Dynamics of abundant solutions to the generalized (3+1)-dimensional B-type Kadomtsev–Petviashvili equation

2020 ◽  
pp. 2150110
Author(s):  
Huanhuan Lu ◽  
Yufeng Zhang
Keyword(s):  
Soliton Solutions ◽  
Point Of View ◽  
Complex Conjugate ◽  
Bell Polynomial ◽  
Localized Solutions ◽  
Petviashvili Equation ◽  
Hirota Bilinear Form ◽  
Hybrid Solutions ◽  
Wave Limit ◽  
Hirota Bilinear

The present paper is devoted to discussion of (3+1)-dimensional generalized B-type Kadomtsev–Petviashvili (gBKP) equation from point of view of their multi-soliton solutions and localized solutions associated with multi-soliton solutions. Firstly, taking advantage of the Bell-polynomial approach, we construct the Hirota bilinear form of (1.1). Based on that, the multi-soliton solutions are also singled out. Subsequently, the (3+1)-dimensional gBKP equation is also found to allow fruitful localized solutions, including breather, lump, rogue wave, and hybrid solutions. These results obtained in this work adequately illustrate the effectiveness of the long wave limit method and complex conjugate technique, which are expected to be employed to obtain more abundant exact solutions.

Characteristics of the breather waves, lump waves and semi-rational solutions in a generalized (2+1)-dimensional asymmetrical Nizhnik–Novikov–Veselov equation

2019 ◽  
Vol 33 (28) ◽  
pp. 1950350 ◽  
Author(s):  
Wei-Qi Peng ◽  
Shou-Fu Tian ◽  
Tian-Tian Zhang
Keyword(s):  
Bilinear Form ◽  
Rational Solutions ◽  
Bell Polynomial ◽  
Long Wave ◽  
Hirota Bilinear Form ◽  
Wave Limit ◽  
Parameter Constraints ◽  
Hirota Bilinear

In this work, we study a generalized (2[Formula: see text]+[Formula: see text]1)-dimensional asymmetrical Nizhnik–Novikov–Veselov (NNV) equation. Its Hirota bilinear form is constructed via the Bell polynomial. Based on the obtained bilinear form, the Nth-order breather waves are derived explicitly under certain parameter constraints. Moreover, we generate the nonsingular Nth-order lump waves through applying the long wave limit method. Additionally, we successfully present the semi-rational waves containing the combination of lump waves and single-soliton waves, the combination of lump waves and breather waves.

Soliton molecules and some novel hybrid solutions for (3+1)-dimensional B-type Kadomtsev–Petviashvili equation

2021 ◽  
pp. 2150388
Author(s):  
Hongcai Ma ◽  
Huaiyu Huang ◽  
Aiping Deng
Keyword(s):  
Soliton Solutions ◽  
Dynamic Graphs ◽  
Bilinear Method ◽  
Resonance Mechanism ◽  
Long Wave ◽  
Petviashvili Equation ◽  
Hybrid Solutions ◽  
Wave Limit ◽  
Bkp Equation

In recent years, soliton molecules have received reinvigorating scientific interests in physics and other fields. Soliton molecules have been successfully found in optical experiments. In this paper, we attribute the solutions of the (3+1)-dimensional B-type Kadomtsev–Petviashvili (BKP) equation by employing the bilinear method. Based on the [Formula: see text]-soliton solutions, we establish the soliton molecules, asymmetric solitons and some novel hybrid solutions of this equation by means of the velocity resonance mechanism and the long wave limit method. Finally, we give dynamic graphs of soliton molecules, asymmetric solitons and some novel hybrid solutions.

Multiple breathers and high-order rational solutions of the (3+1)-dimensional B-type Kadomtsev–Petviashvili equation

2021 ◽  
Author(s):  
Na Liu ◽  
Xinhua Tang ◽  
Weiwei Zhang
Keyword(s):  
Soliton Solutions ◽  
High Order ◽  
Rational Solutions ◽  
Bilinear Method ◽  
Dynamical Characteristics ◽  
Interaction Solutions ◽  
Petviashvili Equation ◽  
Wave Limit ◽  
Hirota Bilinear ◽  
Bkp Equation

This paper is devoted to obtaining the multi-soliton solutions, high-order breather solutions and high-order rational solutions of the (3+1)-dimensional B-type Kadomtsev–Petviashvili (BKP) equation by applying the Hirota bilinear method and the long-wave limit approach. Moreover, the interaction solutions are constructed by choosing appropriate value of parameters, which consist of four waves for lumps, breathers, rouges and solitons. Some dynamical characteristics for the obtained exact solutions are illustrated using figures.

