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arXiv · 1804.10863

Long-time asymptotics for the Nonlocal mKdV equation

Fengjing He·Engui Fan·Jian Xu
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Abstract

In this paper, we study the Cauchy problem with decaying initial data for the nonlocal modified Korteweg-de Vries equation (nonlocal mKdV) \[q_t(x,t)+q_{xxx}(x,t)-6q(x,t)q(-x,-t)q_x(x,t)=0,\] which can be viewed as a generalization of the local classical mKdV equation. We first formulate the Riemann-Hilbert problem associated with the Cauchy problem of the nonlocal mKdV equation. Then we apply the Deift-Zhou nonlinear steepest-descent method to analyze the long-time asymptotics for the solution of the nonlocal mKdV equation. In contrast with the classical mKdV equation, we find some new and different results on long-time asymptotics for the nonlocal mKdV equation and some additional assumptions about the scattering data are made in our main results.

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Fengjing He, Engui Fan, Jian Xu. 2018-04-29. Long-time asymptotics for the Nonlocal mKdV equation. https://doi.org/10.1088/0253-6102%2F71%2F5%2F475

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