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Oscillations of the zero dispersion limit of the Korteweg-de Vries equation. (English) Zbl 0810.35114
Summary: We attack the multiphase averaged systems for the zero dispersion limit of the KdV equation. Attention is paid to the most important case – the single phase oscillations. A scheme is developed to solve the Whitham averaged system (single phase averaged system). This system, under our scheme, is transformed to a linear over-determined system of Euler- Poisson-Darboux type whose solution can be written down explicitly.
We show that, for any smooth initial data which has only one hump or is a nontrivial monotone function, the weak limit has single-phase oscillations within a cusp in the \(x-t\) plane for a short time after the breaking time for the corresponding Burgers equation. Outside the cusp, the limit satisfies the Burgers equation. More surprisingly, we also show that the weak limit has global single-phase oscillations within a cusp for any smooth nontrivial monotone initial data with only one inflection point.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35Q05 Euler-Poisson-Darboux equations
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