Localized waves and interaction solutions to an extended (3+1)-dimensional Kadomtsev–Petviashvili equation

2020 ◽  
Vol 34 (06) ◽  
pp. 2050076 ◽  
Author(s):  
Han-Dong Guo ◽  
Tie-Cheng Xia ◽  
Wen-Xiu Ma
Keyword(s):  
Three Dimensional ◽  
Soliton Solutions ◽  
Rogue Waves ◽  
Dark Soliton ◽  
Complex Conjugate ◽  
Bilinear Method ◽  
Interaction Solutions ◽  
Petviashvili Equation ◽  
Parameter Constraints ◽  
Localized Waves

In this paper, an extended (3[Formula: see text]+[Formula: see text]1)-dimensional Kadomtsev–Petviashvili (KP) equation is studied via the Hirota bilinear derivative method. Soliton, breather, lump and rogue waves, which are four types of localized waves, are obtained. N-soliton solution is derived by employing bilinear method. Then, line or general breathers, two-order line or general breathers, interaction solutions between soliton and line or general breathers are constructed by complex conjugate approach. These breathers own different dynamic behaviors in different planes. Taking the long wave limit method on the multi-soliton solutions under special parameter constraints, lumps, two- and three-lump and interaction solutions between dark soliton and dark lump are constructed, respectively. Finally, dark rogue waves, dark two-order rogue waves and related interaction solutions between dark soliton and dark rogue waves or dark lump are also demonstrated. Moreover, dynamical characteristics of these localized waves and interaction solutions are further vividly demonstrated through lots of three-dimensional graphs.

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.

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.

High-order breathers, lumps and hybrid solutions to the (2+1)-dimensional fifth-order KdV equation

2019 ◽  
Vol 33 (22) ◽  
pp. 1950255 ◽  
Author(s):  
Wen-Tao Li ◽  
Zhao Zhang ◽  
Xiang-Yu Yang ◽  
Biao Li
Keyword(s):  
Kdv Equation ◽  
Perturbation Expansion ◽  
Soliton Solutions ◽  
High Order ◽  
Bilinear Method ◽  
Long Wave ◽  
Hybrid Solutions ◽  
Wave Limit ◽  
Fifth Order ◽  
Parameter Constraints

In this paper, the (2+1)-dimensional fifth-order KdV equation is analytically investigated. By using Hirota’s bilinear method combined with perturbation expansion, the high-order breather solutions of the fifth-order KdV equation are generated. Then, the high-order lump solutions are also derived from the soliton solutions by a long-wave limit method and some suitable parameter constraints. Furthermore, we extend this method to obtain hybrid solutions by taking long-wave limit for partial soliton solutions. Finally, the dynamic behavior of these solutions is presented in the figures.

scholarly journals Lump and Lump-Type Solutions of the Generalized (3+1)-Dimensional Variable-Coefficient B-Type Kadomtsev-Petviashvili Equation

2019 ◽  
Vol 2019 ◽  
pp. 1-5 ◽  
Author(s):  
Yanni Zhang ◽  
Jing Pang
Keyword(s):  
Symbolic Computation ◽  
Bilinear Form ◽  
Variable Coefficient ◽  
Three Dimensional ◽  
Petviashvili Equation ◽  
Hirota Bilinear Form ◽  
Contour Plots ◽  
Dimensional Variable ◽  
Hirota Bilinear

Based on the Hirota bilinear form of the generalized (3+1)-dimensional variable-coefficient B-type Kadomtsev-Petviashvili equation, the lump and lump-type solutions are generated through symbolic computation, whose analyticity can be easily achieved by taking special choices of the involved parameters. The property of solutions is investigated and exhibited vividly by three-dimensional plots and contour plots.

Soliton solutions of the Korteweg de Vries and the Kadomtsev-Petviashvili equations: the Wronskian technique

1983 ◽  
Vol 389 (1797) ◽  
pp. 319-329 ◽  
Keyword(s):  
Evolution Equation ◽  
Soliton Solution ◽  
Soliton Solutions ◽  
Inverse Scattering Method ◽  
Compact Representation ◽  
Scattering Method ◽  
Wronskian Technique ◽  
Hirota Bilinear Form ◽  
Usual Representation ◽  
Hirota Bilinear

The well known soliton solutions of the Kadomtsev-Petviashvili equations are written in terms of determinants of Wronskian form. By using this compact representation together with the Hirota bilinear form of the equations, it is demonstrated by elementary algebraic methods that the N -soliton solution satisfies the evolution equation and the N and N + 1-soliton solutions satisfy the associated Bäcklund transformation. The relation of these results to the eigensolutions of the inverse scattering method and to the more usual representation of the N -soliton solution is also given.

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