Soliton molecules in the (2+1)-dimensional Nizhnik–Novikov–Veselov equation

2021 ◽  
pp. 2150367
Author(s):  
Huiling Wu ◽  
Jinxi Fei ◽  
Zhengyi Ma
Keyword(s):  
Soliton Solution ◽  
Elastic Interaction ◽  
Wave Number ◽  
Explicit Solutions ◽  
Variable Formula ◽  
Structure Interaction ◽  
Molecule Structure ◽  
Long Wave ◽  
Wave Limit ◽  
Line Soliton

The [Formula: see text]-soliton solution of the (2+1)-dimensional Nizhnik–Novikov–Veselov equation is constructed. The line soliton molecule, the breather and the lump soliton are presented successively for [Formula: see text]. The three-soliton molecule structure, interaction of one-soliton molecule and a line soliton, the soliton molecules consisting of a line soliton and the breather/lump soliton of the solution [Formula: see text] are constructed for [Formula: see text]. Moreover, the four-soliton molecule structure, interaction of the soliton molecule and a line soliton, the soliton molecule consisting of the line soliton molecule and a lump soliton, the elastic interaction between the line soliton molecule and a lump soliton, the soliton molecules consisting of the line soliton molecule and the breather, two breather solitons, the breather soliton and a lump of the variable [Formula: see text] for this equation are also derived for [Formula: see text] by applying the velocity resonance, the module resonance of wave number and the long-wave limit ideas. To illustrate these phenomena, the analysis explicit solutions are all given and their dynamics features are all displayed through figures.

Soliton molecules and some related interaction solutions of the (2+1)-dimensional Kadomtsev–Petviashvili hierarchy

2020 ◽  
pp. 2150106
Author(s):  
Jiamin Zhu ◽  
Bo Wang ◽  
Zhengyi Ma ◽  
Jinxi Fei
Keyword(s):  
Resonance Condition ◽  
Soliton Solutions ◽  
Elastic Interaction ◽  
Elastic Collision ◽  
Wave Number ◽  
Interaction Solutions ◽  
Long Wave ◽  
Text Interaction ◽  
Wave Limit ◽  
Line Soliton

The [Formula: see text]-soliton solutions of the (2+1)-dimensional Kadomtsev–Petviashvili hierarchy are first constructed. One soliton molecule satisfies the velocity resonance condition, the breather with the periodic solitary wave, the lump soliton localized in all directions in the space are showed successively for [Formula: see text]. Interaction of one soliton molecule and a line soliton, the soliton molecule hybrid a line soliton with the breather/lump soliton are presented for [Formula: see text]. Moreover, the elastic interaction between two-soliton molecules, the interaction between one soliton molecule, and a breather and the elastic collision between the lump soliton and one soliton molecule are also derived for [Formula: see text] by applying the velocity resonance, the module resonance of wave number, and the long-wave limit ideas. Figures are presented to demonstrate these dynamics features.

scholarly journals Soliton Molecules and Some Novel Types of Hybrid Solutions to (2 + 1)-Dimensional Variable-Coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Shuxin Yang ◽  
Zhao Zhang ◽  
Biao Li
Keyword(s):  
Variable Coefficient ◽  
Resonance Condition ◽  
Soliton Solutions ◽  
Wave Number ◽  
Long Wave ◽  
Hybrid Solutions ◽  
Wave Limit ◽  
Dimensional Variable

Soliton molecules of the (2 + 1)-dimensional variable-coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada equation are derived by N-soliton solutions and a new velocity resonance condition. Moreover, soliton molecules can become asymmetric solitons when the distance between two solitons of the molecule is small enough. Finally, we obtained some novel types of hybrid solutions which are components of soliton molecules, lump waves, and breather waves by applying velocity resonance, module resonance of wave number, and long wave limit method. Some figures are presented to demonstrate clearly dynamics features of these solutions.

