On long-time asymptotic behavior and Painlevé asymptotic to the matrix Hirota equation

Abstract

Abstract The nonlinear descent method is extended to study the long-time asymptotic behavior of the matrix Hirota equation with $4\times 4$ Lax pair in Schwartz space. The implementation of spectral analysis successfully transforms the Cauchy problem of the matrix Hirota equation into the corresponding high-order Riemann–Hilbert (RH) with $4\times 4$ jump matrix, and further analyses the established oscillation RH problem to study the asymptotic behavior of the solution in the space-time plane. Interestingly, the space-time plane $\{(x,t)|-\infty <x<+\infty , t>0\}$ can be divided into three different asymptotic regions based on the phase function and $\xi =x/t$. The first one is the oscillatory region $\xi <\frac{\alpha ^{2}}{3\beta }$, whose leading-term can be approximated applying the Weber equation with an error of $\mathcal{O}(t^{-1}\log t)$. The second region is the Painlevé region $\xi \approx \frac{\alpha ^{2}}{3\beta }$, whose leading-term can be approximated by the coupled Painlevé II equation, which is related to a $4\times 4$ matrix RH problem with an error of $\mathcal{O}(t^{-\frac{2}{3}})$. The last region is the fast decay region $\xi>\frac{\alpha ^{2}}{3\beta }$, which solution is rapidly decreasing as $t\rightarrow \infty $. Our results provide a detailed proof for the asymptotic analysis for the solution of the matrix Hirota equation on the complete space-time plane.