Generating function of the Whitham-KdV hierarchy and effective solution of the Cauchy problem |
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Institution: | Max-Planck-Institute for Aeronomy, D-37191 Katlenburg-Lindau, Germany;Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, United States;Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow, 142190, Russia |
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Abstract: | Generating functions for a complete collection of symmetries of the multiphased averaged KdV equation are constructed. The isospectral generating function has a potential form with one of the canonical basis holomorphic differentials as a potential and possesses some remarkable properties at double points of the hyperelliptic Riemann surface. A new representation for the characteristic speeds of the Whitham-KdV hierarchy is obtained. A global solution to the Whitham system is constructed in an effective form for the case of smooth decreasing initial data with a finite number of inflection points. The large time asymptotics of this solution implies the single-phase limiting behaviour of the oscillations to correlate with the asymptotic predictions of the Lax-Levermore theory. |
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