Two-dimensional new coherent structures of lump-soliton solutions for the Mel’nikov equation

2021 ◽  
pp. 2150416
Author(s):  
Wei Liu ◽  
Yuan Meng ◽  
Xiaoyan Qiao
Keyword(s):  
Coherent Structures ◽  
Dimensional Space ◽  
Soliton Solutions ◽  
Two Dimensional ◽  
Rational Solutions ◽  
Bilinear Method ◽  
Dark Line ◽  
New Novel ◽  
Short Period ◽  
Hirota Bilinear

Under investigation in this paper are new novel coherent structures of two-dimensional lump-soliton for the Mel’nikov equation. The Hirota bilinear method and Kadomtsev–Petviashvili hierarchy reduction method are applied to construct a particular family of determinant semi-rational solutions exhibiting various coherent waves to the Mel’nikov equation. We first investigate some novel coherent waves, [Formula: see text]th-order lumps first appear from the [Formula: see text] dark line solitons and finally disappear into those [Formula: see text] dark line solitons after living on the constant background for a very short period. In contrast to the usual lump, those lumps in the coherent structures of lump-soliton are not only localized in two-dimensional space and but also localized in time.

Applications of mixed lump-solitons solutions and multi-peaks solitons for newly extended (2+1)-dimensional Boussinesq wave equation

2019 ◽  
Vol 33 (29) ◽  
pp. 1950363 ◽  
Author(s):  
Dianchen Lu ◽  
Aly R. Seadawy ◽  
Iftikhar Ahmed
Keyword(s):  
Boussinesq Equation ◽  
Three Dimensional ◽  
Soliton Solutions ◽  
Function Technique ◽  
Two Dimensional ◽  
Hirota Bilinear Method ◽  
Bilinear Method ◽  
Contour Plots ◽  
Hirota Bilinear ◽  
Interaction Phenomena

In this work, based on the Hirota bilinear method, mixed lump-solitons solutions and multi-peaks solitons are derived for a new extended (2[Formula: see text]+[Formula: see text]1)-dimensional Boussinesq equation by using ansatz function technique with symbolic computation. Through the mixed lump-solitons, we obtained two types of interaction phenomena, first from lump-single soliton solution and other from lump-two soliton solutions and their dynamics is given by three-dimensional plots and two-dimensional contour plots by taking appropriate values of given parameters. Furthermore, we obtained new patterns of multi-peaks solitons.

scholarly journals Two-dimensional fractional algebraic internal solitary waves model and its solution

2019 ◽  
Vol 11 (3) ◽  
pp. 168781401983835 ◽  
Author(s):  
Yuqing Li ◽  
Zheyuan Yu ◽  
Yaodeng Chen ◽  
Hongwei Yang
Keyword(s):  
Solitary Waves ◽  
Soliton Solutions ◽  
Internal Solitary Waves ◽  
System Of Equations ◽  
Two Dimensional ◽  
Basic System ◽  
Bilinear Method ◽  
New Model ◽  
Physical Systems ◽  
Hirota Bilinear

With the development of theory and advancement of scientific research, fractional calculus has begun to be considered as a method for the study of physical systems. In this work, based on the basic system of equations for internal solitary waves, we have derived two-dimensional Benjamin–Ono equation. Then, the integer-order two-dimensional Benjamin–Ono equation is transformed into the time-fractional Benjamin–Ono equation. To study the properties of the algebraic internal solitary waves, we discuss the conservation laws of the new model. Also, using the Hirota bilinear method, the derived new model is solved. Finally, we explore the characteristics of motion of the algebraic internal solitary waves with the help of the multi-soliton 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.

Integrability and Multi-Soliton Solutions for a Variable-Coefficient Coupled Gross–Pitaevskii System for Atomic MatterWaves

2012 ◽  
Vol 67 (10-11) ◽  
pp. 525-533
Author(s):  
Zhi-Qiang Lin ◽  
Bo Tian ◽  
Ming Wang ◽  
Xing Lu
Keyword(s):  
Variable Coefficient ◽  
Soliton Solutions ◽  
Variable Coefficients ◽  
Einstein Condensate ◽  
Matter Wave ◽  
Bilinear Method ◽  
Matter Waves ◽  
Bose Einstein Condensate ◽  
Bose Einstein ◽  
Hirota Bilinear

Under investigation in this paper is a variable-coefficient coupled Gross-Pitaevskii (GP) system, which is associated with the studies on atomic matter waves. Through the Painlev´e analysis, we obtain the constraint on the variable coefficients, under which the system is integrable. The bilinear form and multi-soliton solutions are derived with the Hirota bilinear method and symbolic computation. We found that: (i) in the elastic collisions, an external potential can change the propagation of the soliton, and thus the density of the matter wave in the two-species Bose-Einstein condensate (BEC); (ii) in the shape-changing collision, the solitons can exchange energy among different species, leading to the change of soliton amplitudes.We also present the collisions among three solitons of atomic matter waves.

