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The International Conference on Multiphase Systems, ICMS'2000
Ufa, RUSSIA, June 15-17, 2000

Zero-Dispersion Limit to the Korteweg-de Vries Equation: a Dressing Chain Approach

Novokshenov V. Yu.

Institute of Mathematics of R.A.S., Ufa, RUSSIA

An asymptotic solution of the KdV equation with small dispersion is studied for the case of smooth hump-like initial condition with monotonically decreasing slopes. Despite the well-known approaches by Lax-Levermore and Gurevich-Pitaevskii, a new way of constructing the asymptotics is proposed using the inverse scattering transform together with the dressing chain technique recently developed by A.Shabat. It enables to get the Whitham-type approximaton of the leading term by solving the dressing chain through a finite-gap asymptotic ansatz. This yields the Whitham equations on the Riemann invariants together with hodograph transform for solving them. They reproduce uniform in $x$ asymptotics combined by the smooth solution of the Hopf equation outside the oscillating domain and a cnoidal wave modulated by Whitham equations within the domain. Finally, the dressing chain techniqe provides the proof of an asymptotic estimate for the leading term.