Well-posedness for the Schrödinger-Korteweg-de Vries system

A. J. Corcho, F. Linares
2007 Transactions of the American Mathematical Society  
We study well-posedness of the Cauchy problem associated to the Schrödinger-Korteweg-de Vries system. We obtain local well-posedness for weak initial data, where the best result obtained is for data in the Sobolev space 3 4 + . This result implies in particular the global well-posedness in the energy space H 1 (R) × H 1 (R). Both results considerably improve the previous ones by 4089 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 4090
more » ... -terms-of-use 4090 A. J. CORCHO AND F. LINARES following conserved functionals for the flow defined by (1.1): namely, M(t) := +∞ −∞ |u| 2 dx = M(0), (1.2) Q(t) := +∞ −∞ αv 2 + 2γIm(u∂ x u) dx = Q(0), (1.3) and (1.4) E(t) := +∞ −∞ αγv|u| 2 − α 6 v 3 + βγ 2 |u| 4 + α 2 |∂ x v| 2 + γ|∂ x u| 2 dx = E(0).
doi:10.1090/s0002-9947-07-04239-0 fatcat:xo4renzcundi7mtjpv65izvndq