For shifted-step initial data, the long-time solution of the nonlocal focusing NLS splits into 4n+2 sectors: alternating sectors decay to 0 or approach explicit constants.
Long-time asymptotics for the Nonlocal mKdV equation
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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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math.AP 1years
2019 1verdicts
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Long-time asymptotics for the integrable nonlocal focusing nonlinear Schr\"odinger equation for a family of step-like initial data
For shifted-step initial data, the long-time solution of the nonlocal focusing NLS splits into 4n+2 sectors: alternating sectors decay to 0 or approach explicit constants.