Traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity: some results and open problems
Mihai Mariş[1]
- [1] Institut de Mathématiques de Toulouse UMR 5219, Université Paul Sabatier - Toulouse 3, 118, Route de Narbonne, 31062 Toulouse cedex, France
Journées Équations aux dérivées partielles (2010)
- page 1-22
- ISSN: 0752-0360
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topMariş, Mihai. "Traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity: some results and open problems." Journées Équations aux dérivées partielles (2010): 1-22. <http://eudml.org/doc/116380>.
@article{Mariş2010,
abstract = {This text is a survey of recent results on traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity. We present the existence, nonexistence and stability results and we describe the main ideas used in proofs.},
affiliation = {Institut de Mathématiques de Toulouse UMR 5219, Université Paul Sabatier - Toulouse 3, 118, Route de Narbonne, 31062 Toulouse cedex, France},
author = {Mariş, Mihai},
journal = {Journées Équations aux dérivées partielles},
language = {eng},
month = {6},
pages = {1-22},
publisher = {Groupement de recherche 2434 du CNRS},
title = {Traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity: some results and open problems},
url = {http://eudml.org/doc/116380},
year = {2010},
}
TY - JOUR
AU - Mariş, Mihai
TI - Traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity: some results and open problems
JO - Journées Équations aux dérivées partielles
DA - 2010/6//
PB - Groupement de recherche 2434 du CNRS
SP - 1
EP - 22
AB - This text is a survey of recent results on traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity. We present the existence, nonexistence and stability results and we describe the main ideas used in proofs.
LA - eng
UR - http://eudml.org/doc/116380
ER -
References
top- I. V. Barashenkov, V. G. Makhankov, Soliton-like “bubbles” in a system of interacting bosons, Phys. Lett. A 128 (1988), pp. 52-56. MR935918
- I. V. Barashenkov, A. D. Gocheva, V. G. Makhankov, I. V. Puzynin, Stability of soliton-like bubbles, Physica D 34 (1989), pp. 240-254. Zbl0697.35127MR982390
- N. Berloff, Quantised vortices, travelling coherent structures and superfluid turbulence, in Stationary and time dependent Gross-Pitaevskii equations, A. Farina and J.-C. Saut eds., Contemp. Math. Vol. 473, AMS, Providence, RI, 2008, pp. 26-54. Zbl1166.35360MR2522013
- F. Béthuel, P. Gravejat, J.-C. Saut, Travelling-waves for the Gross-Pitaevskii equation II, Comm. Math. Phys. 285, No. 2 (2009), pp. 567-651. Zbl1190.35196MR2461988
- F. Béthuel, P. Gravejat, J.-C. Saut, On the KP I transonic limit of two dimensional Gross-Pitaevskii travelling-waves, Dynamics of PDE 5, No. 3, (2008), pp. 241-280. Zbl1186.35199MR2455894
- F. Béthuel, P. Gravejat, J.-C. Saut, D. Smets, On the Korteweg-de Vries long-wave approximation of the Gross-Pitaevskii equation I, Int. Math. Res. Notes (2009), pp. 2700-2748. Zbl1183.35240MR2520771
- F. Béthuel, P. Gravejat, J.-C. Saut, D. Smets, Orbital stability of the black soliton to the Gross-Pitaevskii equation, Indiana Univ. Math. J. 57, No. 6 (2008), pp. 2611-2642. Zbl1171.35012MR2482993
- F. Béthuel, G. Orlandi, D. Smets, Vortex rings for the Gross-Pitaevskii equation, J. Eur. Math. Soc. (JEMS) 6 (2004), pp. 17-94. Zbl1091.35085MR2041006
