Generalized (3+1)-dimensional Boussinesq equation: Breathers, rogue waves and their dynamics

2019 ◽  
Vol 34 (01) ◽  
pp. 2050003 ◽  
Author(s):  
Xue-Wei Yan
Keyword(s):  
Boussinesq Equation ◽  
Rogue Wave ◽  
Rational Solution ◽  
Soliton Solutions ◽  
Rogue Waves ◽  
Bell Polynomial ◽  
Wave Solutions ◽  
Water Based ◽  
Extreme Behavior ◽  
Polynomial Theory

In this work, we consider the generalized (3[Formula: see text]+[Formula: see text]1)-dimensional Boussinesq equation, which can describe the propagation of gravity wave on the surface of water. Based on the Bell polynomial theory, a powerful technique is employed to explicitly construct its bilinear formalism and two-soliton solutions, based on which the new rational solution is well-constructed. Moreover, the extended homoclinic test approach is presented to succinctly construct the breather wave and rogue wave solutions of the Boussinesq equation. Then the main characteristics of these solutions are graphically discussed. More importantly, they reveal that the extreme behavior of the breather wave can give rise to the rogue wave.

scholarly journals On the breather waves, rogue waves and solitary waves to a generalized (2+1)-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada equation

Filomat ◽  
2018 ◽  
Vol 32 (14) ◽  
pp. 4959-4969 ◽  
Author(s):  
Wei-Qi Peng ◽  
Shou-Fu Tian ◽  
Tian-Tian Zhang
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Bilinear Form ◽  
Rogue Wave ◽  
Rogue Waves ◽  
Solitary Wave Solutions ◽  
Bell Polynomial ◽  
Wave Solutions ◽  
Extreme Behavior ◽  
Natural Way

In this paper, we consider a generalized (2+1)-dimensional Caudrey-Dodd-Gibbon-Kotera- Sawada (CDGKS) equation. By using the Bell polynomial, we derive its bilinear form. Based on the homoclinic breather limit method, we construct the homoclinic breather wave and the rational rogue wave solutions of the equation. Then by using its bilinear form, some solitary wave solutions of the CDGKS equation are provided by a very natural way. Moreover, some prominent characteristics for the dynamic behaviors of these solitons are analyzed by several graphics. Our results show that the breather wave can be transformed into rogue wave under the extreme behavior.

Rational and semi-rational solutions for the (3 + 1)-dimensional B-type Kadomtsev–Petviashvili–Boussinesq equation

2019 ◽  
Vol 33 (25) ◽  
pp. 1950296 ◽  
Author(s):  
Ya-Si Deng ◽  
Bo Tian ◽  
Yan Sun ◽  
Chen-Rong Zhang ◽  
Cong-Cong Hu
Keyword(s):  
Water Waves ◽  
Boussinesq Equation ◽  
Rogue Wave ◽  
Soliton Solutions ◽  
Rogue Waves ◽  
Rational Solutions ◽  
Bilinear Method ◽  
Different Shapes ◽  
Lump Wave ◽  
Kink Soliton

Nonlinear waves are seen in nature, such as the water waves and plasma waves. Investigated in this paper is a (3 + 1)-dimensional B-type Kadomtsev–Petviashvili–Boussinesq equation. Based on the bilinear method, we get the rational solutions, which are different from the published ones, semi-rational solutions and breather-type kink soliton solutions. Through the rational solutions, we observe two types of waves: the lump waves and line rogue waves. The semi-rational solutions depict two types of interactions: (1) The fusion or fission between the lump wave and soliton; (2) The interaction between the line rogue wave and soliton. During the interaction between the line rogue wave and soliton, the line rogue wave evolves with three different shapes: the bright rogue waves, bright–dark rogue waves and dark rogue waves. Via the breather-type kink soliton solutions, we observe the breather-soliton mixture.

The twin properties of rogue waves and homoclinic solutions for some nonlinear wave equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Wei Tan ◽  
Zhao-Yang Yin
Keyword(s):  
Differential Equations ◽  
Wave Equations ◽  
Rogue Wave ◽  
Nonlinear Partial Differential Equations ◽  
Rogue Waves ◽  
Homoclinic Solutions ◽  
Nonlinear Wave Equations ◽  
Bilinear Method ◽  
Wave Solutions ◽  
Rogue Wave Solutions

Abstract The parameter limit method on the basis of Hirota’s bilinear method is proposed to construct the rogue wave solutions for nonlinear partial differential equations (NLPDEs). Some real and complex differential equations are used as concrete examples to illustrate the effectiveness and correctness of the described method. The rogue waves and homoclinic solutions of different structures are obtained and simulated by three-dimensional graphics, respectively. More importantly, we find that rogue wave solutions and homoclinic solutions appear in pairs. That is to say, for some NLPDEs, if there is a homoclinic solution, then there must be a rogue wave solution. The twin phenomenon of rogue wave solutions and homoclinic solutions of a class of NLPDEs is discussed.

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.

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.

