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Existence and uniqueness of the solution to the modified Schrödinger map. (English) Zbl 1082.35140

The author studies the initial value problem for a system of nonlinear Schrödinger equations in two space dimensions (modified Schrödinger map) which is derived from Schrödinger maps from \(\mathbb{R}\times \mathbb{R}^2\) to the unit sphere \(S^2\) or to the hyperbolic space \(\mathbb{H}^2\) by using appropriate gauge change. The existence and uniqueness of the solution was known for data in \(H^s(\mathbb{R}^2)\) with \(s> 1\). In this paper the local existence of the solution is proved for the initial data in \(H^s(\mathbb{R}^2)\) with \(s> 1/2\). The uniqueness of the solution is also proved when the data belong to \(H^1(\mathbb{R}^2)\).

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35G25 Initial value problems for nonlinear higher-order PDEs
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