On well-posedness and wave operator for the gKdV equation

2012/04/25 by Luiz Gustavo Farah, Farah, Luiz Gustavo, Ademir Pastor +1
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #math.AP

paper · pdf · doi:10.48550/arxiv.1204.5688

11 pages. To appear Bulletin des Sciences Mathématiques

arxiv created 2012/04/25 · arxiv updated 2012/04/26

Abstract

We consider the generalized Korteweg-de Vries (gKdV) equation ∂_t u+∂_x³u+μ∂_x(u^k+1)=0, where k>4 is an integer number and μ=±1. We give an alternative proof of the Kenig, Ponce, and Vega result in \citekpv1, which asserts local and global well-posedness in H^s_k(\R), with s_k=(k-4)/2k. A blow-up alternative in suitable Strichatz-type spaces is also established. The main tool is a new linear estimate. As a consequence, we also construct a wave operator in the critical space H^s_k(\R), extending the results of Côte [2].

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