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

Bekiranov, Daniella; Ogawa, Takayoshi; Ponce, Gustavo

doi:10.1090/S0002-9939-97-03941-5

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.

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