zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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.

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