- F. Béthuel, J.-C. Saut, Travelling-waves for the Gross-Pitaevskii equation I, Ann. Inst. H. Poincaré Phys. Théor. 70 (1999), pp. 147-238. Zbl0933.35177MR1669387
- T. Cazenave, P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys. 85, no. 4 (1982), pp. 549-561. Zbl0513.35007MR677997
- D. Chiron, Travelling-waves for the Gross-Pitaevskii equation in dimension larger than two, Nonlinear Analysis 58 (2004), pp. 175-204. Zbl1054.35091MR2070812
- D. Chiron, M. Mariş, Traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity II, in preparation. Zbl1292.35274
- C. Coste, Nonlinear Schrödinger equation and superfluid hydrodynamics, Eur. Phys. J. B 1 (1998), pp. 245-253. MR1646182
- A. Farina, Finite-energy solutions, quantization effects and Liouville-type results for a variant of the Ginzburg-Landau systems in , Diff. Int. Eq. 11, No. 6 (1998), pp. 875-893. Zbl1074.35504MR1659256
- A. Farina, From Ginzburg-Landau to Gross-Pitaevskii, Monatsh. Math. 139 (2003), pp. 265-269. Zbl1126.35063MR2001707
- C. Gallo, Schrödinger group on Zhidkov spaces, Adv. Differential Equations 9, No. 5-6 (2004), pp. 509-538. Zbl1103.35093MR2099970
- C. Gallo, Growth rate of the Schrödinger group on Zhidkov spaces, C. R. Math. Acad. Sci. Paris 342, No. 5 (2006), pp. 319-323. Zbl1094.35115MR2201956
- C. Gallo, The Cauchy problem for defocusing nonlinear Schrödinger equations with non-vanishing initial data at infinity, Comm. Partial Differential Equations 33, No. 4-6 (2008), pp. 729-771. Zbl1156.35086MR2424376
- P. Gérard, The Cauchy Problem for the Gross-Pitaevskii Equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (5) (2006), pp. 765-779. Zbl1122.35133MR2259616
- P. Gérard, The Gross-Pitaevskii equation in the energy space, in Stationary and time-dependent Gross-Pitaevskii equations, A. Farina and J.-C. Saut eds., Contemp. Math. Vol. 473, Amer. Math. Soc., Providence, RI, 2008, pp. 129-148. Zbl1166.35373MR2522016
- P. Gérard, Z. Zhang, Orbital stability of traveling waves for the one-dimensional Gross-Pitaevskii equation, J. Math. Pures Appl. (9) 91, No. 2 (2009), pp. 178–210. Zbl1232.35152MR2498754
- J. Grant, P.H. Roberts, Motions in a Bose condensate III. The structure and effective masses of charged and uncharged impurities, J. Phys. A: Math., Nucl. Gen., 7 (1974), pp. 260-279.
- P. Gravejat, A non-existence result for supersonic travelling-waves in the Gross-Pitaevskii equation, Comm. Math. Phys. 243, No. 1 (2003), pp. 93-103. Zbl1044.35087MR2020221
- P. Gravejat, Decay for travelling waves in the Gross-Pitaevskii equation, Ann. Inst. Henri Poincaré, Analyse non linéeaire 21, No. 5 (2004), pp. 591-637. Zbl1057.35060MR2086751
- P. Gravejat, Limit at infinity and non-existence results for sonic travelling waves in the Gross-Pitaevskii equation, Differential and Integral Equations 17, No. 11-12 (2004), pp. 1213-1232. Zbl1150.35301MR2100023
- P. Gravejat, Asymptotics for the travelling waves in the Gross-Pitaevskii equation, Asymptot. Anal. 45, No. 3-4 (2005), pp. 227-299. Zbl1092.35103MR2191764
- P. Gravejat, First order asymptotics for the travelling waves in the Gross-Pitaevskii equation, Adv. Differential Equations 11, No. 3 (2006), pp. 259-280. Zbl1102.35029MR2221483
- E. P. Gross, Hydrodynamics of a superfluid condensate, J. Math. Phys. 4 (2), (1963), pp. 195-207.
