scholarly journals Controllable symmetry breaking solutions for a nonlocal Boussinesq system

2019 ◽  
Vol 9 (1) ◽  
Author(s):  
Jinxi Fei ◽  
Zhengyi Ma ◽  
Weiping Cao
Keyword(s):  
Symmetry Breaking ◽  
Boussinesq Equation ◽  
Rogue Wave ◽  
Symmetry Transformation ◽  
Boussinesq System ◽  
Residual Symmetry ◽  
Hirota Bilinear Form ◽  
Wave Symmetry ◽  
Generalized Boussinesq Equation ◽  
Hirota Bilinear

AbstractThe generalized Boussinesq equation is a useful model to describe the water wave. In this paper, with the coupled Alice-Bob (AB) systems, the nonlocal Boussinesq system can be obtained via the parity and time reversal symmetry reduction. By introducing an extended Bäcklund transformation, the symmetry breaking rogue wave, symmetry breaking soliton and symmetry breaking breather solutions for a nonlocal Boussinesq system are obtained through the derived Hirota bilinear form. The residual symmetry and finite symmetry transformation of the nonlocal AB-Boussinesq system are also studied.

Analytic rogue wave-type solutions for the generalized (2+1)-dimensional Boussinesq Equation

2021 ◽  
pp. 2150380
Author(s):  
Xiu-Rong Guo
Keyword(s):  
Boussinesq Equation ◽  
Rogue Wave ◽  
Wave Interaction ◽  
Rogue Waves ◽  
Rational Solutions ◽  
Wave Type ◽  
Hirota Bilinear Form ◽  
Basic Facts ◽  
2D And 3D ◽  
Solitary Wave Interaction

Based on the Hirota bilinear form of the generalized (2+1)-dimensional Boussinesq equation, which can be expressed as the shallow water wave mechanism appearing in fluid mechanics, we applied the new polynomial functions to construct the rational solutions and rogue wave-type solutions. Next, the system parameters control on the rational solutions and rogue wave-type solutions were also shown. As a result, we found the following basic facts: (i) these parameters may affect the wave shapes, amplitude, and bright/dark for this considered equation, (ii) the solitary wave interaction rogue waves and triplet rogue wave-type solutions can be viewed on [Formula: see text], [Formula: see text], and [Formula: see text] planes, respectively. Their nonlinear dynamic behaviors were presented by numerical simulation of the 2D- and 3D-plots.

Rogue Waves and Hybrid Solutions of the Boussinesq Equation

2017 ◽  
Vol 72 (4) ◽  
pp. 307-314 ◽  
Author(s):  
Ji-Guang Rao ◽  
Yao-Bin Liu ◽  
Chao Qian ◽  
Jing-Song He
Keyword(s):  
Boussinesq Equation ◽  
Rogue Wave ◽  
Three Dimensional ◽  
Rogue Waves ◽  
Rational Solutions ◽  
Bilinear Method ◽  
Long Wave ◽  
Hybrid Solutions ◽  
Wave Limit ◽  
Hirota Bilinear

AbstractThe rational and semirational solutions in the Boussinesq equation are obtained by the Hirota bilinear method and long wave limit. It is shown that the rational solutions contain dark and bright rogue waves, and their typical dynamics are analysed and illustrated. The semirational solutions possess a range of hybrid solutions, and the hybrid of rogue wave and solitons are demonstrated in detail by the three-dimensional figures. Under certain parameter conditions, a new kind of semirational solutions consisted of rogue waves, breathers and solitons is discovered, which describes the dynamics of the rogue waves interacting with the breathers and solitons at the same time.

A Two-dimensional Boussinesq equation for water waves and some of its solutions

1996 ◽  
Vol 323 ◽  
pp. 65-78 ◽  
Author(s):  
R. S. Johnson
Keyword(s):  
Soliton Solution ◽  
Water Waves ◽  
Boussinesq Equation ◽  
Kdv Equation ◽  
Boussinesq Equations ◽  
Parametric Representation ◽  
Two Dimensional ◽  
Completely Integrable ◽  
Hirota Bilinear Form ◽  
Hirota Bilinear

A two-dimensional Boussinesq equation, \[u_{tt} - u_{xx} + 3(u^2)_{xx} - u_{xxxx} - u_{yy} = 0,\] is introduced to describe the propagation of gravity waves on the surface of water, in particular the head-on collision of oblique waves. This equation combines the two-way propagation of the classical Boussinesq equation with the (weak) dependence on a second spatial variable, as occurs in the two-dimensional Korteweg-de Vries (2D KdV) (or KPII) equation. Exact and general solitary-wave, two-soliton and resonant solutions are obtained from the Hirota bilinear form of the equation. The existence of a distributed-soliton solution is investigated, but it is shown that this is not a possibility. However the connection with the classical 2D KdV equation (which does possess such a solution) is explored via a suitable parametric representation of the dispersion relation.A three-soliton solution is also constructed, but this exists only if an auxiliary constraint among the six parameters is satisfied; thus the two-dimensional Boussinesq equation is not one of the class of completely integrable equations, confirming the analysis of Hietarinta (1987). This constraint is automatically satisfied for the classical Boussinesq equation (which is completely integrable). Graphical reproductions of some of the solutions of the two-dimensional Boussinesq equations are also presented.

