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An approximation scheme for Schrödinger maps. (English) Zbl 1122.35138
The author studies the Cauchy problem for Schrödinger maps from \(\mathbb R^d\times \mathbb R^+\) \((d \geq 2)\) into a target manifold \(N\) with a complex structure \(J\). The Schrödinger map equation is given by \[ \partial _t u = J(u)D_k \partial _k u, \tag{1} \] where \(D\) is the covariant derivative along the curve \(u\). A Schrödinger map is a function \(u:[0,T]\times \mathbb R^d \to N\) that solves the equation (1). To show that solutions of the Cauchy problem for the Schrödinger map equation (1) exist, the author first studies the following approximate equation for any \(\delta > 0\) \[ \delta ^2D_t \partial_t u^\delta - J(u^\delta )\partial_t u^\delta - D_m \partial_m u^\delta = 0. \tag{2} \] The equation (2) is a wave map, for which general existence theory is known. For appropriate initial data there is a sequence of local solutions \(u^\delta\) of equation (2) that exist on the time intervals \([0,T_\delta]\). The limit of these approximate solutions as \(\delta \to 0\) solves the Schrödinger map problem. Energy estimates which bound the norms of solutions \(u^\delta\) imply that \(T_\delta\) is independent of \(\delta\). Then, for some fixed \(T > 0\), the time interval of existence for each solution \(u^\delta\) is \([0,T]\), and their limit \(u\) exist on this same interval. By a standard convergence argument it is proven that \(u\) satisfies the Schrödinger map equation (1). The uniqueness of the Schrödinger map is also shown.

35Q55 NLS equations (nonlinear Schrödinger equations)
35A35 Theoretical approximation in context of PDEs
35G25 Initial value problems for nonlinear higher-order PDEs
Full Text: DOI
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