Long-time asymptotics for the Nonlocal mKdV equation

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.

27 pages

References in corpus (4)

• Long-Time Asymptotics for the Camassa-Holm Equation
• Alice-Bob Physics: Coherent Solutions of Nonlocal KdV Systems
• Long-Time Asymptotics for the Korteweg-de Vries Equation via Nonlinear Steepest Descent
• Long-time asymptotic for the derivative nonlinear Schrödinger equation with decaying initial value

Cited by in corpus (6)

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• Long time asymptotic behavior for the nonlocal mKdV equation in space-time solitonic regions-II
• Superposed periodic kink and pulse solutions of coupled nonlinear equations
• Long time asymptotics for the nonlocal mKdV equation with finite density initial data