Traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity

Abstract

For a large class of nonlinear Schrödinger equations with nonzero conditions at infinity and for any speed $c$ less than the sound velocity, we prove the existence of nontrivial finite energy traveling waves moving with speed $c$ in any space dimension $N\geq 3$. Our results are valid as well for the Gross-Pitaevskii equation and for NLS with cubic-quintic nonlinearity.

  • [AB] Go to document L. Almeida and F. Bethuel, "Topological methods for the Ginzburg-Landau equations," J. Math. Pures Appl., vol. 77, iss. 1, pp. 1-49, 1998.
    @article {AB, MRKEY = {1617594},
      AUTHOR = {Almeida, Lu{\'ı}s and Bethuel, Fabrice},
      TITLE = {Topological methods for the {G}inzburg-{L}andau equations},
      JOURNAL = {J. Math. Pures Appl.},
      FJOURNAL = {Journal de Mathématiques Pures et Appliquées. Neuvième Série},
      VOLUME = {77},
      YEAR = {1998},
      NUMBER = {1},
      PAGES = {1--49},
      ISSN = {0021-7824},
      CODEN = {JMPAAM},
      MRCLASS = {35Q55 (58E50)},
      MRNUMBER = {1617594},
      MRREVIEWER = {Song Mu Zheng},
      DOI = {10.1016/S0021-7824(98)80064-0},
      ZBLNUMBER = {0904.35023},
      }
  • [barashenkov2] Go to document I. V. Barashenkov, A. D. Gocheva, V. G. Makhan$’$kov, and I. V. Puzynin, "Stability of the soliton-like “bubbles”," Phys. D, vol. 34, iss. 1-2, pp. 240-254, 1989.
    @article {barashenkov2, MRKEY = {0982390},
      AUTHOR = {Barashenkov, I. V. and Gocheva, A. D. and Makhan$'$kov, V. G. and Puzynin, I. V.},
      TITLE = {Stability of the soliton-like ``bubbles''},
      JOURNAL = {Phys. D},
      FJOURNAL = {Physica D. Nonlinear Phenomena},
      VOLUME = {34},
      YEAR = {1989},
      NUMBER = {1-2},
      PAGES = {240--254},
      ISSN = {0167-2789},
      CODEN = {PDNPDT},
      MRCLASS = {35Q20 (65P05 81C05 82A99)},
      MRNUMBER = {0982390},
      DOI = {10.1016/0167-2789(89)90237-6},
      ZBLNUMBER = {0697.35127},
     }
  • [barashenkov1] Go to document I. V. Barashenkov and V. G. Makhan$’$kov, "Soliton-like “bubbles” in a system of interacting bosons," Phys. Lett. A, vol. 128, iss. 1-2, pp. 52-56, 1988.
    @article {barashenkov1, MRKEY = {0935918},
      AUTHOR = {Barashenkov, I. V. and Makhan$'$kov, V. G.},
      TITLE = {Soliton-like ``bubbles'' in a system of interacting bosons},
      JOURNAL = {Phys. Lett. A},
      FJOURNAL = {Physics Letters. A},
      VOLUME = {128},
      YEAR = {1988},
      NUMBER = {1-2},
      PAGES = {52--56},
      ISSN = {0375-9601},
      CODEN = {PYLAAG},
      MRCLASS = {82A15 (35Q20)},
      MRNUMBER = {0935918},
      DOI = {10.1016/0375-9601(88)91042-0},
      }
  • [berloff] Go to document N. G. Berloff, "Quantised vortices, travelling coherent structures and superfluid turbulence," in Stationary and Time Dependent Gross-Pitaevskii Equations, Providence, RI: Amer. Math. Soc., 2008, vol. 473, pp. 27-54.
    @incollection {berloff, MRKEY = {2522013},
      AUTHOR = {Berloff, Natalia G.},
      TITLE = {Quantised vortices, travelling coherent structures and superfluid turbulence},
      BOOKTITLE = {Stationary and Time Dependent {G}ross-{P}itaevskii Equations},
      SERIES = {Contemp. Math.},
      VOLUME = {473},
      PAGES = {27--54},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2008},
      MRCLASS = {82D50 (35Q55 76Y05)},
      MRNUMBER = {2522013},
      MRREVIEWER = {Jean-Yves Fortin},
      DOI = {10.1090/conm/473/09223},
      ZBLNUMBER = {1166.35360},
     }
  • [BGS] Go to document F. Béthuel, P. Gravejat, and J. Saut, "Travelling waves for the Gross-Pitaevskii equation. II," Comm. Math. Phys., vol. 285, iss. 2, pp. 567-651, 2009.
