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ISSN 1088-6826(e) ISSN 0002-9939(p)

Weak solvability and well-posedness of a coupled Schrödinger-Korteweg de Vries equation for capillary-gravity wave interactions

Author(s): Daniella Bekiranov; Takayoshi Ogawa; Gustavo Ponce
Journal: Proc. Amer. Math. Soc. 125 (1997), 2907-2919.
MSC (1991): Primary 35Q53, 35Q55, 76B15
MathSciNet review: 1403113
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Abstract: An interaction equation of the capillary-gravity wave is considered. We show that the Cauchy problem of the coupled Schrödinger-KdV equation,

$\begin{equation*}\begin {cases} i \partial _tu + \partial ^2_x u= \alpha vu + \gamma |u|^2u, \qquad t,x\in \Bbb R, \partial _tv + \partial _x^3v + \partial _x v^2 = \beta \partial _x(|u|^2), u(x,0)=u_0(x), v(x,0)=v_0(x), \end {cases} \end{equation*}$

is locally well-posed for weak initial data $u_0\times v_0\in L^2(\Bbb R)\times H^{-1/2}(\Bbb R)$ . We apply the analogous method for estimating the nonlinear coupling terms developed by Bourgain and refined by Kenig, Ponce, and Vega.

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