@article{AIHPC_2008__25_4_697_0, author = {Banica, V. and Vega, L.}, title = {On the {Dirac} delta as initial condition for nonlinear {Schr\"odinger} equations}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {697--711}, publisher = {Elsevier}, volume = {25}, number = {4}, year = {2008}, doi = {10.1016/j.anihpc.2007.03.007}, mrnumber = {2436789}, zbl = {1147.35092}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2007.03.007/} }
TY - JOUR AU - Banica, V. AU - Vega, L. TI - On the Dirac delta as initial condition for nonlinear Schrödinger equations JO - Annales de l'I.H.P. Analyse non linéaire PY - 2008 SP - 697 EP - 711 VL - 25 IS - 4 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2007.03.007/ DO - 10.1016/j.anihpc.2007.03.007 LA - en ID - AIHPC_2008__25_4_697_0 ER -
%0 Journal Article %A Banica, V. %A Vega, L. %T On the Dirac delta as initial condition for nonlinear Schrödinger equations %J Annales de l'I.H.P. Analyse non linéaire %D 2008 %P 697-711 %V 25 %N 4 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2007.03.007/ %R 10.1016/j.anihpc.2007.03.007 %G en %F AIHPC_2008__25_4_697_0
Banica, V.; Vega, L. On the Dirac delta as initial condition for nonlinear Schrödinger equations. Annales de l'I.H.P. Analyse non linéaire, Tome 25 (2008) no. 4, pp. 697-711. doi : 10.1016/j.anihpc.2007.03.007. http://www.numdam.org/articles/10.1016/j.anihpc.2007.03.007/
[1] Travelling waves for the Gross-Pitaevskii equation I, Ann. Inst. H. Poincaré Phys. Theor. 70 (2) (1999) 147-238. | EuDML | Numdam | MR | Zbl
, ,[2] Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pures Appl. (9) 62 (1983) 73-97. | MR | Zbl
, ,[3] The Cauchy problem for the critical nonlinear Schrödinger in , Nonlinear Anal. TMA 14 (1990) 807-836. | MR | Zbl
, ,[4] Au delà des opérateurs pseudo-différentiels, Astérisque, vol. 57, Soc. Math. France, 1978. | Numdam | MR | Zbl
, ,[5] On the motion of an unbounded fluid with a vortex filament of any shape, Rend. Circ. Mat. Palermo 22 (1906) 117.
,[6] Self-similar solutions for the 1-D Schrödinger map on the hyperbolic plane, Math. Z. 257 (2007) 61-80. | MR | Zbl
,[7] A note on NLS and the Schrödinger flow of maps, Phys. Lett. A 248 (1) (1998) 49-56. | Zbl
,[8] Schrödinger group on Zhidkov spaces, Adv. Differential Equations 9 (2004) 509-538. | MR | Zbl
,[9] C. Gallo, The Cauchy problem for defocusing nonlinear Schrödinger equations with non-vanishing initial data at infinity, preprint. | MR | Zbl
[10] The Cauchy problem for the Gross-Pitaevskii equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006) 765-779. | Numdam | MR | Zbl
,[11] Introduction aux équations de Schrödinger non linéaires, Edition de Paris-Sud, 1998.
,[12] On a class of nonlinear Schrödinger equations, J. Funct. Anal. 32 (1979) 1-71. | MR | Zbl
, ,[13] Two remarks on solutions of Gross-Pitaevskii equations on Zhidkov spaces, Monatsh. Math. 151 (2007) 39-44. | MR | Zbl
,[14] A. Grünrock, Abstract in Mathematisches Forschunginstitut Oberwolfach report 50 (2004).
[15] Scattering for the Gross-Pitaevskii equation, Math. Res. Lett. 13 (2-3) (2006) 273-286. | MR | Zbl
, , ,[16] Formation of singularities and self-similar vortex motion under the localized induction approximation, Comm. Partial Differential Equations 28 (2003) 927-968. | MR | Zbl
, , ,[17] On the ill-posedness of some canonical non-linear dispersive equations, Duke Math. J. 106 (3) (2001) 617-633. | MR | Zbl
, , ,[18] N. Kita, Nonlinear Schrödinger equation with triple δ-functions as initial data, in: Sapporo Guest House Symposium 20 “Nonlinear Wave Equations”, 2005.
[19] Vortex filament equation and semilinear Schrödinger equation, in: Nonlinear Waves, Sapporo, 1995, GAKUTO Internat. Ser. Math. Sci. Appl., vol. 10, Gakkōtosho, Tokyo, 1997, pp. 231-236. | MR | Zbl
,[20] Long range scattering for nonlinear Schrödinger equations in one space dimension, Commun. Math. Phys. 139 (3) (1991) 479-493. | MR | Zbl
,[21] Global well-posedness for 1d non-linear Schrödinger equations for data with an infinite norm, J. Math. Pures Appl. 80 (10) (2001) 1029-1044. | MR | Zbl
, ,[22] Asymptotic analysis of an ordinary differential equation and non-uniqueness for a semilinear partial differential equation, Arch. Ration. Mech. Anal. 91 (1986) 231-245. | MR | Zbl
,[23] P.E. Zhidkov, The Cauchy problem for a nonlinear Schrödinger equation, Soobshch. OIYaI R5-87-373, Dubna, 1987.
[24] Korteveg-de-Vries and Nonlinear Schrödinger Equations: Qualitative Theory, Lecture Notes in Math., vol. 1756, Springer-Verlag, 2001. | MR | Zbl
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