TY - JOUR
T1 - A review of dispersive limits of (non)linear schrodinger-type equations
AU - Gasser, Ingenuin
AU - Lin, Chi K.
AU - Markowich, Peter
N1 - Affiliations: Fachbereich Mathematik, Universität Hamburg, Bundesstrabe 55, 20146 Hamburg, Germany; Department of Mathematics, National Cheng Kung University, Tainan 701, Taiwan; Institut für Mathematik, Universität Wien, Boltzmanngasse 9, Vienna A-1090, Austria
Adressen: Gasser, I.; Fachbereich Mathematik; Universität Hamburg; Bundesstrabe 55 20146 Hamburg, Germany; email: [email protected]
Source-File: 506Scopus.csv
Import aus Scopus: 2-s2.0-0000538578
Importdatum: 24.01.2007 11:28:11
22.10.2007: Datenanforderung 1920 (Import Sachbearbeiter)
04.01.2008: Datenanforderung 2054 (Import Sachbearbeiter)
PY - 2000
Y1 - 2000
N2 - In this review paper we present the most important mathematical properties of dispersive limits of (non)linear Schrošdinger type equations. Different formulations are used to study these singular limits, e.g., the kinetic formulation of the linear Schrošdinger equation based on the Wigner transform is well suited for global-in-time analysis without using WKB-(expansion) techniques, while the modified Madelung transformation reformulating Schrošdinger equations in terms of a dispersive perturbation of a quasilinear symmetric hyperbolic system usually only gives local-in-time results due to the hyperbolic nature of the limit equations. Deterministic analogues of turbulence are also discussed. There, turbulent diffusion appears naturally in the zero dispersion limit.
AB - In this review paper we present the most important mathematical properties of dispersive limits of (non)linear Schrošdinger type equations. Different formulations are used to study these singular limits, e.g., the kinetic formulation of the linear Schrošdinger equation based on the Wigner transform is well suited for global-in-time analysis without using WKB-(expansion) techniques, while the modified Madelung transformation reformulating Schrošdinger equations in terms of a dispersive perturbation of a quasilinear symmetric hyperbolic system usually only gives local-in-time results due to the hyperbolic nature of the limit equations. Deterministic analogues of turbulence are also discussed. There, turbulent diffusion appears naturally in the zero dispersion limit.
M3 - Review
SN - 1027-5487
VL - 4
SP - 501
EP - 529
JO - Taiwanese Journal of Mathematics
JF - Taiwanese Journal of Mathematics
IS - 4
ER -