The Large-Time Solution of a Nonlinear 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 nonlinear fourth-order partial differential equation u t + uu x + γ u x x x x = 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) = 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 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.