On the Dirac delta as initial condition for nonlinear Schrödinger equations
Banica, V. ; Vega, L.
Annales de l'I.H.P. Analyse non linéaire, Tome 25 (2008), p. 697-711 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU) 
@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},
     volume = {25},
     year = {2008},
     pages = {697-711},
     doi = {10.1016/j.anihpc.2007.03.007},
     mrnumber = {2436789},
     zbl = {1147.35092},
     language = {en},
     url = {http://dml.mathdoc.fr/item/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) pp. 697-711. doi : 10.1016/j.anihpc.2007.03.007. http://gdmltest.u-ga.fr/item/AIHPC_2008__25_4_697_0/

[1] Bethuel F., Saut J.-C., Travelling waves for the Gross-Pitaevskii equation I, Ann. Inst. H. Poincaré Phys. Theor. 70 (2) (1999) 147-238. | Numdam | MR 1669387 | Zbl 0933.35177

[2] Brézis H., Friedman A., Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pures Appl. (9) 62 (1983) 73-97. | MR 700049 | Zbl 0527.35043

[3] Cazenave T., Weissler F., The Cauchy problem for the critical nonlinear Schrödinger in H s , Nonlinear Anal. TMA 14 (1990) 807-836. | MR 1055532 | Zbl 0706.35127

[4] Coifman R.R., Meyer Y., Au delà des opérateurs pseudo-différentiels, Astérisque, vol. 57, Soc. Math. France, 1978. | MR 518170 | Zbl 0483.35082

[5] Da Rios L.S., On the motion of an unbounded fluid with a vortex filament of any shape, Rend. Circ. Mat. Palermo 22 (1906) 117.

[6] De La Hoz F., Self-similar solutions for the 1-D Schrödinger map on the hyperbolic plane, Math. Z. 257 (2007) 61-80. | MR 2318570 | Zbl 1128.35099

[7] Ding Q., A note on NLS and the Schrödinger flow of maps, Phys. Lett. A 248 (1) (1998) 49-56. | Zbl 1115.35368

[8] Gallo C., Schrödinger group on Zhidkov spaces, Adv. Differential Equations 9 (2004) 509-538. | MR 2099970 | Zbl 1103.35093

[9] C. Gallo, The Cauchy problem for defocusing nonlinear Schrödinger equations with non-vanishing initial data at infinity, preprint. | MR 2424376 | Zbl 1156.35086

[10] Gérard P., The Cauchy problem for the Gross-Pitaevskii equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006) 765-779. | Numdam | MR 2259616 | Zbl 1122.35133

[11] Ginibre J., Introduction aux équations de Schrödinger non linéaires, Edition de Paris-Sud, 1998.

[12] Ginibre J., Velo G., On a class of nonlinear Schrödinger equations, J. Funct. Anal. 32 (1979) 1-71. | MR 533219 | Zbl 0396.35029

[13] Goubet O., Two remarks on solutions of Gross-Pitaevskii equations on Zhidkov spaces, Monatsh. Math. 151 (2007) 39-44. | MR 2317389 | Zbl 1128.35096

[14] A. Grünrock, Abstract in Mathematisches Forschunginstitut Oberwolfach report 50 (2004).

[15] Gustafson S., Nakanishi K., Tsai T.-P., Scattering for the Gross-Pitaevskii equation, Math. Res. Lett. 13 (2-3) (2006) 273-286. | MR 2231117 | Zbl 1119.35084

[16] Gutiérrez S., Rivas J., Vega L., Formation of singularities and self-similar vortex motion under the localized induction approximation, Comm. Partial Differential Equations 28 (2003) 927-968. | MR 1986056 | Zbl 1044.35089

[17] Kenig C., Ponce G., Vega L., On the ill-posedness of some canonical non-linear dispersive equations, Duke Math. J. 106 (3) (2001) 617-633. | MR 1813239 | Zbl 1034.35145

[18] N. Kita, Nonlinear Schrödinger equation with triple δ-functions as initial data, in: Sapporo Guest House Symposium 20 “Nonlinear Wave Equations”, 2005.

[19] Koiso N., 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 1602745 | Zbl 0890.35144

[20] Ozawa T., Long range scattering for nonlinear Schrödinger equations in one space dimension, Commun. Math. Phys. 139 (3) (1991) 479-493. | MR 1121130 | Zbl 0742.35043

[21] Vargas A., Vega L., Global well-posedness for 1d non-linear Schrödinger equations for data with an infinite L 2 norm, J. Math. Pures Appl. 80 (10) (2001) 1029-1044. | MR 1876762 | Zbl 1027.35134

[22] Weissler F., 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 806003 | Zbl 0614.35043

[23] P.E. Zhidkov, The Cauchy problem for a nonlinear Schrödinger equation, Soobshch. OIYaI R5-87-373, Dubna, 1987.

[24] Zhidkov P.E., Korteveg-de-Vries and Nonlinear Schrödinger Equations: Qualitative Theory, Lecture Notes in Math., vol. 1756, Springer-Verlag, 2001. | MR 1831831 | Zbl 0987.35001