Exact, multiple soliton solutions of the double sine Gordon equation

1978 ◽  
Vol 359 (1699) ◽  
pp. 479-495 ◽  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Gordon Equation ◽  
Dimensional Space ◽  
Nonlinear Partial Differential Equations ◽  
Soliton Solutions ◽  
Sine Gordon Equation ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Method Of Solution

Exact, particular solutions of the double sine Gordon equation in n dimensional space are constructed. Under certain restrictions these solutions are N solitons, where N ≤ 2 q — 1 and q is the dimensionality of spacetime. The method of solution, known as the base equation technique, relates solutions of nonlinear partial differential equations to solutions of linear partial differential equations. This method is reviewed and its applicability to the double sine Gordon equation shown explicitly. The N soliton solutions have the remarkable property that they collapse to a single soliton when the wave vectors are parallel.

scholarly journals Abundant Interaction Solutions of Sine-Gordon Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
DaZhao Lü ◽  
YanYing Cui ◽  
ChangHe Liu ◽  
ShangWen Wu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Symbolic Computation ◽  
Gordon Equation ◽  
Expansion Method ◽  
Sine Gordon Equation ◽  
Partial Differential ◽  
Interaction Solutions ◽  
Linear Partial Differential Equations

With the help of computer symbolic computation software (e.g.,Maple), abundant interaction solutions of sine-Gordon equation are obtained by means of a constructed Wronskian form expansion method. The method is based upon the forms and structures of Wronskian solutions of sine-Gordon equation, and the functions used in the Wronskian determinants do not satisfy linear partial differential equations. Such interaction solutions are difficultly obtained via other methods. And the method can be automatically carried out in computer.

scholarly journals A Boiti-Leon Pimpinelli equations with time-conformable derivative

2019 ◽  
Vol 9 (3) ◽  
pp. 95-101 ◽  
Author(s):  
Kamal Ait Touchent ◽  
Zakia Hammouch ◽  
Toufik Mekkaoui ◽  
Canan Unlu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Gordon Equation ◽  
Simple Technique ◽  
Nonlinear Partial Differential Equations ◽  
Soliton Solutions ◽  
Conformable Derivative ◽  
Hyperbolic Functions ◽  
Partial Differential ◽  
Expansion Technique

In this paper, we derive some new soliton solutions to $(2+1)$-Boiti-Leon Pempinelli equations with conformable derivative by using an expansion technique based on the Sinh-Gordon equation. The obtained solutions have different expression such as trigonometric, complex and hyperbolic functions. This powerful and simple technique can be used to investigate solutions of other  nonlinear partial differential equations.

Lyapunov-type inequalities for partial differential equations with 𝑝-Laplacian

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Robert Stegliński
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Nonlinear Partial Differential Equations ◽  
Dirichlet Boundary Conditions ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Lyapunov Inequalities ◽  
Funct Anal

Abstract The aim of this paper is to extend results from [A. Cañada, J. A. Montero and S. Villegas, Lyapunov inequalities for partial differential equations, J. Funct. Anal. 237 (2006), 1, 176–193] about Lyapunov-type inequalities for linear partial differential equations to nonlinear partial differential equations with 𝑝-Laplacian with zero Neumann or Dirichlet boundary conditions.

Solution Non Linear Partial Differential Equations By ZMA Decomposition Method

2021 ◽  
Vol 20 ◽  
pp. 712-716
Author(s):  
Zainab Mohammed Alwan
Keyword(s):  
Exact Solution ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Decomposition Method ◽  
Integral Transformation ◽  
Nonlinear Partial Differential Equations ◽  
Adomian Decomposition Method ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Adomian Decomposition

In this survey, viewed integral transformation (IT) combined with Adomian decomposition method (ADM) as ZMA- transform (ZMAT) coupled with (ADM) in which said ZMA decomposition method has been utilized to solve nonlinear partial differential equations (NPDE's).This work is very useful for finding the exact solution of (NPDE's) and this result is more accurate obtained with compared the exact solution obtained in the literature.

scholarly journals Integrable Equations and Their Evolutions Based on Intrinsic Geometry of Riemann Spaces

2009 ◽  
Vol 2009 ◽  
pp. 1-16
Author(s):  
Paul Bracken
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Soliton Solutions ◽  
Partial Differential ◽  
Intrinsic Geometry ◽  
Integrable Equations ◽  
Higher Dimensional ◽  
Riemannian Spaces

The intrinsic geometry of surfaces and Riemannian spaces will be investigated. It is shown that many nonlinear partial differential equations with physical applications and soliton solutions can be determined from the components of the relevant metric for the space. The manifolds of interest are surfaces and higher-dimensional Riemannian spaces. Methods for specifying integrable evolutions of surfaces by means of these equations will also be presented.

scholarly journals The New Simulation of Quasiperiodic Wave, Periodic Wave, and Soliton Solutions of the KdV-mKdV Equation via a Deep Learning Method

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Yong Zhang ◽  
Huanhe Dong ◽  
Jiuyun Sun ◽  
Zhen Wang ◽  
Yong Fang ◽  
...  
Keyword(s):  
Deep Learning ◽  
Partial Differential Equations ◽  
Initial Data ◽  
Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Soliton Solutions ◽  
Periodic Wave ◽  
Mkdv Equation ◽  
Learning Method ◽  
Partial Differential

How to solve the numerical solution of nonlinear partial differential equations efficiently and conveniently has always been a difficult and meaningful problem. In this paper, the data-driven quasiperiodic wave, periodic wave, and soliton solutions of the KdV-mKdV equation are simulated by the multilayer physics-informed neural networks (PINNs) and compared with the exact solution obtained by the generalized Jacobi elliptic function method. Firstly, the different types of solitary wave solutions are used as initial data to train the PINNs. At the same time, the different PINNs are applied to learn the same initial data by selecting the different numbers of initial points sampled, residual collocation points sampled, network layers, and neurons per hidden layer, respectively. The result shows that the PINNs well reconstruct the dynamical behaviors of the quasiperiodic wave, periodic wave, and soliton solutions for the KdV-mKdV equation, which gives a good way to simulate the solutions of nonlinear partial differential equations via one deep learning method.

scholarly journals Optical Solutions of Fokas-Lenells Equation via (m-1/G') - Expansion Method

2020 ◽  
Vol 7 ◽  
pp. 20-24
Author(s):  
Hasan Bulut ◽  
Khalid ◽  
Ban Jamal
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Efficient Method ◽  
Nonlinear Partial Differential Equations ◽  
Soliton Solutions ◽  
Expansion Method ◽  
Three Dimensions ◽  
Partial Differential ◽  
Wide Range

In this research paper, we investigate some novel soliton solutions to the perturbed Fokas-Lenells equation by using the (m + 1/G') expansion method. Some new solutions are obtained and they are plotted in two and three dimensions. This technique appears as a suitable, applicable, and efficient method to search for the exact solutions of nonlinear partial differential equations in a wide range. All gained optical soliton solutions are substituted into the FokasLenells equation and they verify it. The constraint conditions are also given.

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