    @article {BGS, MRKEY = {2461988},
      AUTHOR = {B{é}thuel, Fabrice and Gravejat, Philippe and Saut, Jean-Claude},
      TITLE = {Travelling waves for the {G}ross-{P}itaevskii equation. {II}},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {285},
      YEAR = {2009},
      NUMBER = {2},
      PAGES = {567--651},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {35Q55 (35C07 47J30 49J10)},
      MRNUMBER = {2461988},
      MRREVIEWER = {Anjan Kundu},
      DOI = {10.1007/s00220-008-0614-2},
      ZBLNUMBER = {1190.35196},
      }
  • [BGSS] Go to document F. Béthuel, P. Gravejat, J. Saut, and D. Smets, "Orbital stability of the black soliton for the Gross-Pitaevskii equation," Indiana Univ. Math. J., vol. 57, iss. 6, pp. 2611-2642, 2008.
    @article {BGSS, MRKEY = {2482993},
      AUTHOR = {B{é}thuel, Fabrice and Gravejat, Philippe and Saut, Jean-Claude and Smets, Didier},
      TITLE = {Orbital stability of the black soliton for the {G}ross-{P}itaevskii equation},
      JOURNAL = {Indiana Univ. Math. J.},
      FJOURNAL = {Indiana University Mathematics Journal},
      VOLUME = {57},
      YEAR = {2008},
      NUMBER = {6},
      PAGES = {2611--2642},
      ISSN = {0022-2518},
      CODEN = {IUMJAB},
      MRCLASS = {35Q55 (35B35 35C07 35C08)},
      MRNUMBER = {2482993},
      MRREVIEWER = {Hiroaki Kikuchi},
      DOI = {10.1512/iumj.2008.57.3632},
      ZBLNUMBER = {1171.35012},
      }
  • [BOS] Go to document F. Béthuel, G. Orlandi, and D. Smets, "Vortex rings for the Gross-Pitaevskii equation," J. Eur. Math. Soc. $($JEMS$)$, vol. 6, iss. 1, pp. 17-94, 2004.
    @article {BOS, MRKEY = {2041006},
      AUTHOR = {B{é}thuel, Fabrice and Orlandi, G. and Smets, D.},
      TITLE = {Vortex rings for the {G}ross-{P}itaevskii equation},
      JOURNAL = {J. Eur. Math. Soc. $($JEMS$)$},
      FJOURNAL = {Journal of the European Mathematical Society (JEMS)},
      VOLUME = {6},
      YEAR = {2004},
      NUMBER = {1},
      PAGES = {17--94},
      ISSN = {1435-9855},
      MRCLASS = {35Q55 (58E50 82D55)},
      MRNUMBER = {2041006},
      MRREVIEWER = {Bruno Scheurer},
      URL = {http://link.springer.de/cgi/linkref?issn=1435-9855&year=04&volume=6&page=17},
      ZBLNUMBER = {1091.35085},
      }
  • [BS] Go to document F. Béthuel and J. Saut, "Travelling waves for the Gross-Pitaevskii equation. I," Ann. Inst. H. Poincaré Phys. Théor., vol. 70, iss. 2, pp. 147-238, 1999.
    @article {BS, MRKEY = {1669387},
      AUTHOR = {B{é}thuel, Fabrice and Saut, Jean-Claude},
      TITLE = {Travelling waves for the {G}ross-{P}itaevskii equation. {I}},
      JOURNAL = {Ann. Inst. H. Poincaré Phys. Théor.},
      FJOURNAL = {Annales de l'Institut Henri Poincaré. Physique Théorique},
      VOLUME = {70},
      YEAR = {1999},
      NUMBER = {2},
      PAGES = {147--238},
      ISSN = {0246-0211},
      CODEN = {AHPAAO},
      MRCLASS = {35Q55 (35D05 58E50)},
      MRNUMBER = {1669387},
      MRREVIEWER = {Anjan Kundu},
      URL = {http://www.numdam.org/item?id=AIHPA_1999__70_2_147_0},
      ZBLNUMBER = {0933.35177},
      }
  • [brezis] H. Brezis, Analyse Fonctionnelle, Paris: Masson, 1983.
    @book {brezis, MRKEY = {0697382},
      AUTHOR = {Brezis, Ha{ï}m},
      TITLE = {Analyse Fonctionnelle},
      SERIES = {Collect. Math. Appl. pour la Ma\^\i trise. [Collect. Appl. Math. for the Master's Degree]},
      PUBLISHER = {Masson},
      ADDRESS = {Paris},
      YEAR = {1983},
      PAGES = {xiv+234},
      ISBN = {2-225-77198-7},
      MRCLASS = {46-01 (47-01)},
      MRNUMBER = {0697382},
      ZBLNUMBER = {0511.46001},
      }
  • [BBM] Go to document J. Bourgain, H. Brezis, and P. Mironescu, "Lifting in Sobolev spaces," J. Anal. Math., vol. 80, pp. 37-86, 2000.
