Long-time asymptotic behavior for an extended modified Korteweg-de Vries equation

Abstract

We investigate an integrable extended modified Korteweg-de Vries equation on the line with the initial value belonging to the Schwartz space. By performing the nonlinear steepest descent analysis of an associated matrix Riemann--Hilbert problem, we obtain the explicit leading-order asymptotics of the solution of this initial value problem as time $t$ goes to infinity. For a special case $\alpha=0$, we present the asymptotic formula of the solution to the extended modified Korteweg-de Vries equation in region $\mathcal{P}=\{(x,t)\in\bfR^2|0<x\leq Mt^{\frac{1}{5}},t\geq3\}$ in terms of the solution of a fourth order Painlev\'e II equation.