Riemann–Hilbert approach and long-time asymptotics of the positive flow short-pulse equation

Abstract

In this paper, we study the Cauchy problem of the positive flow short-pulse equation by using the Riemann–Hilbert approach, from which we obtain multi-soliton formulas of the Cauchy problem under the reflectionless case and the long-time asymptotic behavior of the solution of the Cauchy problem in the solitonless sector region. Starting with the spectral analysis of the Lax pair related to this equation, a Riemann–Hilbert problem is established by introducing some proper spectral function transformations and variable transformations, and the solution of the Cauchy problem for the positive flow short-pulse equation is transformed into the solution of the corresponding Riemann–Hilbert problem. Then we give the soliton classification of the positive flow short-pulse equation without reflection. Various Deift–Zhou contour deformations and the motivation behind them are given. We finally obtain the long-time asymptotics for the positive flow short-pulse equation in the solitonless sector region by resorting to the Deift–Zhou nonlinear steepest descent method.