Long-time asymptotics for the short pulse equation

Abstract

In this paper, we analyze the long-time behavior of the solution of the initial value problem (IVP) for the short pulse (SP) equation. As the SP equation is a completely integrable system, which posses a Wadati–Konno–Ichikawa (WKI)-type Lax pair, we formulate a 2×2 matrix Riemann–Hilbert problem to this IVP by using the inverse scattering method. Since the spectral variable k is the same order in the WKI-type Lax pair, we construct the solution of this IVP parametrically in the new scale (y,t), whereas the original scale (x,t) is given in terms of functions in the new scale, in terms of the solution of this Riemann–Hilbert problem. However, by employing the nonlinear steepest descent method of Deift and Zhou for oscillatory Riemann–Hilbert problems, we can get the explicit leading order asymptotic of the solution of the short pulse equation in the original scale (x,t) as time t goes to infinity.