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Approximation of the Korteweg-de Vries equation by the nonlinear Schrödinger equation. (English) Zbl 0940.35179

In 1968, Zakharov derived the nonlinear Schrödinger equation as an approximation equation for the water wave problem. Naturally, one wonders if solutions of the water wave problem can actually be rigorously approximated by solutions of the nonlinear Schrödinger equation. This paper considers solutions of the Korteweg-de Vries equation, which describes water waves with small amplitude in a long and narrow canal. Let $\psi$ be the approximation wave constructed from a fixed $C\left(\left[0,T\right],{H}^{s+5}\left(ℝ,ℂ\right)\right)$-solution of the nonlinear Schrödinger equation, where $s\ge 3$ and $T>0$. Then, the following result is shown: There are $C$, ${\epsilon }_{0}>0$ such that for each $\epsilon \in \left(0,{\epsilon }_{0}\right)$, the Korteweg-de Vries equation has a solution $u$ satisfying

$\underset{t\in \left[0,T/{\epsilon }^{2}\right]}{sup}{\parallel u-\psi \parallel }_{{H}^{s}\left(ℝ,ℝ\right)}\le C{\epsilon }^{3/2}·$

##### MSC:
 35Q53 KdV-like (Korteweg-de Vries) equations 35Q55 NLS-like (nonlinear Schrödinger) equations 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
##### Keywords:
water wave problem; KdV equation; NLS equation; approximation