- S. Gustafson, K. Nakanishi, T.-P. Tsai, Scattering for the Gross-Pitaevskii equation, Math. Res. Lett. 13, No. 2-3 (2006), pp. 273-285. Zbl1119.35084MR2231117
- S. Gustafson, K. Nakanishi, T.-P. Tsai, Global dispersive solutions for the Gross-Pitaevskii equation in two and three dimensions, Ann. Inst. H. Poincaré 8, No. 7 (2007), pp. 1303-1331. Zbl05218113MR2360438
- S. Gustafson, K. Nakanishi, T.-P. Tsai, Scattering theory for the Gross-Pitaevskii equation in three dimensions, Commun. Contemp. Math. 11, No. 4 (2009), pp. 657-707. Zbl1180.35481MR2559713
- S. V. Iordanskii, A. V. Smirnov, Three-dimensional solitons in He II, JETP Lett. 27 (10) (1978), pp. 535-538.
- C. A. Jones, P. H. Roberts, Motions in a Bose condensate IV, Axisymmetric solitary waves, J. Phys A: Math. Gen. 15 (1982), pp. 2599-2619.
- C. A. Jones, S. J. Putterman, P. H. Roberts, Motions in a Bose condensate V. Stability of wave solutions of nonlinear Schrödinger equations in two and three dimensions, J. Phys A: Math. Gen. 19 (1986), pp. 2991-3011.
- T. Kato, Schrödinger operators with singular potentials, Israel J. Math. 13 (1972), pp. 135-148. Zbl0246.35025MR333833
- Y. S. Kivshar, B. Luther-Davies, Dark optical solitons: physics and applications, Phys. Rep. 298 (1998), pp. 81-197.
- Y. S. Kivshar, D. E. Pelinovsky, Y. A. Stepanyants, Self-focusing of plane dark solitons in nonlinear defocusing media, Phys. Rev. E 51 (5) (1995), pp. 5016-5026. MR1383087
- E. A. Kuznetsov, J. J. Rasmussen, Instability of two-dimensional solitons and vortices in defocusing media, Phys. Rev. E 51, No. 5 (1995), pp. 4479-4484.
- E. A. Kuznetsov, J. J. Rasmussen, Self-defocusing instability of two-dimensional solitons and vortices, JETP Lett. 62, No. 2 (1995), pp. 105-112.
- E. H. Lieb, On the lowest eigenvalue of the Laplacian for the intersection of two domains, Invent. Math. 74 (1983), pp. 441-448. Zbl0538.35058MR724014
- Z. Lin, Stability and instability of traveling solitonic bubbles, Adv. Differential Equations 7 (2002), No. 8, pp. 897-918. Zbl1033.35117MR1895111
- P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, part I, Ann. Inst. H. Poincaré, Anal. non linéaire, 1 (1984), pp. 109-145. Zbl0541.49009MR778970
- M. Mariş, Existence of nonstationary bubbles in higher dimensions, Journal des Mathématiques Pures et Appliquées 81 (2002), pp. 1207-1239. Zbl1040.35116MR1952162
- M. Mariş, Nonexistence of supersonic traveling waves for nonlinear Schrödinger equations with non-zero conditions at infinity, SIAM Journal on Mathematical Analysis, Vol. 40, No. 3 (2008), pp. 1076-1103. Zbl1167.35518MR2452881
- M. Mariş, On the symmetry of minimizers, Archive for Rational Mechanics and Analysis 192, No. 2 (2009), pp. 311-330. Zbl1159.49005MR2486598
- M. Mariş, Traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity, preprint, arXiv: 0903.0354. Zbl1315.35207
- M. Mariş, On some minimization problems, in preparation.
- L. M. Pismen, A. A. Nepomnyashchy, Stability of vortex maps in a model of superflow, Physica D 69 (1993), pp. 173-175. Zbl0791.35128MR1245660
- L. M. Pismen, J. Rubinstein, Motion of vortex lines in the Ginzburg-Landau model, Physica D 47 (1991), pp. 353-360. Zbl0728.35090MR1098255
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