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 complete integrable system, which posses a Wadati-Konno-Ichikawa (WKI)-type Lax pair, we formulate a $2\times 2$ 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 problem, 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.

References in corpus (2)

• The short pulse equation is integrable
• On integrability of the vector short pulse equation

Cited by in corpus (3)

• Global solution of the Wadati-Konno-Ichikawa equation with small initial data
• Long-time asymptotic behavior for the complex short pulse equation
• An initial-boundary value problem for the coupled focusing-defocusing complex short pulse equation with a $4\times 4$ Lax pair