Asymptotic analysis of a perturbation problem

AbstractAn asymptotic expansion is constructed for the solution of the initial-value problemutt-uxx+u=ε(ut-13ut3),-∞<x<∞,t⩾0,u(x,0)=sinkx,ut(x,0)=0,when t is restricted to the interval [0,T/ε], where T is any given number. Our analysis is mathematically rigorous; that is, we show that the difference between the true solution u(t,x;ε) and the Nth partial sum of the asymptotic series is bounded by εN+1 multiplied by a constant depending on T but not on x and t