THE LARGE-TIME SOLUTION OF A NONLINEAR FOURTH-ORDER EQUATION INITIAL-VALUE PROBLEM I. INITIAL DATA WITH A DISCONTINUOUS EXPANSIVE STEP

Abstract

AbstractIn this paper we consider an initial-value problem for the nonlinear fourth-order partial differential equation ut+uux+γuxxxx=0, −∞<x<∞, t>0, where x and t represent dimensionless distance and time respectively and γ is a negative constant. In particular, we consider the case when the initial data has a discontinuous expansive step so that u(x,0)=u0(>0) for x≥0 and u(x,0)=0 for x<0. The method of matched asymptotic expansions is used to obtain the large-time asymptotic structure of the solution to this problem which exhibits the formation of an expansion wave. Whilst most physical applications of this type of equation have γ>0, our calculations show how it is possible to infer the large-time structure of a whole family of solutions for a range of related equations.