Lump and interaction solutions of nonlinear partial differential equations

2019 ◽  
Vol 33 (11) ◽  
pp. 1950133 ◽  
Author(s):  
Yong-Li Sun ◽  
Wen-Xiu Ma ◽  
Jian-Ping Yu ◽  
Bo Ren ◽  
Chaudry Masood Khaliqu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Hirota Bilinear Method ◽  
Kp Equation ◽  
Bilinear Method ◽  
Partial Differential ◽  
Interaction Solutions ◽  
Kink Waves ◽  
Hirota Bilinear

In this paper, lump solutions of nonlinear partial differential equations, the generalized (2[Formula: see text]+[Formula: see text]1)-dimensional KP equation and the Jimbo–Miwa equation, are studied by using the Hirota bilinear method and carrying out symbolic computations in Maple. Moreover, the interaction solutions, i.e. collisions between lump waves and kink waves, are investigated. A group of graphs are plotted to illustrate the dynamics of the obtained results.

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.

scholarly journals Lumps, breathers, and interaction solutions of a (3+1)-dimensional generalized Kadovtsev–Petviashvili equation

2020 ◽  
pp. 2150041
Author(s):  
Xi Ma ◽  
Tie-Cheng Xia ◽  
Handong Guo
Keyword(s):  
Soliton Solution ◽  
Hirota Bilinear Method ◽  
Kp Equation ◽  
Bilinear Method ◽  
Test Technique ◽  
Interaction Solutions ◽  
Long Wave ◽  
Petviashvili Equation ◽  
Wave Limit ◽  
Hirota Bilinear

In this paper, we use the Hirota bilinear method to find the [Formula: see text]-soliton solution of a [Formula: see text]-dimensional generalized Kadovtsev–Petviashvili (KP) equation. Then, we obtain the [Formula: see text]-order breathers of the equation, and combine the long-wave limit method to give the [Formula: see text]-order lumps. Resorting to the extended homoclinic test technique, we obtain the breather-kink solutions for the equation. Last, the interaction solution composed of the [Formula: see text]-soliton solution, [Formula: see text]-breathers, and [Formula: see text]-lumps for the [Formula: see text]-dimensional generalized KP equation is constructed.

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.

scholarly journals Analytical mathematical schemes: Circular rod grounded via transverse Poisson’s effect and extensive wave propagation on the surface of water

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 545-554
Author(s):  
Asghar Ali ◽  
Aly R. Seadawy ◽  
Dumitru Baleanu
Keyword(s):  
Wave Propagation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Longitudinal Wave ◽  
Symbolic Computation ◽  
Boussinesq Equation ◽  
Nonlinear Partial Differential Equations ◽  
Wave Model ◽  
Simple Equation ◽  
Partial Differential

AbstractThis article scrutinizes the efficacy of analytical mathematical schemes, improved simple equation and exp(-\text{Ψ}(\xi ))-expansion techniques for solving the well-known nonlinear partial differential equations. A longitudinal wave model is used for the description of the dispersion in the circular rod grounded via transverse Poisson’s effect; similarly, the Boussinesq equation is used for extensive wave propagation on the surface of water. Many other such types of equations are also solved with these techniques. Hence, our methods appear easier and faster via symbolic computation.

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