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An approximation scheme for Schrödinger maps. (English) Zbl 1122.35138

The author studies the Cauchy problem for Schrödinger maps from d × + (d2) into a target manifold N with a complex structure J. The Schrödinger map equation is given by

t u=J(u)D k k u,(1)

where D is the covariant derivative along the curve u. A Schrödinger map is a function u:[0,T]× d 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 δ>0

δ 2 D t t u δ -J(u δ ) t u δ -D m m u δ =0·(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 δ of equation (2) that exist on the time intervals [0,T δ ]. The limit of these approximate solutions as δ0 solves the Schrödinger map problem. Energy estimates which bound the norms of solutions u δ imply that T δ is independent of δ. Then, for some fixed T>0, the time interval of existence for each solution u δ 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.

MSC:
35Q55NLS-like (nonlinear Schrödinger) equations
35A35Theoretical approximation to solutions of PDE
35G25Initial value problems for nonlinear higher-order PDE