scholarly journals Residual Symmetry, Bäcklund Transformation, and Soliton Solutions of the Higher-Order Broer-Kaup System

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Yarong Xia ◽  
Ruoxia Yao ◽  
Xiangpeng Xin
Keyword(s):  
Soliton Solutions ◽  
Expansion Method ◽  
Higher Order ◽  
Group Invariant ◽  
Residual Symmetry ◽  
Nonlinear Excitations ◽  
Group Invariant Solutions ◽  
Symmetry Transformations ◽  
Bidirectional Propagation ◽  
Truncated Painlevé Expansion

Under investigation in this paper is the higher-order Broer-Kaup(HBK) system, which describes the bidirectional propagation of long waves in shallow water. Via the standard truncated Painlevé expansion method, the residual symmetry of this system is derived. By introducing an appropriate auxiliary-dependent variable, the residual symmetry is successfully localized to Lie point symmetries. Via solving the initial value problems, the finite symmetry transformations are presented. However, the solution which obtained from the residual symmetry is a special group invariant solutions. In order to find more general solution of HBK system, we further generalize the residual symmetry method to the consistent tanh expansion (CTE) method and prove that the HBK system is CTE solvable, then the resonant soliton solutions and interaction solutions among different nonlinear excitations are obtained by the CET method.

New bright soliton solutions for Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equations and bidirectional propagation of water wave surface

2021 ◽  
Author(s):  
S. Saha Ray ◽  
Shailendra Singh
Keyword(s):  
Water Wave ◽  
Soliton Solutions ◽  
Fluid Flows ◽  
Wave Model ◽  
Expansion Method ◽  
Bright Soliton ◽  
Wave Surface ◽  
Model Equations ◽  
Bidirectional Propagation ◽  
Truncated Painlevé Expansion

The governing equations for fluid flows, i.e. Kadomtsev–Petviashvili–Benjamin–Bona–Mahony (KP-BBM) model equations represent a water wave model. These model equations describe the bidirectional propagating water wave surface. In this paper, an auto-Bäcklund transformation is being generated by utilizing truncated Painlevé expansion method for the considered equation. This paper determines the new bright soliton solutions for [Formula: see text] and [Formula: see text]-dimensional nonlinear KP-BBM equations. The simplified version of Hirota’s technique is utilized to infer new bright soliton solutions. The results are plotted graphically to understand the physical behavior of solutions.

scholarly journals Solutions and Painlevé Property for the KdV Equation with Self-Consistent Source

2018 ◽  
Vol 2018 ◽  
pp. 1-16
Author(s):  
Yali Shen ◽  
Ruoxia Yao
Keyword(s):  
Kdv Equation ◽  
Traveling Wave Solutions ◽  
Elliptic Functions ◽  
Expansion Method ◽  
Hyperbolic Function ◽  
Group Invariant ◽  
Painlevé Property ◽  
The Kdv Equation ◽  
Self Consistent ◽  
Group Invariant Solutions

In this paper, the polynomial solutions in terms of Jacobi’s elliptic functions of the KdV equation with a self-consistent source (KdV-SCS) are presented. The extended (G′/G)-expansion method is utilized to obtain exact traveling wave solutions of the KdV-SCS, which finally are expressed in terms of the hyperbolic function, the trigonometric function, and the rational function. Meanwhile we find the Lie point symmetry and Lie symmetry group and give several group-invariant solutions for the KdV-SCS. Finally, we supplement the results of the Painlevé property in our previous work and get the Bäcklund transformations of the KdV-SCS.

scholarly journals Symmetry Reductions of (2 + 1)-Dimensional CDGKS Equation and Its Reduced Lax Pairs

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Na Lv ◽  
Xuegang Yuan ◽  
Jinzhi Wang
Keyword(s):  
Direct Method ◽  
Lax Pair ◽  
Group Method ◽  
Symmetry Groups ◽  
Invariant Solutions ◽  
Lax Pairs ◽  
Group Invariant ◽  
Lie Group Method ◽  
Group Invariant Solutions ◽  
Symmetry Transformations

With the aid of symbolic computation, we obtain the symmetry transformations of the (2 + 1)-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada (CDGKS) equation by Lou’s direct method which is based on Lax pairs. Moreover, we use the classical Lie group method to seek the symmetry groups of both the CDGKS equation and its Lax pair and then reduce them by the obtained symmetries. In particular, we consider the reductions of the Lax pair completely. As a result, three reduced (1 + 1)-dimensional equations with their new Lax pairs are presented and some group-invariant solutions of the equation are given.

New symmetries, group-invariant solutions, linear differential constraints of a generalized Burgers-KdV equation and its reduction

2020 ◽  
pp. 2150031
Author(s):  
Huanhuan Lu ◽  
Yufeng Zhang
Keyword(s):  
Burgers Equation ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Lie Point Symmetries ◽  
Differential Constraints ◽  
Nonlocal Symmetries ◽  
Group Invariant ◽  
Liquid Moving ◽  
Group Invariant Solutions ◽  
Linear Differential

All we know that the Burgers-KdV equation is extensively used to study the liquid flow with bubbles and the liquid moving flow in the elastic pipes. In this paper, we obtain the Lie point symmetries, self-nonlinear adjointness of a generalized Burgers-KdV equation (GB-KdVE) are obtained, it follows that the conservation laws are worked out. As a reduction of the GB-KdVE, a Burgers equation with general coefficients is presented, whose new strong symmetry and new nonlocal symmetries are generated, respectively. Furthermore, the noninvariant solutions of the GB-KdVE are produced as well. Finally, we propose the double linear differential constraints for GB-KdVE-type so that some soliton solutions are singled out.