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

2021 ◽  
Author(s):  
Hongcai Ma ◽  
Shupan Yue ◽  
Yidan Gao ◽  
Aiping Deng
Keyword(s):  
Evolution Equation ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Three Dimensional ◽  
Soliton Solutions ◽  
Bilinear Method ◽  
Test Function ◽  
Lump Solutions ◽  
Hirota Bilinear ◽  
Test Function Method

Abstract 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.

Families of exact solutions of the generalized (3+1)-dimensional nonlinear-wave equation

2018 ◽  
Vol 32 (29) ◽  
pp. 1850359 ◽  
Author(s):  
Wenhao Liu ◽  
Yufeng Zhang
Keyword(s):  
Wave Equation ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Soliton Solutions ◽  
Rogue Waves ◽  
Lump Solution ◽  
Rational Solutions ◽  
Bilinear Method ◽  
The One ◽  
Wave Limit

In this paper, the traveling wave method is employed to investigate the one-soliton solutions to two different types of bright solutions for the generalized (3[Formula: see text]+[Formula: see text]1)-dimensional nonlinear-wave equation, primarily. In the following parts, we derive the breathers and rational solutions by using the Hirota bilinear method and long-wave limit. More specifically, we discuss the lump solution and rogue wave solution, in which their trajectory will be changed by varying the corresponding coefficient or coordinate axis. On the one hand, the breathers express the form of periodic line waves in different planes, on the other hand, rogue waves are localized in time.

Y-shaped soliton solutions for the (2+1)-dimensional bidirectional Sawada–Kotera equation

2021 ◽  
Author(s):  
Shuxin Yang ◽  
Zhao Zhang ◽  
Biao Li
Keyword(s):  
Soliton Solutions ◽  
Complex Interaction ◽  
High Order ◽  
Hirota Bilinear Method ◽  
Bilinear Method ◽  
Dynamic Behaviors ◽  
Interaction Solutions ◽  
Localized Waves ◽  
Hirota Bilinear ◽  
Interaction Phenomenon

On the basis of the Hirota bilinear method, resonance Y-shaped soliton and its interaction with other localized waves of (2+1)-dimensional bidirectional Sawada–Kotera equation are derived by introducing the constraint conditions. These types of mixed soliton solutions exhibit complex interaction phenomenon between the resonance Y-shaped solitons and line waves, breather waves, and high-order lump waves. The dynamic behaviors of the interaction solutions are analyzed and illustrated.

MULTI-SOLITON SOLUTIONS AND THEIR INTERACTIONS FOR THE (2+1)-DIMENSIONAL SAWADA-KOTERA MODEL WITH TRUNCATED PAINLEVÉ EXPANSION, HIROTA BILINEAR METHOD AND SYMBOLIC COMPUTATION

2009 ◽  
Vol 23 (25) ◽  
pp. 5003-5015 ◽  
Author(s):  
XING LÜ ◽  
TAO GENG ◽  
CHENG ZHANG ◽  
HONG-WU ZHU ◽  
XIANG-HUA MENG ◽  
...  
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Symbolic Computation ◽  
Bilinear Form ◽  
Soliton Solutions ◽  
Hirota Bilinear Method ◽  
Bilinear Method ◽  
Truncated Painlevé Expansion ◽  
Hirota Bilinear ◽  
Bilinear Equations

In this paper, the (2+1)-dimensional Sawada-Kotera equation is studied by the truncated Painlevé expansion and Hirota bilinear method. Firstly, based on the truncation of the Painlevé series we obtain two distinct transformations which can transform the (2+1)-dimensional Sawada-Kotera equation into two bilinear equations of different forms (which are shown to be equivalent). Then employing Hirota bilinear method, we derive the analytic one-, two- and three-soliton solutions for the bilinear equations via symbolic computation. A formula which denotes the N-soliton solution is given simultaneously. At last, the evolutions and interactions of the multi-soliton solutions are graphically discussed as well. It is worthy to be noted that the truncated Painlevé expansion provides a useful dependent variable transformation which transforms a partial differential equation into its bilinear form and by means of the bilinear form, further study of the original partial differential equation can be conducted.

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.

Multiple-soliton solutions for coupled KdV and coupled KP systems

2009 ◽  
Vol 87 (12) ◽  
pp. 1227-1232 ◽  
Author(s):  
Abdul-Majid Wazwaz
Keyword(s):  
Soliton Solutions ◽  
Resonance Phenomenon ◽  
Hirota Bilinear Method ◽  
Bilinear Method ◽  
Multiple Soliton Solutions ◽  
Two Systems ◽  
Kp Equations ◽  
Hirota Bilinear ◽  
Singular Soliton ◽  
Multiple Singular Soliton Solutions

In this work we study two systems of coupled KdV and coupled KP equations. The Hirota bilinear method is applied to show that these two systems are completely integrable. Multiple-soliton solutions and multiple singular-soliton solutions are derived for each system. The resonance phenomenon is examined as well.

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