Lump, rogue wave, multi-waves and Homoclinic breather solutions for (2+1)-Modified Veronese Web equation

2021 ◽  
Vol 35 (04) ◽  
pp. 2150055
Author(s):  
S. T. R. Rizvi ◽  
Aly R. Seadawy ◽  
S. Ahmed ◽  
M. Younis ◽  
K. Ali
Keyword(s):  
Rogue Wave ◽  
Nonlinear Partial Differential Equation ◽  
Quadratic Function ◽  
Wave Transformation ◽  
Physical Parameters ◽  
Wave Solutions ◽  
Hirota Bilinear Form ◽  
Wave Phenomena ◽  
Hirota Bilinear ◽  
Linearly Degenerate

This work addresses the four main inducements: Lump, rogue wave, Homoclinic breather and multi-wave solutions for (2+1)-Modified Veronese Web (MVW) equation via Hirota bilinear approach and the ansatz technique. This model is a linearly degenerate integrable nonlinear partial differential equation (NLPDE) and can also be used to admit a differential covering with nonremoval physical parameters. By assuming the function [Formula: see text] in the Hirota bilinear form of the presented model as the general quadratic function, trigonometric function and exponential function form, also with appropriate set of parameters, we have prevented the lump, rogue wave, breather and multi-wave solutions successfully. A precise compatible wave transformation is utilized to obtain multi-wave solutions of governing model. Also, the motion track of the lump, Rogue wave and multi-waves is also explained both physically and theoretically. These new results contain some special arbitrary constants that can be useful to spell out diversity in qualitative features of wave phenomena.

scholarly journals Abundant Symmetry-Breaking Solutions of the Nonlocal Alice–Bob Benjamin–Ono System

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Wang Shen ◽  
Zheng-Yi Ma ◽  
Jin-Xi Fei ◽  
Quan-Yong Zhu ◽  
Jun-Chao Chen
Keyword(s):  
Symmetry Breaking ◽  
Gravity Waves ◽  
Bilinear Form ◽  
Time Reversal ◽  
Internal Gravity Waves ◽  
Explicit Solutions ◽  
Time Reversal Symmetry ◽  
Lump Solutions ◽  
Hirota Bilinear Form ◽  
Hirota Bilinear

The Benjamin–Ono equation is a useful model to describe the long internal gravity waves in deep stratified fluids. In this paper, the nonlocal Alice–Bob Benjamin–Ono system is induced via the parity and time-reversal symmetry reduction. By introducing an extended Bäcklund transformation, the symmetry-breaking soliton, breather, and lump solutions for this system are obtained through the derived Hirota bilinear form. By taking suitable constants in the involved ansatz functions, abundant fascinating symmetry-breaking structures of the related explicit solutions are shown.

scholarly journals Symmetry Breaking Soliton, Breather, and Lump Solutions of a Nonlocal Kadomtsev–Petviashvili System

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Hong-Yu Wu ◽  
Jin-Xi Fei ◽  
Zheng-Yi Ma ◽  
Jun-Chao Chen ◽  
Wen-Xiu Ma
Keyword(s):  
Symmetry Breaking ◽  
Nonlinear Waves ◽  
Time Reversal ◽  
Space Variable ◽  
Dispersive Media ◽  
Explicit Solutions ◽  
Lump Solutions ◽  
Petviashvili Equation ◽  
Hirota Bilinear Form ◽  
Hirota Bilinear

The Kadomtsev–Petviashvili equation is one of the well-studied models of nonlinear waves in dispersive media and in multicomponent plasmas. In this paper, the coupled Alice-Bob system of the Kadomtsev–Petviashvili equation is first constructed via the parity with a shift of the space variable x and time reversal with a delay. By introducing an extended Bäcklund transformation, symmetry breaking soliton, symmetry breaking breather, and symmetry breaking lump solutions for this system are presented through the established Hirota bilinear form. According to the corresponding constants in the involved ansatz function, a few fascinating symmetry breaking structures of the presented explicit solutions are shown.

Abundant rogue wave solutions for the (2 + 1)-dimensional generalized Korteweg–de Vries equation

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Huanhuan Lu ◽  
Yufeng Zhang
Keyword(s):  
Vries Equation ◽  
Rogue Wave ◽  
Rogue Waves ◽  
Third Order ◽  
First Order ◽  
Wave Solutions ◽  
De Vries ◽  
Rogue Wave Solutions ◽  
Bilinear Formulation ◽  
Hirota Bilinear

AbstractIn this paper, we analyse two types of rogue wave solutions generated from two improved ansatzs, to the (2 + 1)-dimensional generalized Korteweg–de Vries equation. With symbolic computation, the first-order rogue waves, second-order rogue waves, third-order rogue waves are generated directly from the first ansatz. Based on the Hirota bilinear formulation, another type of one-rogue waves and two-rogue waves can be obtained from the second ansatz. In addition, the dynamic behaviours of obtained rogue wave solutions are illustrated graphically.

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