The Cauchy problem for the Schrödinger-KdV system |
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Authors: | Hua Wang Shangbin Cui |
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Institution: | a Department of Mathematics, Huazhong Normal University, Wuhan, Hubei 430079, China b Département de Mathématiques et Applications, École Normale Supérieure, 45 rue d?Ulm, F 75230, Paris, cedex 05, France c Department of Mathematics, Sun Yat-Sen University, Guangzhou, Guangdong 510275, China |
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Abstract: | In this paper we prove that in the general case (i.e. β not necessarily vanishing) the Cauchy problem for the Schrödinger-Korteweg-de Vries system is locally well-posed in , and if β=0 then it is locally well-posed in with . These results improve the corresponding results of Corcho and Linares (2007) 5]. Idea of the proof is to establish some bilinear and trilinear estimates in the space Gs×Fs, where Gs and Fs are dyadic Bourgain-type spaces related to the Schrödinger operator and the Airy operator , respectively, but with a modification on Fs in low frequency part of functions with a weaker structure related to the maximal function estimate of the Airy operator. |
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Keywords: | 35Q55 35Q60 35B65 |
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