Explicit traveling wave solutions of five kinds of nonlinear evolution equations

Abstract

First of all, by using Bernoulli equations, we develop some technical lemmas. Then, we establish the explicit traveling wave solutions of five kinds of nonlinear evolution equations: nonlinear convection diffusion equations (including Burgers equations), nonlinear dispersive wave equations (including Korteweg-de Vries equations), nonlinear dissipative dispersive wave equations (including Ginzburg-Landau equation, Korteweg-de Vries-Burgers equation and Benjamin-Bona-Mahony-Burgers equation), nonlinear hyperbolic equations (including Sine-Gordon equation) and nonlinear reaction diffusion equations (including Belousov-Zhabotinskii system of reaction diffusion equations).

Persistence of periodic traveling waves and Abelian integrals

“…In these systems the phenomena of dispersion, dissipation, diffusion, reaction and convection are the fundamental physical common facts. We refer the reader to some interesting sources to know more details about the first appearance of this kind of solutions in the works of Russell (1834), Boussinesq (1877), Korteweg and de Vries (1895), Luther (1906), Fisher (1937), Kolmogorov, Petrovskii and Piskunov (1937), and to find several examples of applications and further motivation to study them: see [12,14,15,16,18,20,23,26,28] and the references therein.…”

Discontinuous Galerkin hp-adaptive methods for multiscale chemical reactors: Quiescent reactors

“…Finally, we should note that under special circumstances the full system (2.11) admits traveling-wave solutions that model important physical features of 2.11), such as phase transitions, oscillatory chemical reactions, and action potential formation, e.g. [68,75,90]. The class of traveling-wave solutions allows under certain circumstances (2.11) to be transformed into a system of second order (though often still highly nonlinear) ordinary differential equations (ODEs).…”