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

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. |