Multiple rogue and soliton wave solutions to the generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt equation arising in fluid mechanics and plasma physics

2021 ◽  
pp. 2150383
Author(s):  
Onur Alp Ilhan ◽  
Sadiq Taha Abdulazeez ◽  
Jalil Manafian ◽  
Hooshmand Azizi ◽  
Subhiya M. Zeynalli
Keyword(s):  
Rogue Wave ◽  
Three Dimensional ◽  
Soliton Solutions ◽  
The Novel ◽  
Algebraic Equations ◽  
Bilinear Method ◽  
Dynamical Characteristics ◽  
Wave Solutions ◽  
Multiple Soliton Solutions ◽  
Soliton Wave Solutions

Under investigation in this paper is the generalized Konopelchenko–Dubrovsky–Kaup-Kupershmidt equation. Based on bilinear method, the multiple rogue wave (RW) solutions and the novel multiple soliton solutions are constructed by giving some specific activation functions for the considered model. By means of symbolic computation, these analytical solutions and corresponding rogue wave solutions are obtained via Maple 18 software. The exact lump and RW solutions, by solving the under-determined nonlinear system of algebraic equations for the specified parameters, will be constructed. Via various three-dimensional plots and density plots, dynamical characteristics of these waves are exhibited.

Breather wave, rogue wave and lump wave solutions for a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation in fluid

2018 ◽  
Vol 32 (20) ◽  
pp. 1850223 ◽  
Author(s):  
Ming-Zhen Li ◽  
Bo Tian ◽  
Yan Sun ◽  
Xiao-Yu Wu ◽  
Chen-Rong Zhang
Keyword(s):  
Wave Velocity ◽  
Water Waves ◽  
Rogue Wave ◽  
Rogue Waves ◽  
Dispersion Effect ◽  
Wave Solutions ◽  
Weak Perturbation ◽  
Petviashvili Equation ◽  
Nonlinearity Effect ◽  
Lump Wave

Under investigation in this paper is a (3[Formula: see text]+[Formula: see text]1)-dimensional generalized Kadomtsev–Petviashvili equation, which describes the long water waves and small-amplitude surface waves with the weak nonlinearity, weak dispersion and weak perturbation in a fluid. Via the Hirota method and symbolic computation, the lump wave, breather wave and rogue wave solutions are obtained. We graphically present the lump waves under the influence of the dispersion effect, nonlinearity effect, disturbed wave velocity effects and perturbed effects: Decreasing value of the dispersion effect can lead to the range of the lump wave decreases, but has no effect on the amplitude. When the value of the nonlinearity effect or disturbed wave velocity effects increases respectively, lump wave’s amplitude decreases but lump wave’s location keeps unchanged. Amplitudes of the lump waves are independent of the perturbed effects. Breather waves and rogue waves are displayed: Rogue waves emerge when the periods of the breather waves go to the infinity. When the value of the dispersion effect decreases, range of the rogue wave increases. When the value of the nonlinearity effect or disturbed wave velocity effects decreases respectively, rogue wave’s amplitude decreases. Value changes of the perturbed effects cannot influence the rogue wave.

scholarly journals General rogue wave solutions of the coupled Fokas–Lenells equations and non-recursive Darboux transformation

2019 ◽  
Vol 475 (2224) ◽  
pp. 20180806 ◽  
Author(s):  
Yanlin Ye ◽  
Yi Zhou ◽  
Shihua Chen ◽  
Fabio Baronio ◽  
Philippe Grelu
Keyword(s):  
Darboux Transformation ◽  
Rogue Wave ◽  
Rogue Waves ◽  
Background Field ◽  
Double Root ◽  
Transformation Technique ◽  
Wave Dynamics ◽  
Wave Solutions ◽  
Future Experimental Study ◽  
Rogue Wave Solutions

We formulate a non-recursive Darboux transformation technique to obtain the general n th-order rational rogue wave solutions to the coupled Fokas–Lenells system, which is an integrable extension of the noted Manakov system, by considering both the double-root and triple-root situations of the spectral characteristic equation. Based on the explicit fundamental and second-order rogue wave solutions, we demonstrate several interesting rogue wave dynamics, among which are coexisting rogue waves and anomalous Peregrine solitons. Our solutions are generalized to include the complete background-field parameters and therefore helpful for future experimental study.

A variable coefficient nonlinear Schrödinger equation: localized waves on the plane wave background and their dynamics

2019 ◽  
Vol 33 (10) ◽  
pp. 1850121 ◽  
Author(s):  
Xiu-Bin Wang ◽  
Bo Han
Keyword(s):  
Variable Coefficient ◽  
Rogue Wave ◽  
Rogue Waves ◽  
Rational Solutions ◽  
Nls Equation ◽  
Nonlinear Schrödinger ◽  
Wave Solutions ◽  
Plane Wave Background ◽  
Localized Waves ◽  
Similarity Reductions

In this work, a variable coefficient nonlinear Schrödinger (vc-NLS) equation is under investigation, which can describe the amplification or absorption of pulses propagating in an optical fiber with distributed dispersion and nonlinearity. By means of similarity reductions, a similar transformation helps us to relate certain class of solutions of the standard NLS equation to the solutions of integrable vc-NLS equation. Furthermore, we analytically consider nonautonomous breather wave, rogue wave solutions and their interactions in the vc-NLS equation, which possess complicated wave propagation in time and differ from the usual breather waves and rogue waves. Finally, the main characteristics of the rational solutions are graphically discussed. The parameters in the solutions can be used to control the shape, amplitude and scale of the rogue waves.

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.

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