Communications in Mathematical Sciences

Volume 17 (2019)

Number 7

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

Pages: 1877 – 1913



Nan Liu (Institute of Applied Physics and Computational Mathematics, Beijing, China)

Boling Guo (Institute of Applied Physics and Computational Mathematics, Beijing, China)

Dengshan Wang (School of Applied Science, Beijing Information Science and Technology University, Beijing, China)

Yufeng Wang (College of Science, Minzu University of China, Beijing, China)


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 $\mathcal{P} = \lbrace (x,t) \in \mathbb{R}^2 \vert 0 \lt x \leq Mt^{\frac{1}{5}} , t \geq 3 \rbrace$ in terms of the solution of a fourth order Painlevé II equation.


extended modified Korteweg–de Vries equation, Riemann–Hilbert problem, nonlinear steepest descent method, long-time asymptotics

2010 Mathematics Subject Classification

35G25, 35Q15, 37K40

Received 27 June 2018

Accepted 20 June 2019

Published 6 January 2020