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On quadratic Schrödinger equations in R 1+1 : a normal form approach
Author(s) -
Oh Seungly,
Stefanov Atanas
Publication year - 2012
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jds016
Subject(s) - lipschitz continuity , corollary , quadratic equation , mathematics , operator (biology) , schrödinger equation , mathematical physics , mathematical analysis , transformation (genetics) , property (philosophy) , nonlinear system , pure mathematics , physics , quantum mechanics , geometry , biochemistry , chemistry , philosophy , epistemology , repressor , transcription factor , gene
For the Schrödinger equation u t + iu xx = 〈▿〉 β [ u 2 ], β ∈ (0, ½), we establish local well‐posedness in H β −1+ (note that if β =0, this matches, up to an endpoint, the sharp result of Bejenaru–Tao [Sharp well‐posedness and ill‐posedness results for a quadratic non‐linear Schrödinger equation, J. Funct. Anal. 233 (2006) 228–259.]). Our approach differs significantly from the previous one, we use normal form transformation to analyze the worst interacting terms in the nonlinearity and then show that the remaining terms are (much) smoother. In particular, this allows us to conclude that u − e − it ∂ x 2 u (0) ∈ H −1/2 ( R 1 ), even though u (0) ∈ H β −1+ . In addition, as a byproduct of our normal form analysis, we obtain a Lipschitz continuity property in H −1/2 of the solution operator (which originally acts on H β −1+ ), which is new even in the case β =0. As an easy corollary, we obtain local well‐posedness results for u t + iu xx = 〈▿〉 β z 〈▿〉 β z . Finally, we sketch an approach to obtain similar results for the equations u t + iu xx = 〈▿〉 β [ u ū ] and u t + iu xx = 〈▿〉 β [ ū 2 ].