    @article {BBM, MRKEY = {1771523},
      AUTHOR = {Bourgain, Jean and Brezis, Haim and Mironescu, Petru},
      TITLE = {Lifting in {S}obolev spaces},
      JOURNAL = {J. Anal. Math.},
      FJOURNAL = {Journal d'Analyse Mathématique},
      VOLUME = {80},
      YEAR = {2000},
      PAGES = {37--86},
      ISSN = {0021-7670},
      CODEN = {JOAMAV},
      MRCLASS = {46E35 (35J20 58D15 58E20)},
      MRNUMBER = {1771523},
      MRREVIEWER = {Piotr Haj{\l}asz},
      DOI = {10.1007/BF02791533},
      ZBLNUMBER = {0967.46026},
      }
  • [brezis-lieb] Go to document H. Brezis and E. H. Lieb, "Minimum action solutions of some vector field equations," Comm. Math. Phys., vol. 96, iss. 1, pp. 97-113, 1984.
    @article {brezis-lieb, MRKEY = {0765961},
      AUTHOR = {Brezis, Ha{ï}m and Lieb, Elliott H.},
      TITLE = {Minimum action solutions of some vector field equations},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {96},
      YEAR = {1984},
      NUMBER = {1},
      PAGES = {97--113},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {35J60 (49A22)},
      MRNUMBER = {0765961},
      MRREVIEWER = {William D. Collins},
      DOI = {10.1007/BF01217349},
      ZBLNUMBER = {0579.35025},
      }
  • [chiron] Go to document D. Chiron, "Travelling waves for the Gross-Pitaevskii equation in dimension larger than two," Nonlinear Anal., vol. 58, iss. 1-2, pp. 175-204, 2004.
    @article {chiron, MRKEY = {2070812},
      AUTHOR = {Chiron, David},
      TITLE = {Travelling waves for the {G}ross-{P}itaevskii equation in dimension larger than two},
      JOURNAL = {Nonlinear Anal.},
      FJOURNAL = {Nonlinear Analysis. Theory, Methods \& Applications. An International Multidisciplinary Journal. Series A: Theory and Methods},
      VOLUME = {58},
      YEAR = {2004},
      NUMBER = {1-2},
      PAGES = {175--204},
      ISSN = {0362-546X},
      CODEN = {NOANDD},
      MRCLASS = {35Q55 (82D50)},
      MRNUMBER = {2070812},
      MRREVIEWER = {Barbara Prinari},
      DOI = {10.1016/j.na.2003.10.028},
      ZBLNUMBER = {1054.35091},
      }
  • [CM1] D. Chiron and M. Marics, Traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity, II.
    @misc{CM1,
      author={Chiron, David and Mari\c{s},
      M.},
      TITLE={Traveling waves for nonlinear {S}chrödinger equations with nonzero conditions at infinity, {II}},
      ARXIV={1203.1912},
      SORTYEAR={2010},
     }
  • [CM2] D. Chiron and M. Marics, Rarefaction pulses for the nonlinear Schrödinger equation in the transonic limit.
    @misc{CM2,
      author={Chiron, David and Mari\c{s},
      M.},
      TITLE={Rarefaction pulses for the nonlinear {S}chrödinger equation in the transonic limit},
      ARXIV={1210.1315},
      SORTYEAR={2011},
     }
  • [coste] Go to document C. Coste, "Nonlinear Schrödinger equation and superfluid hydrodynamics," Eur. Phys. J. B Condens. Matter Phys., vol. 1, iss. 2, pp. 245-253, 1998.
    @article {coste, MRKEY = {1646182},
      AUTHOR = {Coste, C.},
      TITLE = {Nonlinear {S}chrödinger equation and superfluid hydrodynamics},
      JOURNAL = {Eur. Phys. J. B Condens. Matter Phys.},
      FJOURNAL = {The European Physical Journal. B. Condensed Matter Physics},
      VOLUME = {1},
      YEAR = {1998},
      NUMBER = {2},
      PAGES = {245--253},
      ISSN = {1434-6028},
      MRCLASS = {82D50 (35Q55)},
      MRNUMBER = {1646182},
      DOI = {10.1007/s100510050178},
      }
  • [AdB] Go to document A. de Bouard, "Instability of stationary bubbles," SIAM J. Math. Anal., vol. 26, iss. 3, pp. 566-582, 1995.
