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On Schrödinger maps
Author(s) -
Nahmod Andrea,
Stefanov Atanas,
Uhlenbeck Karen
Publication year - 2003
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.10054
Subject(s) - mathematics , initial value problem , schrödinger's cat , space (punctuation) , cauchy distribution , nonlinear system , cauchy problem , mathematical analysis , pure mathematics , computer science , physics , quantum mechanics , operating system
Abstract We study the question of well‐posedness of the Cauchy problem for Schrödinger maps from ℝ × ℝ 2 to the sphere 2 or to ℍ 2 , the hyperbolic space. The idea is to choose an appropriate gauge change so that the derivatives of the map will satisfy a certain nonlinear Schrödinger system of equations and then study this modified Schrödinger map system (MSM). We then prove local well‐posedness of the Cauchy problem for the MSM with minimal regularity assumptions on the data and outline a method to derive well‐posedness of the Schrödinger map itself from it. In proving well‐posedness of the MSM, the heart of the matter is resolved by considering truly quatrilinear forms of weighted L 2 ‐functions. © 2002 Wiley Periodicals, Inc.