Shock wave solutions of the variants of the Kadomtsev–Petviashvili equation

2011 ◽  
Vol 89 (9) ◽  
pp. 979-984 ◽  
Author(s):  
Houria Triki ◽  
B.J.M. Sturdevant ◽  
T. Hayat ◽  
O.M. Aldossary ◽  
A. Biswas
Keyword(s):  
Shock Wave ◽  
Wave Solution ◽  
Wave Solutions ◽  
Petviashvili Equation ◽  
Shock Wave Solution

This study obtained the shock wave or kink solutions of the variants of the Kadomtsev–Petviashvili equation with generalized evolution. There are three types of variants of this equation that were considered. The relation between the parameters and the constraint conditions will naturally fall out as a consequence of the derivation of the shock wave solution.

scholarly journals A Class of Shock Wave Solution to Singularly Perturbed Nonlinear Time-Delay Evolution Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Yi-Hu Feng ◽  
Lei Hou
Keyword(s):  
Shock Wave ◽  
Time Delay ◽  
Wave Solution ◽  
Evolution Equations ◽  
Multiple Scales ◽  
Expansion Method ◽  
Initial Boundary ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problem ◽  
Shock Wave Solution

Nonlinear singularly perturbed problem for time-delay evolution equation with two parameters is studied. Using the variables of the multiple scales method, homogeneous equilibrium method, and approximation expansion method from the singularly perturbed theories, the structure of the solution to the time-delay problem with two small parameters is discussed. Under suitable conditions, first, the outer solution to the time-delay initial boundary value problem is given. Second, the multiple scales variables are introduced to obtain the shock wave solution and boundary layer corrective terms for the solution. Then, the stretched variable is applied to get the initial layer correction terms. Finally, using the singularly perturbed theory and the fixed point theorem from functional analysis, the uniform validity of asymptotic expansion solution to the problem is proved. In addition, the proposed method possesses the advantages of being very convenient to use.

scholarly journals Regularization of the Shock Wave Solution to the Riemann Problem for the Relativistic Burgers Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Ting Zhang ◽  
Chun Shen
Keyword(s):  
Shock Wave ◽  
Initial Data ◽  
Riemann Problem ◽  
Wave Solution ◽  
Burgers Equation ◽  
Explicit Computation ◽  
Convective Term ◽  
Regularization Technique ◽  
Shock Wave Solution

The regularization of the shock wave solution to the Riemann problem for the relativistic Burgers equation is considered. For Riemann initial data consisting of a single decreasing jump, we find that the regularization of nonlinear convective term cannot capture the correct shock wave solution. In order to overcome it, we consider a new regularization technique called the observable divergence method introduced by Mohseni and discover that it can capture the correct shock wave solution. In addition, we take the Helmholtz filter for the fully explicit computation.

scholarly journals Lump-Type Wave and Interaction Solutions of the Bogoyavlenskii–Kadomtsev–Petviashvili Equation

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Chuanjian Wang ◽  
Hui Fang
Keyword(s):  
Wave Solution ◽  
Solitary Wave Solutions ◽  
Kink Wave Solution ◽  
Type Wave ◽  
Interaction Solutions ◽  
Dynamical Behaviors ◽  
Wave Solutions ◽  
Petviashvili Equation ◽  
Kink Wave ◽  
Hermitian Quadratic Form

Lump-type wave solution of the Bogoyavlenskii–Kadomtsev–Petviashvili equation is constructed by using the bilinear structure and Hermitian quadratic form. The dynamical behaviors of lump-type wave solution are investigated and presented analytically and graphically. Furthermore, we discuss the interaction between a lump-type wave and a kink wave solution. Absorb-emit interaction between two kinds of solitary wave solutions is shown. This kind of interaction solution can be regarded as a lump-type wave which propagates on the kink wave background.

scholarly journals Shock Wave Solution for a Nonlinear Partial Differential Equation Arising in the Study of a Non-Newtonian Fourth Grade Fluid Model

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Taha Aziz ◽  
A. Fatima ◽  
F. M. Mahomed
Keyword(s):  
Differential Equation ◽  
Shock Wave ◽  
Partial Differential Equation ◽  
Wave Solution ◽  
Fluid Model ◽  
Nonlinear Partial Differential Equation ◽  
Fourth Grade ◽  
Shock Wave Solution ◽  
Grade Fluid ◽  
Fourth Grade Fluid

This study focuses on obtaining a new class of closed-form shock wave solution also known as soliton solution for a nonlinear partial differential equation which governs the unsteady magnetohydrodynamics (MHD) flow of an incompressible fourth grade fluid model. The travelling wave symmetry formulation of the model leads to a shock wave solution of the problem. The restriction on the physical parameters of the flow problem also falls out naturally in the course of derivation of the solution.

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