Weak solvability and well-posedness of a coupled Schrödinger-Korteweg de Vries equation for capillary-gravity wave interactions
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- by Daniella Bekiranov, Takayoshi Ogawa and Gustavo Ponce PDF
- Proc. Amer. Math. Soc. 125 (1997), 2907-2919 Request permission
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.References
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Additional Information
- Daniella Bekiranov
- Affiliation: Department of Mathematics, Florida International University, Miami, Florida 33199
- Email: bekiranov@fiu.edu
- Takayoshi Ogawa
- Affiliation: Graduate School of Polymathematics, Nagoya University, Nagoya, 464-01 Japan
- MR Author ID: 289654
- Email: ogawa@math.nagoya-u.ac.jp
- Gustavo Ponce
- Affiliation: Department of Mathematics, University of California Santa Barbara, Santa Barbara, California 93106
- MR Author ID: 204988
- Email: ponce@math.ucsb.edu
- Received by editor(s): April 24, 1996
- Additional Notes: The third author was partially supported by an NSF grant.
- Communicated by: J. Marshall Ash
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 125 (1997), 2907-2919
- MSC (1991): Primary 35Q53, 35Q55, 76B15
- DOI: https://doi.org/10.1090/S0002-9939-97-03941-5
- MathSciNet review: 1403113