+1)-dimensional Shallow Water Wave Equation

Residual Symmetry, Bäcklund Transformation, and Soliton Solutions of the Higher-Order Broer-Kaup System

Lump Solutions, Multi Lump Solutions and More Soliton Solutions of a Novel (2+1)-dimensional Nonlinear Evolution Equation

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