    @article {AdB, MRKEY = {1325903},
      AUTHOR = {de Bouard, Anne},
      TITLE = {Instability of stationary bubbles},
      JOURNAL = {SIAM J. Math. Anal.},
      FJOURNAL = {SIAM Journal on Mathematical Analysis},
      VOLUME = {26},
      YEAR = {1995},
      NUMBER = {3},
      PAGES = {566--582},
      ISSN = {0036-1410},
      CODEN = {SJMAAH},
      MRCLASS = {35Q55 (35B35 35K99)},
      MRNUMBER = {1325903},
      DOI = {10.1137/S0036141092237029},
      ZBLNUMBER = {0823.35017},
      }
  • [deLaire] Go to document A. de Laire, "Non-existence for travelling waves with small energy for the Gross-Pitaevskii equation in dimension $N\geq 3$," C. R. Math. Acad. Sci. Paris, vol. 347, iss. 7-8, pp. 375-380, 2009.
    @article {deLaire, MRKEY = {2537233},
      AUTHOR = {de Laire, Andr{é}},
      TITLE = {Non-existence for travelling waves with small energy for the {G}ross-{P}itaevskii equation in dimension {$N\geq 3$}},
      JOURNAL = {C. R. Math. Acad. Sci. Paris},
      FJOURNAL = {Comptes Rendus Mathématique. Académie des Sciences. Paris},
      VOLUME = {347},
      YEAR = {2009},
      NUMBER = {7-8},
      PAGES = {375--380},
      ISSN = {1631-073X},
      MRCLASS = {35Q55 (35C05)},
      MRNUMBER = {2537233},
      DOI = {10.1016/j.crma.2009.02.006},
      ZBLNUMBER = {1165.35047},
      }
  • [EG] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Boca Raton, FL: CRC Press, 1992.
    @book {EG, MRKEY = {1158660},
      AUTHOR = {Evans, Lawrence C. and Gariepy, Ronald F.},
      TITLE = {Measure Theory and Fine Properties of Functions},
      SERIES = {Stud. Adv. Math.},
      PUBLISHER = {CRC Press},
      ADDRESS = {Boca Raton, FL},
      YEAR = {1992},
      PAGES = {viii+268},
      ISBN = {0-8493-7157-0},
      MRCLASS = {28-02 (26-02 26Bxx 46E35)},
      MRNUMBER = {1158660},
      MRREVIEWER = {R. G. Bartle},
      ZBLNUMBER = {0804.28001},
      }
  • [farina1] A. Farina, "Finite-energy solutions, quantization effects and Liouville-type results for a variant of the Ginzburg-Landau systems in ${\bf R}^K$," Differential Integral Equations, vol. 11, iss. 6, pp. 875-893, 1998.
    @article {farina1, MRKEY = {1659256},
      AUTHOR = {Farina, Alberto},
      TITLE = {Finite-energy solutions, quantization effects and {L}iouville-type results for a variant of the {G}inzburg-{L}andau systems in {${\bf R}\sp K$}},
      JOURNAL = {Differential Integral Equations},
      FJOURNAL = {Differential and Integral Equations. An International Journal for Theory \& Applications},
      VOLUME = {11},
      YEAR = {1998},
      NUMBER = {6},
      PAGES = {875--893},
      ISSN = {0893-4983},
      MRCLASS = {35Q55 (58E50)},
      MRNUMBER = {1659256},
      ZBLNUMBER = {1074.35504},
      }
  • [farina2] Go to document A. Farina, "From Ginzburg-Landau to Gross-Pitaevskii," Monatsh. Math., vol. 139, iss. 4, pp. 265-269, 2003.
    @article {farina2, MRKEY = {2001707},
      AUTHOR = {Farina, Alberto},
      TITLE = {From {G}inzburg-{L}andau to {G}ross-{P}itaevskii},
      JOURNAL = {Monatsh. Math.},
      FJOURNAL = {Monatshefte für Mathematik},
      VOLUME = {139},
      YEAR = {2003},
      NUMBER = {4},
      PAGES = {265--269},
      ISSN = {0026-9255},
      CODEN = {MNMTA2},
      MRCLASS = {35Q55 (35B45 35B65 35J60 35Q51)},
      MRNUMBER = {2001707},
      MRREVIEWER = {Mihai Mari{\c{s}}},
      DOI = {10.1007/s00605-002-0514-z},
      ZBLNUMBER = {1126.35063},
      }
  • [gallo] Go to document C. Gallo, "The Cauchy problem for defocusing nonlinear Schrödinger equations with non-vanishing initial data at infinity," Comm. Partial Differential Equations, vol. 33, iss. 4-6, pp. 729-771, 2008.
