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The long-wave limit for the water wave problem. I: The case of zero surface tension. (English) Zbl 1034.76011
The paper revisits the classical problem of the description of long small-amplitude weakly nonlinear weakly dispersive surface waves in a long channel, neglecting the capillary effects. The analysis starts with a system of equations for two-dimensional irrotational inviscid flow. It is proved that, on a relatively short time scale, 1/ϵ, a general initial perturbation, in the form of a localized pulse, splits into two pulses which travel in opposite directions. At a longer time scale, ϵ -3 , each of the two pulses splits into an array of solitons obeying Korteweg-de Vries (KdV) equation. At the latter time scale, it is also proved that collisions between solitons can be described asymptotically correctly by KdV equation. The proofs are based on estimates of the difference between the solutions to the full water-wave system of equations and the asymptotic KdV equation. A novelty in comparison with previous works analyzing the rigorous correspondence between the full system and the approximation based on the KdV equation is that the class of functions admitted in the analysis does not exclude solitons and soliton trains.

MSC:
76B25Solitary waves (inviscid fluids)
35Q51Soliton-like equations
76M45Asymptotic methods, singular perturbations (fluid mechanics)