Evans functions and bifurcations of nonlinear waves of some nonlinear reaction diffusion equations

Abstract

The main purposes of this paper are to accomplish the existence, stability, instability and bifurcation of the nonlinear waves of the nonlinear system of reaction diffusion equations ut=uxx+α[βH(u−θ)−u]−w, wt=ε(u−γw) and to establish the existence, stability, instability and bifurcation of the nonlinear waves of the nonlinear scalar reaction diffusion equation ut=uxx+α[βH(u−θ)−u], under different conditions on the model constants.To establish the bifurcation for the system, we will study the existence and instability of a standing pulse solution if 0<2(1+αγ)θ<αβγ; the existence and stability of two standing wave fronts if 2(1+αγ)θ=αβγ and γ2ε>1; the existence and instability of two standing wave fronts if 2(1+αγ)θ=αβγ and 0<γ2ε<1; the existence and instability of an upside down standing pulse solution if 0<(1+αγ)θ<αβγ<2(1+αγ)θ. To establish the bifurcation for the scalar equation, we will study the existence and stability of a traveling wave front as well as the existence and instability of a standing pulse solution if 0<2θ<β; the existence and stability of two standing wave fronts if 2θ=β; the existence and stability of a traveling wave front as well as the existence and instability of an upside down standing pulse solution if 0<θ<β<2θ. By the way, we will also study the existence and stability of a traveling wave back of the nonlinear scalar reaction diffusion equation ut=uxx+α[βH(u−θ)−u]−w0, where w0=α(β−2θ)>0 is a positive constant, if 0<2θ<β.To achieve the main goals, we will make complete use of the special structures of the model equations and we will construct Evans functions and apply them to study the eigenvalues and eigenfunctions of several eigenvalue problems associated with several linear differential operators. It turns out that a complex number λ0 is an eigenvalue of the linear differential operator, if and only if λ0 is a zero of the Evans function. The stability, instability and bifurcations of the nonlinear waves follow from the zeros of the Evans functions.A very important motivation to study the existence, stability, instability and bifurcations of the nonlinear waves is to study the existence and stability/instability of infinitely many fast/slow multiple traveling pulse solutions of the nonlinear system of reaction diffusion equations. The existence and stability of infinitely many fast multiple traveling pulse solutions are of great interests in mathematical neuroscience.