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.

Lump and interaction solutions to linear PDEs in 2+1 dimensions via symbolic computation

2019 ◽  
Vol 33 (36) ◽  
pp. 1950457 ◽  
Author(s):  
Wen-Xiu Ma
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Symbolic Computation ◽  
Rogue Wave ◽  
Soliton Solutions ◽  
Partial Differential ◽  
Interaction Solutions ◽  
Linear Partial Differential Equations ◽  
Linear Pdes

The aim of this paper is to show that there exist lump solutions and interaction solutions to linear partial differential equations in 2[Formula: see text]+[Formula: see text]1 dimensions. Through symbolic computations with Maple, we exhibit a great variety of exact solutions to a class of (2[Formula: see text]+[Formula: see text]1)-dimensional linear partial differential equations, and present a specific example which possesses lump, lump-kink and lump-soliton solutions. This supplements the study on lump, rogue wave and breather solutions and their interaction solutions to nonlinear integrable equations.

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.

New types of interaction solutions to the (3 + 1)-dimensional generalized Kadomtsev-Petviashvili equation

2020 ◽  
Vol 34 (23) ◽  
pp. 2050237
Author(s):  
Yuexing Bai ◽  
Temuerchaolu ◽  
Yan Li ◽  
Sudao Bilige
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Symbolic Computation ◽  
Nonlinear Partial Differential Equations ◽  
Lump Solution ◽  
Biological Engineering ◽  
Partial Differential ◽  
Lump Solutions ◽  
Interaction Solutions ◽  
Petviashvili Equation

In this paper, with the aid of symbolic computation system Maple, and based on the simplified Hirota method and ansatz technique, we discussed the (3+1)-dimensional generalized Kadomtsev–Petviashvili equation with [Formula: see text] to obtain lump solutions, lump–kink solutions and three classes of interaction solutions. Comparing our new results with other researchers’ results shows that using this method gives the more opportunity to solve the nonlinear partial differential equations that appear in mathematics, physics, biological engineering and other fields. We also presented profiles of new lump solution, lump–kink solutions and interaction solutions as illustrative examples.

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