Diverse solitons and interaction solutions for the (2+1)-dimensional CDGKS equation

2019 ◽  
Vol 33 (16) ◽  
pp. 1950174 ◽  
Author(s):  
Jian-Hong Zhuang ◽  
Yaqing Liu ◽  
Xin Chen ◽  
Juan-Juan Wu ◽  
Xiao-Yong Wen
Keyword(s):  
Soliton Solutions ◽  
Propagation Direction ◽  
Soliton Interaction ◽  
Bilinear Method ◽  
Interaction Solutions ◽  
Dynamical Behaviors ◽  
Long Wave ◽  
Wave Limit ◽  
Hirota Bilinear ◽  
Line Soliton

In this paper, the (2[Formula: see text]+[Formula: see text]1)-dimensional CDGKS equation is studied and its diverse soliton solutions consisting of line soliton, periodic soliton and lump soliton with different parameters are derived based on the Hirota bilinear method and long-wave limit method. Based on exact solution formulae with different parameters, the interaction between line soliton and periodic soliton, the interaction between line soliton and lump soliton, as well as the interaction between periodic soliton and lump soliton are illustrated. According to the dynamical behaviors, it can be found that the effects of different parameters are on the propagation direction and shapes. Novel soliton interaction phenomena are also observed.

scholarly journals Lumps, breathers, and interaction solutions of a (3+1)-dimensional generalized Kadovtsev–Petviashvili equation

2020 ◽  
pp. 2150041
Author(s):  
Xi Ma ◽  
Tie-Cheng Xia ◽  
Handong Guo
Keyword(s):  
Soliton Solution ◽  
Hirota Bilinear Method ◽  
Kp Equation ◽  
Bilinear Method ◽  
Test Technique ◽  
Interaction Solutions ◽  
Long Wave ◽  
Petviashvili Equation ◽  
Wave Limit ◽  
Hirota Bilinear

In this paper, we use the Hirota bilinear method to find the [Formula: see text]-soliton solution of a [Formula: see text]-dimensional generalized Kadovtsev–Petviashvili (KP) equation. Then, we obtain the [Formula: see text]-order breathers of the equation, and combine the long-wave limit method to give the [Formula: see text]-order lumps. Resorting to the extended homoclinic test technique, we obtain the breather-kink solutions for the equation. Last, the interaction solution composed of the [Formula: see text]-soliton solution, [Formula: see text]-breathers, and [Formula: see text]-lumps for the [Formula: see text]-dimensional generalized KP equation is constructed.

scholarly journals Solitons, Breathers, and Lump Solutions to the (2 + 1)-Dimensional Generalized Calogero–Bogoyavlenskii–Schiff Equation

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Hongcai Ma ◽  
Qiaoxin Cheng ◽  
Aiping Deng
Keyword(s):  
Exact Solutions ◽  
Soliton Solution ◽  
Dynamic Characteristics ◽  
Hirota Bilinear Method ◽  
Bilinear Method ◽  
Lump Solutions ◽  
Long Wave ◽  
Wave Limit ◽  
Localized Waves ◽  
Hirota Bilinear

In this paper, a generalized (2 + 1)-dimensional Calogero–Bogoyavlenskii–Schiff equation is considered. Based on the Hirota bilinear method, three kinds of exact solutions, soliton solution, breather solutions, and lump solutions, are obtained. Breathers can be obtained by choosing suitable parameters on the 2-soliton solution, and lump solutions are constructed via the long wave limit method. Figures are given out to reveal the dynamic characteristics on the presented solutions. Results obtained in this work may be conducive to understanding the propagation of localized waves.

On periodic water-waves and their convergence to solitary waves in the long-wave limit

1981 ◽  
Vol 303 (1481) ◽  
pp. 633-669 ◽  
Keyword(s):  
Integral Equation ◽  
Solitary Waves ◽  
Water Waves ◽  
Periodic Problem ◽  
Existence Theory ◽  
Periodic Waves ◽  
Equation Formulation ◽  
Long Wave ◽  
Integral Equation Formulation ◽  
Wave Limit

A detailed discussion of Nekrasov’s approach to the steady water-wave problems leads to a new integral equation formulation of the periodic problem. This development allows the adaptation of the methods of Amick & Toland (1981) to show the convergence of periodic waves to solitary waves in the long-wave limit. In addition, it is shown how the classical integral equation formulation due to Nekrasov leads, via the Maximum Principle, to new results about qualitative features of periodic waves for which there has long been a global existence theory (Krasovskii 1961, Keady & Norbury 1978).

