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

Authors: Daniella Bekiranov, Takayoshi Ogawa and 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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Additional Information

Daniella Bekiranov
Affiliation: Department of Mathematics, Florida International University, Miami, Florida 33199

Takayoshi Ogawa
Affiliation: Graduate School of Polymathematics, Nagoya University, Nagoya, 464-01 Japan
MR Author ID: 289654

Gustavo Ponce
Affiliation: Department of Mathematics, University of California Santa Barbara, Santa Barbara, California 93106
MR Author ID: 204988

Keywords: Capillary-gravity wave, nonlinear Schrödinger, KdV, well-posedness
Received by editor(s): April 24, 1996
Additional Notes: The third author was partially supported by an NSF grant.
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1997 American Mathematical Society