scholarly journals Existence and stability of traveling wave solutions to first-order quasilinear hyperbolic systems

2013 ◽  
Vol 100 (1) ◽  
pp. 34-68 ◽  
Author(s):  
Cunming Liu ◽  
Peng Qu
Keyword(s):  
Traveling Wave ◽  
Hyperbolic Systems ◽  
Traveling Wave Solutions ◽  
Quasilinear Hyperbolic Systems ◽  
First Order ◽  
Wave Solutions ◽  
Existence And Stability

scholarly journals Integrability via Functional Expansion for the KMN Model

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1819
Author(s):  
Radu Constantinescu ◽  
Aurelia Florian
Keyword(s):  
Differential Equations ◽  
Traveling Wave ◽  
Nonlinear Differential Equations ◽  
Traveling Wave Solutions ◽  
Auxiliary Equation ◽  
First Order ◽  
Second Order Differential Equations ◽  
Functional Expansion ◽  
Wave Solutions ◽  
First Time

This paper considers issues such as integrability and how to get specific classes of solutions for nonlinear differential equations. The nonlinear Kundu–Mukherjee–Naskar (KMN) equation is chosen as a model, and its traveling wave solutions are investigated by using a direct solving method. It is a quite recent proposed approach called the functional expansion and it is based on the use of auxiliary equations. The main objectives are to provide arguments that the functional expansion offers more general solutions, and to point out how these solutions depend on the choice of the auxiliary equation. To see that, two different equations are considered, one first order and one second order differential equations. A large variety of KMN solutions are generated, part of them listed for the first time. Comments and remarks on the dependence of these solutions on the solving method and on form of the auxiliary equation, are included.

TRAVELING WAVE SOLUTIONS FOR DISCRETE-TIME MODEL OF DELAYED CELLULAR NEURAL NETWORKS

2013 ◽  
Vol 23 (06) ◽  
pp. 1350107 ◽  
Author(s):  
CHENG-HSIUNG HSU ◽  
JIAN-JHONG LIN
Keyword(s):  
Neural Networks ◽  
Discrete Time ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Cellular Neural Networks ◽  
Time Model ◽  
Discrete Time Model ◽  
Characteristic Roots ◽  
Wave Solutions ◽  
Existence And Stability

The aim of this work is to study the existence and stability of traveling wave solutions for discrete-time model of delayed cellular neural networks distributed in the one-dimensional integer lattice ℤ1. Since the dynamics of each given cell depends on its left and right neighboring cells, it is not easy to construct the traveling wave solutions. Using the method of step along with positive characteristic roots of the equations, we successfully prove the existence of traveling wave solutions. Moreover, we show that all the traveling wave solutions are unstable. We also provide some numerical results to support our results, and point out the different structures of traveling wave solutions between the continuous-time and discrete-time models.

Sign in / Sign up

Export Citation Format

Share Document