Long-time asymptotics for the modified complex short pulse equation

Abstract

<p style='text-indent:20px;'>Based on the spectral analysis and the inverse scattering method, by introducing some spectral function transformations and variable transformations, the initial value problem for the modified complex short pulse (mCSP) equation is transformed into a <inline-formula><tex-math id="M1">\begin{document}$ 2\times2 $\end{document}</tex-math></inline-formula> matrix Riemann-Hilbert problem. It is proved that the solution of the initial value problem for the mCSP equation has a parametric expression related to the solution of the matrix Riemann-Hilbert problem. Various Deift-Zhou contour deformations and the motivation behind them are given. Through several appropriate transformations and strict error estimates, the original matrix Riemann-Hilbert problem can be reduced to the model Riemann-Hilbert problem, whose solution can be solved explicitly in terms of the parabolic cylinder functions. Finally, the long-time asymptotics of the solution of the initial value problem for the mCSP equation is obtained by using the nonlinear steepest decent method.</p>