scholarly journals Exact Solutions of Newell-Whitehead-Segel Equations Using Symmetry Transformations

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Khudija Bibi ◽  
Khalil Ahmad
Keyword(s):  
Exact Solutions ◽  
Transformation Group ◽  
Symmetry Transformation ◽  
Discrete Symmetry ◽  
Transformation Groups ◽  
Invariant Solutions ◽  
Group Invariant ◽  
Group Invariant Solutions ◽  
Symmetry Transformations ◽  
Linear And Nonlinear

In this article, Lie and discrete symmetry transformation groups of linear and nonlinear Newell-Whitehead-Segel (NWS) equations are obtained. By using these symmetry transformation groups, several group invariant solutions of considered NWS equations have been constructed. Furthermore, some more group invariant solutions are generated by using discrete symmetry transformation group. Graphical representations of some obtained solutions are also presented.

Analytic study of solutions for a two-component Novikov system

2020 ◽  
pp. 2150025
Author(s):  
Hui Gao ◽  
Gangwei Wang
Keyword(s):  
Exact Solutions ◽  
Stationary Solutions ◽  
Invariant Solutions ◽  
Group Invariant ◽  
New Exact Solutions ◽  
Two Component ◽  
Group Invariant Solutions ◽  
Symmetry Transformations ◽  
Symmetry Method ◽  
Similarity Reductions

Under investigation in this paper is a two-component Novikov system (also called Geng-Xue equation), which was proposed by Geng and Xue in 2009. Firstly, via the Lie symmetry method, infinitesimal generators, commutator table of Lie algebra and symmetry groups of the two-component Novikov system are presented. At the same time, some group invariant solutions are computed through similarity reductions. In particular, we construct peakon solution by applying the distribution theory. In addition, based on obtained group invariant solutions and symmetry transformations, we derive some new exact solutions, which include stationary solutions, smooth solutions, and a weak solution. The analytical properties to some of group invariant solutions and new exact solutions are discussed, such as decay, asymptotic behavior, and boundedness.

Lump, mixed lump-soliton, and periodic lump solutions of a (2+1)-dimensional extended higher-order Broer–Kaup System

2020 ◽  
Vol 34 (33) ◽  
pp. 2050384
Author(s):  
Fan Guo ◽  
Ji Lin
Keyword(s):  
Rogue Wave ◽  
Soliton Solutions ◽  
Elastic Collision ◽  
Auxiliary Function ◽  
Higher Order ◽  
Trigonometric Functions ◽  
Resonant Line ◽  
Lump Solutions ◽  
Truncated Painlevé Expansion ◽  
Line Soliton

In this paper, a (2+1)-dimensional extended higher-order Broer–Kaup system is introduced and its bilinear form is presented from the truncated Painlevé expansion. By taking the auxiliary function as the ansatzs including quadratic, exponential, and trigonometric functions, lump, mixed lump-soliton, and periodic lump solutions are derived. The mixed lump-soliton solutions are classified into two cases: the first one describes the non-elastic collision between one lump and one line soliton, which exhibits fission and fusion phenomena. The second one depicts the interaction consisting of one lump and two line soliton, which generates a rogue wave excited from two resonant line solitons.

scholarly journals N-soliton solutions and the Hirota conditions in (1 + 1)-dimensions

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Wen-Xiu Ma
Keyword(s):  
Common Factors ◽  
Soliton Solutions ◽  
Higher Order ◽  
Kdv Equations ◽  
The Common ◽  
Bilinear Formulation ◽  
Generalized Kdv Equations ◽  
Hirota Bilinear

Abstract We analyze N-soliton solutions and explore the Hirota N-soliton conditions for scalar (1 + 1)-dimensional equations, within the Hirota bilinear formulation. An algorithm to verify the Hirota conditions is proposed by factoring out common factors out of the Hirota function in N wave vectors and comparing degrees of the involved polynomials containing the common factors. Applications to a class of generalized KdV equations and a class of generalized higher-order KdV equations are made, together with all proofs of the existence of N-soliton solutions to all equations in two classes.

BILINEAR FORM AND SOLITON SOLUTIONS FOR THE COUPLED NONLINEAR KLEIN–GORDON EQUATIONS

2012 ◽  
Vol 26 (15) ◽  
pp. 1250057
Author(s):  
HE LI ◽  
XIANG-HUA MENG ◽  
BO TIAN
Keyword(s):  
Field Theory ◽  
Bilinear Form ◽  
Gordon Equation ◽  
Relativistic Quantum ◽  
Soliton Solutions ◽  
Relativistic Quantum Mechanics ◽  
Klein Gordon ◽  
Truncated Painlevé Expansion ◽  
Klein Gordon Equation ◽  
Singular Manifolds

With the coupling of a scalar field, a generalization of the nonlinear Klein–Gordon equation which arises in the relativistic quantum mechanics and field theory, i.e., the coupled nonlinear Klein–Gordon equations, is investigated via the Hirota method. With the truncated Painlevé expansion at the constant level term with two singular manifolds, the coupled nonlinear Klein–Gordon equations are transformed to a bilinear form. Starting from the bilinear form, with symbolic computation, we obtain the N-soliton solutions for the coupled nonlinear Klein–Gordon equations.

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