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The collisionless shock region for the long-time behavior of solutions of the KdV equation. (English) Zbl 0797.35143

In earlier work of the first and third authors, a general nonlinear steepest descent method was introduced to evaluate the asymptotics of oscillatory Riemann-Hilbert problems. The method has since been applied to the study of the long time asymptotics of the MKdV, NLS and Toda equations, to the long time behavior of the autocorrelation function of the transverse Ising chain at the critical value of the magnetic field, and also to the spatial asymptotics of the Painlevé II equation.
In this paper the authors further develop the method to give a full description of the collisionless shock region for solutions of the KdV equation with decaying initial data. In developing the method new phenomena and new technical issues arise.
Reviewer: P.Deift (New York)

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35B40 Asymptotic behavior of solutions to PDEs
76L05 Shock waves and blast waves in fluid mechanics
35Q15 Riemann-Hilbert problems in context of PDEs
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