The large-time solution of a non-linear fourth-order equation initial-value problem I. Initial data with a discontinuous expansive step

Abstract

In this paper we consider an initial-value problem for the non-linear fourth order partial differential equation $u_t + uu_x + \gamma u_{xxxx}=0$, $-\infty< x < \infty$, t > 0, where x and t represent dimensionless distance and time respectively and \gamma is a negative constant. In particular, we consider the case when the initial data has a discontinuous expansive step so that $u(x,0)=u_0$(>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 $\gamma>0$, 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. doi:10.1017/S1446181110000015