Asymptotic analysis of a perturbation problem

An asymptotic expansion is constructed for the solution of the initial-value problem u tt - u xx + u = ε ( u t - 1 3 u t 3 ) , - ∞ < x < ∞ , t ⩾ 0 , u ( x , 0 ) = sin kx , u t ( 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 ε N + 1 multiplied by a constant depending on T but not on x and t.