A nonlocal Boussinesq equation: multiple-soliton solutions and symmetry analysis

2021 ◽  
Author(s):  
Xi-zhong Liu ◽  
Jun Yu
Keyword(s):  
Exact Solutions ◽  
Boussinesq Equation ◽  
Soliton Solutions ◽  
Symmetry Reduction ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Lie Symmetry Method ◽  
Multiple Soliton Solutions ◽  
Symmetry Method

Abstract A nonlocal Boussinesq equation is deduced from the local one by using consistent correlated bang method. To study various exact solutions of the nonlocal Boussinesq equation, it is converted into two local equations which contain the local Boussinesq equation. From the N-soliton solutions of the local Boussinesq equation, the N-soliton solutions of the nonlocal Boussinesq equation are obtained, among which the (N = 2, 3, 4)-soliton solutions are analyzed with graphs. Some periodic and traveling solutions of the nonlocal Boussinesq equation are derived directly from known solutions of the local Boussinesq equation. Symmetry reduction solutions of the nonlocal Boussinesq equation are also obtained by using the classical Lie symmetry method.

New soliton solutions of the CH–DP equation using Lie symmetry method

2018 ◽  
Vol 32 (20) ◽  
pp. 1850234 ◽  
Author(s):  
A. H. Abdel Kader ◽  
M. S. Abdel Latif
Keyword(s):  
Traveling Wave ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Elliptic Functions ◽  
Dark Soliton ◽  
Lie Symmetry ◽  
Jacobi Elliptic Functions ◽  
Lie Symmetry Method ◽  
Wave Solutions ◽  
Symmetry Method

In this paper, using Lie symmetry method, we obtain some new exact traveling wave solutions of the Camassa–Holm–Degasperis–Procesi (CH–DP) equation. Some new bright and dark soliton solutions are obtained. Also, some new doubly periodic solutions in the form of Jacobi elliptic functions and Weierstrass elliptic functions are obtained.

scholarly journals Residual Symmetry Reduction and Consistent Riccati Expansion to a Nonlinear Evolution Equation

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-9 ◽  
Author(s):  
Lamine Thiam ◽  
Xi-zhong Liu
Keyword(s):  
Evolution Equation ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Symmetry Reduction ◽  
Lie Symmetry ◽  
Residual Symmetry ◽  
Interaction Solutions ◽  
Lie Symmetry Method ◽  
Consistent Riccati Expansion ◽  
Symmetry Method

The residual symmetry of a (1 + 1)-dimensional nonlinear evolution equation (NLEE) ut+uxxx−6u2ux+6λux=0 is obtained through Painlevé expansion. By introducing a new dependent variable, the residual symmetry is localized into Lie point symmetry in an enlarged system, and the related symmetry reduction solutions are obtained using the standard Lie symmetry method. Furthermore, the (1 + 1)-dimensional NLEE equation is proved to be integrable in the sense of having a consistent Riccati expansion (CRE), and some new Bäcklund transformations (BTs) are given. In addition, some explicitly expressed solutions including interaction solutions between soliton and cnoidal waves are derived from these BTs.

scholarly journals Time fractional Kupershmidt equation: symmetry analysis and explicit series solution with convergence analysis

2019 ◽  
Vol 27 (2) ◽  
pp. 171-185 ◽  
Author(s):  
Astha Chauhan ◽  
Rajan Arora
Keyword(s):  
Differential Equation ◽  
Power Series ◽  
Analytic Solutions ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Power Series Method ◽  
Lie Symmetry Method ◽  
Liouville Fractional Derivative ◽  
Symmetry Generators ◽  
Symmetry Method

AbstractIn this work, the fractional Lie symmetry method is applied for symmetry analysis of time fractional Kupershmidt equation. Using the Lie symmetry method, the symmetry generators for time fractional Kupershmidt equation are obtained with Riemann-Liouville fractional derivative. With the help of symmetry generators, the fractional partial differential equation is reduced into the fractional ordinary differential equation using Erdélyi-Kober fractional differential operator. The conservation laws are determined for the time fractional Kupershmidt equation with the help of new conservation theorem and fractional Noether operators. The explicit analytic solutions of fractional Kupershmidt equation are obtained using the power series method. Also, the convergence of the power series solutions is discussed by using the implicit function theorem.

