Volume 3 (2020)
Colin, Mathieu;  Watanabe, Tatsuya
Cauchy problem for the nonlinear Schrödinger equation coupled with the Maxwell equation
Annales Henri Lebesgue, Volume 3 (2020), pp. 67-85.

Metadata

DOI10.5802/ahl.27
Keywords Schrödinger–Maxwell system, Cauchy problem, symmetric hyperbolic system, energy method

Abstract

In this paper, we study the nonlinear Schrödinger equation coupled with the Maxwell equation. Using energy methods, we obtain a local existence result for the Cauchy problem.


References

[ADM17] Antonelli, Paolo; D’Amico, Michele; Marcati, Pierangelo Nonlinear Maxwell-Schrödinger system and quantum magneto-hydrodynamics in 3-D, Commun. Math. Sci., Volume 15 (2017), pp. 451-479 | DOI | Zbl

[AG91] Alinhac, Serge; Gérard, Patrick Opérateurs pseudo-différentiels et théorème de Nash–Moser, Savoirs Actuels, Éditions du CNRS, 1991 | Zbl

[AP08] Azzollini, Antonio; Pomponio, Alessio Ground state solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., Volume 345 (2008), pp. 90-108 | DOI | Zbl

[BdBS95] Bergé, L.; de Bouard, Anne; Saut, Jean-Claude Blowing up time-dependent solutions of the planar Chern-Simons gauged nonlinear Schrödinger equation, Nonlinearity, Volume 8 (1995) no. 2, pp. 235-253 | DOI | Zbl

[BF14] Benci, Vieri; Fortunato, Donato Solitons in Schrödinger-Maxwell equations, J. Fixed Point Theory Appl., Volume 15 (2014) no. 1, pp. 101-132 | DOI | Zbl

[BS11] Bellazzini, Jacopo; Siciliano, Gaetano Stable standing waves for a class of nonlinear Schrödinger-Poisson equations, Z. Angew. Math. Phys., Volume 62 (2011), pp. 267-280 | DOI | Zbl

[BT09] Bejenaru, Ioan; Tataru, Daniel Global wellposedness in the energy space for the Maxwell-Schrödinger system, Commun. Math. Phys., Volume 288 (2009) no. 1, pp. 145-198 | DOI | Zbl

[CC04] Colin, Mathieu; Colin, Thierry On a quasilinear Zakharov system describing laser-plasma interactions, Differ. Integral Equ., Volume 17 (2004) no. 3-4, pp. 297-330 | MR | Zbl

[CDSS13] Catto, Isabelle; Dolbeault, Jean; Sánchez, Oscar D.; Soler, Juan S. Existence of steady states for the Maxwell-Schrödinger-Poisson system: exploring the applicability of the concentration-compactness principle, Math. Models Methods Appl. Sci., Volume 23 (2013) no. 10, pp. 1915-1938 | DOI | Zbl

[CE88] Cazenave, Thierry; Esteban, Maria J. On the stability of stationary states for nonlinear Schrödinger equations with an external magnetic field, Mat. Apl. Comput., Volume 7 (1988) no. 3, pp. 155-168 | Zbl

[CG01] Colin, Thierry; Ghidaglia, Jean-Michel An initial-boundary value problem for the Korteweg-de-Vries equation posed on a finite interval, Adv. Differ. Equ., Volume 6 (2001) no. 12, pp. 1463-1492 | MR | Zbl

[Col02] Colin, Mathieu On the local well-posedness of quasilinear Schrödinger equation in arbitrary space dimension, Commun. Partial Differ. Equations, Volume 27 (2002) no. 1-2, pp. 325-354 | DOI | Zbl

[CW16] Colin, Mathieu; Watanabe, Tatsuya Cauchy problem for the nonlinear Klein-Gordon equation coupled with the Maxwell equation, J. Math. Anal. Appl., Volume 443 (2016) no. 2, pp. 778-796 | DOI | MR | Zbl

[CW17] Colin, Mathieu; Watanabe, Tatsuya Standing waves for the nonlinear Schrödinger equation coupled with the Maxwell equation, Nonlinearity, Volume 30 (2017) no. 5, pp. 1920-1947 | DOI | Zbl

[DFVV10] D’Ancona, Piero; Fanelli, Luca; Vega, Luis; Visciglia, Nicola Endpoint Strichartz estimates for the magnetic Schrödinger equation, J. Funct. Anal., Volume 258 (2010) no. 10, pp. 3227-3240 | DOI | Zbl

[Fel98] Felsager, Bjørn Geometry, particles and fields, Graduate Texts in Contemporary Physics, Springer, 1998 | MR | Zbl

[GR91] Gonçalves Ribeiro, J. M. Instability of symmetric stationary states for some nonlinear Schrödinger equations with an external magnetic field, Ann. Inst. Henri Poincaré, Phys. Théor., Volume 54 (1991) no. 4, pp. 403-433 | Numdam | MR | Zbl

[Kik07] Kikuchi, Hiroaki Existence and stability of standing waves for Schrödinger–Poisson–Salter equation, Adv. Nonlinear Stud., Volume 7 (2007) no. 3, pp. 403-437 | Zbl

[Mic08] Michel, Laurent Remarks on non-linear Schrödinger equation with magnetic fields, Commun. Partial Differ. Equations, Volume 33 (2008) no. 7, pp. 1198-1215 | DOI | MR | Zbl

[NT86] Nakamitsu, Kuniaki; Tsutsumi, Masayoshi The Cauchy problem for the coupled Maxwell–Schrödinger equations, J. Math. Phys., Volume 27 (1986), pp. 211-216 | DOI | Zbl

[NW07] Nakamura, Makoto; Wada, Takeshi Global existence and uniqueness of solutions to the Maxwell–Schrödinger equations, Commun. Math. Phys., Volume 276 (2007) no. 2, pp. 315-339 | DOI | Zbl

[OT92] Ozawa, Tohru; Tsutsumi, Yoshio Existence and smoothing effect of solution for the Zakharov equations, Publ. Res. Inst. Math. Sci., Volume 28 (1992) no. 3, pp. 329-361 | DOI | MR | Zbl

[RV08] Radu, Eugen; Volkov, Mikhail S. Stationary ring solitons in field theory - Knots and vortons, Phys. Rep., Volume 468 (2008) no. 4, pp. 101-151 | DOI | MR