    @article {gallo, MRKEY = {2424376},
      AUTHOR = {Gallo, Cl{é}ment},
      TITLE = {The {C}auchy problem for defocusing nonlinear {S}chrödinger equations with non-vanishing initial data at infinity},
      JOURNAL = {Comm. Partial Differential Equations},
      FJOURNAL = {Communications in Partial Differential Equations},
      VOLUME = {33},
      YEAR = {2008},
      NUMBER = {4-6},
      PAGES = {729--771},
      ISSN = {0360-5302},
      CODEN = {CPDIDZ},
      MRCLASS = {35Q55 (35B30)},
      MRNUMBER = {2424376},
      MRREVIEWER = {Tohru Ozawa},
      DOI = {10.1080/03605300802031614},
      ZBLNUMBER= {1156.35086},
     }
  • [PG] Go to document P. Gérard, "The Cauchy problem for the Gross-Pitaevskii equation," Ann. Inst. H. Poincaré Anal. Non Linéaire, vol. 23, iss. 5, pp. 765-779, 2006.
    @article {PG, MRKEY = {2259616},
      AUTHOR = {G{é}rard, P.},
      TITLE = {The {C}auchy problem for the {G}ross-{P}itaevskii equation},
      JOURNAL = {Ann. Inst. H. Poincaré Anal. Non Linéaire},
      FJOURNAL = {Annales de l'Institut Henri Poincaré. Analyse Non Linéaire},
      VOLUME = {23},
      YEAR = {2006},
      NUMBER = {5},
      PAGES = {765--779},
      ISSN = {0294-1449},
      MRCLASS = {35Q55 (35A15 35B30 35J20 81Q20)},
      MRNUMBER = {2259616},
      MRREVIEWER = {Zhi-Qiang Wang},
      DOI = {10.1016/j.anihpc.2005.09.004},
      ZBLNUMBER = {1122.35133},
      }
  • [PG2] Go to document P. Gérard, "The Gross-Pitaevskii equation in the energy space," in Stationary and Time Dependent Gross-Pitaevskii Equations, Providence, RI: Amer. Math. Soc., 2008, vol. 473, pp. 129-148.
    @incollection {PG2, MRKEY = {2522016},
      AUTHOR = {G{é}rard, Patrick},
      TITLE = {The {G}ross-{P}itaevskii equation in the energy space},
      BOOKTITLE = {Stationary and Time Dependent {G}ross-{P}itaevskii Equations},
      SERIES = {Contemp. Math.},
      VOLUME = {473},
      PAGES = {129--148},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2008},
      MRCLASS = {35Q55 (35J20)},
      MRNUMBER = {2522016},
      MRREVIEWER = {Ayman Kachmar},
      DOI = {10.1090/conm/473/09226},
      ZBLNUMBER = {1166.35373},
      }
  • [GZ] Go to document P. Gérard and Z. Zhang, "Orbital stability of traveling waves for the one-dimensional Gross-Pitaevskii equation," J. Math. Pures Appl., vol. 91, iss. 2, pp. 178-210, 2009.
    @article {GZ, MRKEY = {2498754},
      AUTHOR = {G{é}rard, Patrick and Zhang, Zhifei},
      TITLE = {Orbital stability of traveling waves for the one-dimensional {G}ross-{P}itaevskii equation},
      JOURNAL = {J. Math. Pures Appl.},
      FJOURNAL = {Journal de Mathématiques Pures et Appliquées. Neuvième Série},
      VOLUME = {91},
      YEAR = {2009},
      NUMBER = {2},
      PAGES = {178--210},
      ISSN = {0021-7824},
      CODEN = {JMPAAM},
      MRCLASS = {35Q55 (35B35 35Q51 37K15 37K45)},
      MRNUMBER = {2498754},
      MRREVIEWER = {Ademir Pastor Ferreira},
      DOI = {10.1016/j.matpur.2008.09.009},
      ZBLNUMBER = {1232.35152},
      }
  • [GT] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 3rd ed., New York: Springer-Verlag, 2001.
    @book {GT, MRKEY = {1814364},
      AUTHOR = {Gilbarg, David and Trudinger, Neil S.},
      TITLE = {Elliptic Partial Differential Equations of Second Order},
      SERIES = {Classics in Math.},
      EDITION={3rd},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {2001},
      PAGES = {xiv+517},
      ISBN = {3-540-41160-7},
      MRCLASS = {35-02 (35Jxx)},
      MRNUMBER = {1814364},
      ZBLNUMBER = {1042.35002},
      }
  • [GR] Go to document J. Grant and 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., vol. 7, pp. 260-279, 1974.
