×

zbMATH — the first resource for mathematics

Geometric optics and boundary layers for nonlinear-Schrödinger equations. (English) Zbl 1179.35303
Authors’ abstract: We justify supercritical geometric optics in small time for the defocusing semiclassical nonlinear Schrödinger equation for a large class of non-necessarily homogeneous nonlinearities. The case of a half-space with Neumann boundary condition is also studied.

MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
78A05 Geometric optics
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Alazard T., Carles R.: Loss of regularity for supercritical nonlinear Schrödinger equations. Math. Ann. 343(2), 397–420 (2009) · Zbl 1161.35047 · doi:10.1007/s00208-008-0276-6
[2] Alazard, T., Carles, R.: Supercritical geometric optics for nonlinear Schrödinger equations. Arch. Rat. Mech. Anal., to appear, doi: 10.1007s00205-008-0176-7 , 2008 · Zbl 1179.35302
[3] Anton R.: Global existence for defocusing cubic NLS and Gross-Pitaevskii equations in exterior domains. J. Math. Pures Appl. (9) 89(4), 335–354 (2008) · Zbl 1148.35081
[4] Brenier Y.: Convergence of the Vlasov-Poisson system to the incompressible Euler equations. Comm. Part. Diff. Eqs. 25(3-4), 737–754 (2000) · Zbl 0970.35110 · doi:10.1080/03605300008821529
[5] Cazenave, T.: Semilinear Schrödinger equations. Courant Lecture Notes in Mathematics, Vol. 10. New York: New York University, Courant Institute of Mathematical Sciences, 2003 · Zbl 1055.35003
[6] Colliander J., Keel M., Staffilani G., Takaoka H., Tao T.: Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in \({\mathbb{R}^3}\) . Ann. of Math. (2) 167(3), 767–865 (2008) · Zbl 1178.35345 · doi:10.4007/annals.2008.167.767
[7] Gérard, P.: Remarques sur l’analyse semi-classique de l’équation de Schrödinger non linéaire. Séminaire sur les Equations aux Dérivées Partielles, Ecole Polytechnique, Palaiseau, 1992-1993, Exp. No. XIII, 13 pp.
[8] Ginibre J., Velo G.: The global Cauchy problem for the nonlinear Schrödinger equation revisited. Ann. Inst. H. Poincaré Anal. Non Linéaire 2(4), 309–327 (1985) · Zbl 0586.35042
[9] Grenier E.: Semiclassical limit of the nonlinear Schrödinger equation in small time. Proc. Amer. Math. Soc. 126(2), 523–530 (1998) · Zbl 0910.35115 · doi:10.1090/S0002-9939-98-04164-1
[10] Grenier E.: On the derivation of homogeneous hydrostatic equations. M2AN Math. Model. Numer. Anal. 33(5), 965–970 (1999) · Zbl 0947.76013 · doi:10.1051/m2an:1999128
[11] Grenier E., Guès O.: Boundary layers of viscous perturbations of noncharacteristic quasilinear hyperbolic problems. J. Diff. Eqs. 143(1), 110–146 (1998) · Zbl 0896.35078 · doi:10.1006/jdeq.1997.3364
[12] Kivshar Y.S., Luther-Davies B.: Dark optical solitons: physics and applications. Physics Reports 298, 81–197 (1998) · doi:10.1016/S0370-1573(97)00073-2
[13] Kolomeisky E.B., Newman T.J., Straley X., Qi J.P. : Low-Dimensional Bose Liquids: Beyond the Gross-Pitaevskii Approximation. Phys. Rev. Lett. 85, 1146–1149 (2000) · doi:10.1103/PhysRevLett.85.1146
[14] Lin F., Zhang P.: Semiclassical limit of the Gross-Pitaevskii equation in an exterior domain. Arch. Rat. Mech. Anal. 179(1), 79–107 (2006) · Zbl 1079.76016 · doi:10.1007/s00205-005-0383-4
[15] Makino T., Ukai S., Kawashima S.: Sur la solution à support compact de l’équations d’Euler compressible. Japan J. Appl. Math. 3(2), 249–257 (1986) · Zbl 0637.76065 · doi:10.1007/BF03167100
[16] Pham C.-T., Nore C., Brachet M.-E.: Boundary layers and emitted excitations in nonlinear Schrödinger superflow past a disk. Phys. D 210(3-4), 203–226 (2005) · Zbl 1078.35116 · doi:10.1016/j.physd.2005.07.013
[17] Rauch J.: Symmetric positive systems with boundary characteristic of constant multiplicity. Trans. Amer. Math. Soc. 291(1), 167–187 (1985) · Zbl 0549.35099 · doi:10.1090/S0002-9947-1985-0797053-4
[18] Taylor, M.: Partial Differential Equations. (III), Applied Mathematical Sciences, 117. New-York: Springer-Verlag, 1997
[19] Zhang P.: Semiclassical limit of nonlinear Schrödinger equation. II. J. Part. Diff. Eqs. 15(2), 83–96 (2002) · Zbl 1003.35116
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.