Lie symmetry analysis and some exact solutions for modified Zakharov–Kuznetsov equation

2018 ◽  
Vol 32 (31) ◽  
pp. 1850383 ◽  
Author(s):  
Xuan Zhou ◽  
Wenrui Shan ◽  
Zhilei Niu ◽  
Pengcheng Xiao ◽  
Ying Wang
Keyword(s):  
Exact Solutions ◽  
Detailed Analysis ◽  
Symmetry Group ◽  
Nonlinear Waves ◽  
Lie Symmetry ◽  
Lie Symmetry Analysis ◽  
One Dimensional ◽  
Infinitesimal Generators ◽  
Lie Symmetry Method ◽  
Symmetry Method

In this study, the Lie symmetry method is used to perform detailed analysis on the modified Zakharov–Kuznetsov equation. We have obtained the infinitesimal generators, commutator table of Lie algebra and symmetry group. In addition to that, optimal system of one-dimensional subalgebras up to conjugacy is derived and used to construct distinct exact solutions. These solutions describe the dynamics of nonlinear waves in isothermal multicomponent magnetized plasmas.

scholarly journals Parametric conditions and exact solution for the Duffing-Van der Pol class of equations

2018 ◽  
Vol 40 (3) ◽  
pp. 251-264
Author(s):  
Dao Huy Bich ◽  
Nguyen Dang Bich
Keyword(s):  
Exact Solution ◽  
Exact Solutions ◽  
Lower Order ◽  
Main Idea ◽  
Lie Symmetry ◽  
First Integrals ◽  
Original Equation ◽  
Van Der Pol ◽  
Lie Symmetry Method ◽  
Symmetry Method

This paper presents a methodology to find the exact solution and respective parametric conditions to the Duffing-Van der Pol class of equations. The supposed method in this paper is different from the Prelle and Singer method and the Lie symmetry method. The main idea of the supposed method is implemented in finding the first integrals of the original equation and leading this equation to a solved equation of lower order to which the exact solution can be obtained. As results the parametric conditions and the exact solutions in parametric forms are indicated. The algorithm for determining integral constants and the investigation of solution characteristics are considered.

Symmetry analysis and invariant solutions of (3 + 1)-dimensional Kadomtsev–Petviashvili equation

2018 ◽  
Vol 15 (08) ◽  
pp. 1850125 ◽  
Author(s):  
Vishakha Jadaun ◽  
Sachin Kumar
Keyword(s):  
Lie Algebra ◽  
Zeta Functions ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Lie Symmetry Analysis ◽  
Power Series Method ◽  
Lie Symmetry Method ◽  
Petviashvili Equation ◽  
Symmetry Method ◽  
Similarity Reductions

Based on Lie symmetry analysis, we study nonlinear waves in fluid mechanics with strong spatial dispersion. The similarity reductions and exact solutions are obtained based on the optimal system and power series method. We obtain the infinitesimal generators, commutator table of Lie algebra, symmetry group and similarity reductions for the [Formula: see text]-dimensional Kadomtsev–Petviashvili equation. For different Lie algebra, Lie symmetry method reduces Kadomtsev–Petviashvili equation into various ordinary differential equations (ODEs). Some of the solutions of [Formula: see text]-dimensional Kadomtsev–Petviashvili equation are of the forms — traveling waves, Weierstrass’s elliptic and Zeta functions and exponential functions.

scholarly journals Exact Solutions of Generalized Boussinesq-Burgers Equations and (2+1)-Dimensional Davey-Stewartson Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Isaiah Elvis Mhlanga ◽  
Chaudry Masood Khalique
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Travelling Wave ◽  
Lie Symmetry ◽  
Coupled Systems ◽  
Burgers Equations ◽  
Lie Symmetry Method ◽  
Symmetry Method

We study two coupled systems of nonlinear partial differential equations, namely, generalized Boussinesq-Burgers equations and (2+1)-dimensional Davey-Stewartson equations. The Lie symmetry method is utilized to obtain exact solutions of the generalized Boussinesq-Burgers equations. The travelling wave hypothesis approach is used to find exact solutions of the (2+1)-dimensional Davey-Stewartson equations.

scholarly journals Symmetries, Traveling Wave Solutions, and Conservation Laws of a(3+1)-Dimensional Boussinesq Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Letlhogonolo Daddy Moleleki ◽  
Chaudry Masood Khalique
Keyword(s):  
Conservation Laws ◽  
Traveling Wave ◽  
Boussinesq Equation ◽  
Traveling Wave Solutions ◽  
Lie Symmetry ◽  
Lie Symmetry Method ◽  
Wave Solutions ◽  
Simplest Equation Method ◽  
Symmetry Method ◽  
Conservation Theorem

We analyze the(3+1)-dimensional Boussinesq equation, which has applications in fluid mechanics. We find exact solutions of the(3+1)-dimensional Boussinesq equation by utilizing the Lie symmetry method along with the simplest equation method. The solutions obtained are traveling wave solutions. Moreover, we construct the conservation laws of the(3+1)-dimensional Boussinesq equation using the new conservation theorem, which is due to Ibragimov.

Exact solutions of Zabolotskaya–Khokhlov equation using optimal system

2018 ◽  
Vol 15 (07) ◽  
pp. 1850111
Author(s):  
Mohsin Umair ◽  
Tooba Feroze
Keyword(s):  
Exact Solutions ◽  
Lie Symmetry ◽  
Optimal System ◽  
Lie Symmetry Method ◽  
Symmetry Method

In this paper, using the Lie symmetry method, we obtain optimal system, group invariants and exact solutions of [Formula: see text] dimensional Zabolotskaya–Khokhlov equation.

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