Asymptotic Behavior Of Traveling Waves To The Cauchy Problem For The Generalized Korteweg-de Vries-Burgers Equation

Korteweg-de Vries-Burgers equation is one of the most improtant research objects of nonlinear evolution equation.It not only can be used to explain physical phenomena,such as water waves and sound waves,but also can be used as a mathematical model for studying hydrodynamics.At present,most of the study of the Cauchy problem to the Korteweg-de Vries-Burgers equation are focused on the nonlinear stability and the large time behavior of rarefaction waves,which can refer to[1,2],but there is less research on traveling waves.In this paper,we consider the global stability and the large time behavior of traveling wave solutions to the Cauchy problem for the generalized Korteweg-de Vries-Burgers equationHere f(u)is a smooth sufficiently convex function,u_-?u_+are two given constants and the constants ? > 0 and ??0 stand for dissipative coefficient and chromatic dispersion coefficient respectively.For the above Cauchy problem,firstly,by employing the phase-plane analysis method,we give a limiting condition to guarantee the existence of monotonic traveling wave solution.Then based on the contraction mapping priniciple,when the initial perturbation is sufficiently small,we establish the local existence of the smooth solution and give a priori estimates by using the L~2-energy method.Thus we can use the continuity argument to show the global existence and nonlinear stability of traveling wave solution to the Cauchy problem(I).Futhermore,by employing the space-time weighted energy method,the algebraic decay rates of the solutions of the Cauchy problem(I)to the traveling waves are obtained in some proper weighted space to discuss the asymptotic behavior of traveling wave solutions to the Korteweg-de Vries-Burgers equation.