    @article{GR,
      author={Grant, J. and Roberts, P. H.},
      TITLE={Motions in a {B}ose condensate. {III. T}he structure and effective masses of charged and uncharged impurities},
      JOURNAL={J. Phys. A: Math., Nucl. Gen.},
      VOLUME={7},
      YEAR={1974},
      PAGES={260--279},
      DOI = {10.1088/0305-4470/7/2/012},
     }
  • [gravejat1] Go to document P. Gravejat, "A non-existence result for supersonic travelling waves in the Gross-Pitaevskii equation," Comm. Math. Phys., vol. 243, iss. 1, pp. 93-103, 2003.
    @article {gravejat1, MRKEY = {2020221},
      AUTHOR = {Gravejat, Philippe},
      TITLE = {A non-existence result for supersonic travelling waves in the {G}ross-{P}itaevskii equation},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {243},
      YEAR = {2003},
      NUMBER = {1},
      PAGES = {93--103},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {35Q55 (35J60 35Q40 35Q51)},
      MRNUMBER = {2020221},
      MRREVIEWER = {Mihai Mari{\c{s}}},
      DOI = {10.1007/s00220-003-0961-y},
      ZBLNUMBER = {1044.35087},
      }
  • [gravejat2] P. Gravejat, "Asymptotics for the travelling waves in the Gross-Pitaevskii equation," Asymptot. Anal., vol. 45, iss. 3-4, pp. 227-299, 2005.
    @article {gravejat2, MRKEY = {2191764},
      AUTHOR = {Gravejat, Philippe},
      TITLE = {Asymptotics for the travelling waves in the {G}ross-{P}itaevskii equation},
      JOURNAL = {Asymptot. Anal.},
      FJOURNAL = {Asymptotic Analysis},
      VOLUME = {45},
      YEAR = {2005},
      NUMBER = {3-4},
      PAGES = {227--299},
      ISSN = {0921-7134},
      CODEN = {ASANEZ},
      MRCLASS = {35Q55 (35B40 82C10)},
      MRNUMBER = {2191764},
      MRREVIEWER = {John Albert},
      ZBLNUMBER = {1092.35103},
      }
  • [gross] Go to document E. P. Gross, "Hydrodynamics of a superfluid condensate," J. Math. Phys., vol. 4, pp. 195-207, 1963.
    @article{gross,
      author={Gross, E. P.},
      TITLE={Hydrodynamics of a superfluid condensate},
      JOURNAL={J. Math. Phys.},
      VOLUME={4},
      YEAR={1963},
      PAGES={195--207},
      DOI = {10.1063/1.1703944},
     }
  • [GNT1] S. Gustafson, K. Nakanishi, and T. Tsai, "Scattering for the Gross-Pitaevskii equation," Math. Res. Lett., vol. 13, iss. 2-3, pp. 273-285, 2006.
    @article {GNT1, MRKEY = {2231117},
      AUTHOR = {Gustafson, Stephen and Nakanishi, Kenji and Tsai, Tai-Peng},
      TITLE = {Scattering for the {G}ross-{P}itaevskii equation},
      JOURNAL = {Math. Res. Lett.},
      FJOURNAL = {Mathematical Research Letters},
      VOLUME = {13},
      YEAR = {2006},
      NUMBER = {2-3},
      PAGES = {273--285},
      ISSN = {1073-2780},
      MRCLASS = {35Q55 (35B40 35P25)},
      MRNUMBER = {2231117},
      MRREVIEWER = {Tohru Ozawa},
      ZBLNUMBER = {1119.35084},
      }
  • [GNT2] Go to document S. Gustafson, K. Nakanishi, and T. Tsai, "Scattering theory for the Gross-Pitaevskii equation in three dimensions," Commun. Contemp. Math., vol. 11, iss. 4, pp. 657-707, 2009.
    @article {GNT2, MRKEY = {2559713},
      AUTHOR = {Gustafson, Stephen and Nakanishi, Kenji and Tsai, Tai-Peng},
      TITLE = {Scattering theory for the {G}ross-{P}itaevskii equation in three dimensions},
      JOURNAL = {Commun. Contemp. Math.},
      FJOURNAL = {Communications in Contemporary Mathematics},
      VOLUME = {11},
      YEAR = {2009},
      NUMBER = {4},
      PAGES = {657--707},
      ISSN = {0219-1997},
      MRCLASS = {35Q55 (35B40 35P25)},
      MRNUMBER = {2559713},
      MRREVIEWER = {R{é}mi Carles},
      DOI = {10.1142/S0219199709003491},
      ZBLNUMBER = {1180.35481},
      }
  • [IS] Go to document S. V. Iordanskii and A. V. Smirnov, "Three-dimensional solitons in He II," JETP Lett., vol. 27, pp. 535-538, 1978.
