Long-time asymptotics for the short pulse equation

Abstract

In this paper, we consider the initial value problem for the complex short pulse equation with a Wadati-Konno-Ichikawa type Lax pair. We show that the solution to the initial value problem has a parametric expression in terms of the solution of 2×2-matrix Riemann-Hilbert problem, from which an implicit onesoliton solution is obtained on the discrete spectrum. While on the continuous spectrum we further establish the explicit long-time asymptotic behavior of the non-soliton solution by using Deift-Zhou nonlinear steepest descent method.

Multi-Cuspon Solutions of the Wadati-Konno-Ichikawa Equation by Riemann-Hilbert Problem Method

“…In this paper, the solutions of the WKI equation are recovered in terms of the solution of the matrix Riemann-Hilbert problem from the order ( ) O λ at 0 λ → , like the case of the SP equation [9]. However, the novelty of our paper Open Journal of Applied Sciences is that 0 λ = is non-singularity of the WKI equation.…”

New Soliton Solutions to the Initial Value Problem for the Two-Component Short Pulse Equation

“…, , e ; b, c and 0 y are real constants; denoting [6], since the parameter chosen satisfies the symmetry condition of SP equation as [9].…”

“…Here, in this paper, we consider the initial value problem for the 2SPE 4with the initial value data ( ) Organization of the paper: In Section 2, we formulate the associated Riemann-Hilbert problem by performing the similar spectral analysis to the Equation (4) as [9]. In Section 3, to obtain the soliton solutions of the Equation 4, we need analyze the residue conditions of the Riemann-Hilbert problem.…”

The short pulse equation: Bäcklund transformations and applications

“…Saleem and Hassan considered the Darboux transformation in terms of quasi‐determinants 10 and Liu et al showed that the quasi‐determinants can be avoided and the ‐fold Darboux transformation may be given by determinants 11 . Recently, a Riemann‐Hilbert approach was applied to the SP equation 12,13 and the long‐time behavior of the solution was studied. It is interesting to note that Matsuno obtained the periodic solutions for the SP equation 14 …”