Asymptotic stability of traveling waves for a dissipative nonlinear evolution system

This paper is concerned with the existence and the nonlinear asymptotic stability of traveling wave solutions to the Cauchy problem for a system of dissipative evolution equations {ξt=−θx+βξxx,θt=νξx+(ξθ)x+αθxx,with initial data and end states (ξ,θ)(x,0)=(ξ0,θ0)(x)→(ξ±,θ±) as x→±∞.(ξ,θ)(x,0)=(ξ0,θ0)(x)→(ξ±,θ±) as x→±∞.We obtain the existence of traveling wave solutions by phase plane analysis and show the asymptotic nonlinear stability of traveling wave solutions without restrictions on the coefficients a and ν by the method of energy estimates.