    @article{IS,
      author={Iordanskii, S. V. and Smirnov, A. V.},
      TITLE={Three-dimensional solitons in {He} {II}},
      JOURNAL={JETP Lett.},
      VOLUME={27},
      YEAR={1978},
      PAGES={535--538},
      URL = {http://www.jetpletters.ac.ru/ps/1553/article_23773.shtml},
     }
  • [JR] Go to document C. A. Jones and P. H. Roberts, "Motions in a Bose condensate IV, Axisymmetric solitary waves," J. Phys A: Math. Gen., vol. 15, pp. 2599-2619, 1982.
    @article{JR,
      author={Jones, C. A. and Roberts, P. H.},
      TITLE={Motions in a {B}ose condensate {IV, A}xisymmetric solitary waves},
      JOURNAL={J. Phys A: Math. Gen.},
      VOLUME={15},
      YEAR={1982},
      PAGES={2599--2619},
      DOI = {10.1088/0305-4470/15/8/036},
     }
  • [JPR] C. A. Jones, S. J. Putterman, and P. H. Roberts, "Stability of wave solutions of nonlinear Schrödinger equations in two and three dimensions," J. Phys A: Math. Gen., vol. 19, pp. 2991-3011, 1986.
    @article{JPR,
      author={Jones, C. A. and Putterman, S. J. and Roberts, P. H.},
      TITLE={Stability of wave solutions of nonlinear {S}chrödinger equations in two and three dimensions},
      JOURNAL={J. Phys A: Math. Gen.},
      VOLUME={19},
      YEAR={1986},
      PAGES={2991--3011},
     }
  • [kavian] O. Kavian, Introduction à la Théorie des Points Critiques et Applications aux Problèmes Elliptiques, New York: Springer-Verlag, 1993, vol. 13.
    @book {kavian, MRKEY = {1276944},
      AUTHOR = {Kavian, Otared},
      TITLE = {Introduction à la Théorie des Points Critiques et Applications aux Problèmes Elliptiques},
      SERIES = {Math. Appl. (Berlin)},
      VOLUME = {13},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1993},
      PAGES = {viii+325},
      ISBN = {2-287-00410-6},
      MRCLASS = {58E05 (35J60 47H15 47N20 49K20)},
      MRNUMBER = {1276944},
      MRREVIEWER = {Jean Mawhin},
      ZBLNUMBER = {0797.58005},
      }
  • [KOPV] R. Killip, T. Oh, O. Pocovnicu, and M. Vicsan, "Global well-posedness of the Gross–Pitaevskii and cubic-quintic nonlinear Schrödinger equations with non-vanishing boundary conditions," Math. Res. Lett., vol. 19, pp. 969-986, 2012.
    @article{KOPV,
      author={Killip, R. and Oh, T. and Pocovnicu, O. and Vi{\c{s}}an, M.},
      TITLE={Global well-posedness of the {G}ross--{P}itaevskii and cubic-quintic nonlinear {S}chrödinger equations with non-vanishing boundary conditions},
      JOURNAL={Math. Res. Lett.},
      VOLUME={19},
      YEAR={2012},
      PAGES={969--986},
     }
  • [KL] Go to document Y. S. Kivshar and B. Luther-Davies, "Dark optical solitons: physics and applications," Phys. Rep., vol. 298, pp. 81-197, 1998.
    @article{KL,
      author={Kivshar, Y. S. and Luther-Davies, B.},
      TITLE={Dark optical solitons: physics and applications},
      JOURNAL={Phys. Rep.},
      VOLUME={298},
      YEAR={1998},
      PAGES={81--197},
      DOI = {10.1016/S0370-1573(97)00073-2},
     }
  • [KPS] Go to document D. E. Pelinovsky, Y. A. Stepanyants, and Y. S. Kivshar, "Self-focusing of plane dark solitons in nonlinear defocusing media," Phys. Rev. E, vol. 51, iss. 5, part B, pp. 5016-5026, 1995.
    @article {KPS, MRKEY = {1383087},
      AUTHOR = {Pelinovsky, Dmitry E. and Stepanyants, Yuri A. and Kivshar, Yuri S.},
      TITLE = {Self-focusing of plane dark solitons in nonlinear defocusing media},
      JOURNAL = {Phys. Rev. E},
      FJOURNAL = {Physical Review E. Statistical, Nonlinear, and Soft Matter Physics},
      VOLUME = {51},
      YEAR = {1995},
      NUMBER = {5, part B},
      PAGES = {5016--5026},
      ISSN = {1539-3755},
      CODEN = {PLEEE8},
      MRCLASS = {78A60 (35Q55)},
      MRNUMBER = {1383087},
      DOI = {10.1103/PhysRevE.51.5016},
      }
  • [lieb] Go to document E. H. Lieb, "On the lowest eigenvalue of the Laplacian for the intersection of two domains," Invent. Math., vol. 74, iss. 3, pp. 441-448, 1983.
