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

Kato, Jun

Math. Res. Lett. 12, No. 2-3, 171-186 (2005).

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 \(H^1(\mathbb{R}^2)\).

Reviewer: Viorel Iftimie (Bucureşti)

MSC:

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

Keywords:

nonlinear Schrödinger equations; modified Schrödinger map; initial value problem