On the modulation of water waves on shear flows

1976 ◽  
Vol 347 (1651) ◽  
pp. 537-546 ◽  
Keyword(s):  
Water Waves ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Vries Equation ◽  
Harmonic Wave ◽  
Short Wave ◽  
Long Wave ◽  
Stokes Waves ◽  
The Stability ◽  
Wave Limit

The method of multiple scales is used to examine the slow modulation of a harmonic wave moving over the surface of a two dimensional channel. The flow is assumed inviscid and incompressible, but the basic flow takes the form of an arbitrary shear. The appropriate nonlinear Schrödinger equation is derived with coefficients that depend, in a complicated way, on the shear. It is shown that this equation agrees with previous work for the case of no shear; it also agrees in the long wave limit with the appropriate short wave limit of the Korteweg-de Vries equation, the shear being arbitrary. Finally, it is remarked that the stability of Stokes waves over any shear can be examined by using the results derived here.

N-soliton solutions and localized wave interaction solutions of a (3 + 1)-dimensional potential-Yu–Toda–Sasa–Fukuyamaf equation

2021 ◽  
pp. 2150277
Author(s):  
Hongcai Ma ◽  
Qiaoxin Cheng ◽  
Aiping Deng
Keyword(s):  
Dynamic Behavior ◽  
Soliton Solution ◽  
Soliton Solutions ◽  
Wave Interaction ◽  
Bilinear Transformation ◽  
Interaction Solution ◽  
Interaction Solutions ◽  
Ytsf Equation ◽  
Line Soliton

[Formula: see text]-soliton solutions are derived for a (3 + 1)-dimensional potential-Yu–Toda–Sasa–Fukuyama (YTSF) equation by using bilinear transformation. Some local waves such as period soliton, line soliton, lump soliton and their interaction are constructed by selecting specific parameters on the multi-soliton solutions. By selecting special constraints on the two soliton solutions, period and lump soliton solution can be obtained; three solitons can reduce to the interaction solution between period soliton and line soliton or lump soliton and line soliton under special parameters; the interaction solution among period soliton and two line solitons, or the interaction solution for two period solitons or two lump solitons via taking specific constraints from four soliton solutions. Finally, some images of the results are drawn, and their dynamic behavior is analyzed.

Lumps, breathers, rogue waves and interaction solutions to a (3+1)-dimensional Kudryashov–Sinelshchikov equation

2020 ◽  
Vol 34 (12) ◽  
pp. 2050117 ◽  
Author(s):  
Xianglong Tang ◽  
Yong Chen
Keyword(s):  
Rogue Waves ◽  
Hirota Bilinear Method ◽  
Bilinear Method ◽  
Lump Solutions ◽  
Interaction Solutions ◽  
Long Wave ◽  
Wave Limit ◽  
Hirota Bilinear

Utilizing the Hirota bilinear method, the lump solutions, the interaction solutions with the lump and the stripe solitons, the breathers and the rogue waves for a (3[Formula: see text]+[Formula: see text]1)-dimensional Kudryashov–Sinelshchikov equation are constructed. Two types of interaction solutions between the lumps and the stripe solitons are exhibited. Some different breathers are given by choosing special parameters in the expressions of the solitons. Through a long wave limit of breathers, the lumps and rogue waves are derived.

Lump and interactional solutions of the (2+1)-dimensional generalized breaking soliton equation

2019 ◽  
Vol 34 (03) ◽  
pp. 2050037
Author(s):  
Yu-Pei Fan ◽  
Ai-Hua Chen
Keyword(s):  
Solitary Wave ◽  
Asymptotic Analysis ◽  
Bilinear Form ◽  
Lump Solution ◽  
Soliton Equation ◽  
Long Wave ◽  
Wave Limit

In this paper, by using the long wave limit method, we study lump solution and interactional solution of the (2[Formula: see text]+[Formula: see text]1)-dimensional generalized breaking soliton equation without using bilinear form. The moving properties of the lump solution, and the interactional properties of a lump and a solitary wave, are analyzed theoretically and graphically with asymptotic analysis.

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