    @article {lieb, MRKEY = {0724014},
      AUTHOR = {Lieb, Elliott H.},
      TITLE = {On the lowest eigenvalue of the {L}aplacian for the intersection of two domains},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {74},
      YEAR = {1983},
      NUMBER = {3},
      PAGES = {441--448},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {35P15 (47F05)},
      MRNUMBER = {0724014},
      MRREVIEWER = {Catherine Bandle},
      DOI = {10.1007/BF01394245},
      ZBLNUMBER = {0538.35058},
      }
  • [lin] Z. Lin, "Stability and instability of traveling solitonic bubbles," Adv. Differential Equations, vol. 7, iss. 8, pp. 897-918, 2002.
    @article {lin, MRKEY = {1895111},
      AUTHOR = {Lin, Zhiwu},
      TITLE = {Stability and instability of traveling solitonic bubbles},
      JOURNAL = {Adv. Differential Equations},
      FJOURNAL = {Advances in Differential Equations},
      VOLUME = {7},
      YEAR = {2002},
      NUMBER = {8},
      PAGES = {897--918},
      ISSN = {1079-9389},
      MRCLASS = {35Q55 (35B35)},
      MRNUMBER = {1895111},
      MRREVIEWER = {Giulio Soliani},
      ZBLNUMBER = {1033.35117},
      }
  • [lions] Go to document P. -L. Lions, "The concentration-compactness principle in the calculus of variations. The locally compact case. I," Ann. Inst. H. Poincaré Anal. Non Linéaire, vol. 1, iss. 2, pp. 109-145, 1984.
    @article {lions, MRKEY = {0778970},
      AUTHOR = {Lions, P.-L.},
      TITLE = {The concentration-compactness principle in the calculus of variations. {T}he locally compact case. {I}},
      JOURNAL = {Ann. Inst. H. Poincaré Anal. Non Linéaire},
      FJOURNAL = {Annales de l'Institut Henri Poincaré. Analyse Non Linéaire},
      VOLUME = {1},
      YEAR = {1984},
      NUMBER = {2},
      PAGES = {109--145},
      ISSN = {0294-1449},
      MRCLASS = {49A50 (49A22)},
      MRNUMBER = {0778970},
      MRREVIEWER = {Gianfranco Bottaro},
      URL = {http://www.numdam.org/item?id=AIHPC_1984__1_2_109_0},
      ZBLNUMBER = {0541.49009},
      }
  • [M3] Go to document M. Marics, "Existence of nonstationary bubbles in higher dimensions," J. Math. Pures Appl., vol. 81, iss. 12, pp. 1207-1239, 2002.
    @article {M3, MRKEY = {1952162},
      AUTHOR = {Mari{\c{s}},
      Mihai},
      TITLE = {Existence of nonstationary bubbles in higher dimensions},
      JOURNAL = {J. Math. Pures Appl.},
      FJOURNAL = {Journal de Mathématiques Pures et Appliquées. Neuvième Série},
      VOLUME = {81},
      YEAR = {2002},
      NUMBER = {12},
      PAGES = {1207--1239},
      ISSN = {0021-7824},
      CODEN = {JMPAAM},
      MRCLASS = {35Q55 (47J30 58E05)},
      MRNUMBER = {1952162},
      MRREVIEWER = {Holger Teismann},
      DOI = {10.1016/S0021-7824(02)01274-6},
      ZBLNUMBER = {1040.35116},
      }
  • [M7] Go to document M. Marics, "On the symmetry of minimizers," Arch. Ration. Mech. Anal., vol. 192, iss. 2, pp. 311-330, 2009.
    @article {M7, MRKEY = {2486598},
      AUTHOR = {Mari{\c{s}},
      Mihai},
      TITLE = {On the symmetry of minimizers},
      JOURNAL = {Arch. Ration. Mech. Anal.},
      FJOURNAL = {Archive for Rational Mechanics and Analysis},
      VOLUME = {192},
      YEAR = {2009},
      NUMBER = {2},
      PAGES = {311--330},
      ISSN = {0003-9527},
      MRCLASS = {49J10 (49K30)},
      MRNUMBER = {2486598},
      MRREVIEWER = {Anna Verde},
      DOI = {10.1007/s00205-008-0136-2},
      ZBLNUMBER = {1159.49005},
      }
  • [M8] Go to document M. Marics, "Nonexistence of supersonic traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity," SIAM J. Math. Anal., vol. 40, iss. 3, pp. 1076-1103, 2008.
    @article {M8, MRKEY = {2452881},
      AUTHOR = {Mari{\c{s}},
      Mihai},
      TITLE = {Nonexistence of supersonic traveling waves for nonlinear {S}chrödinger equations with nonzero conditions at infinity},
      JOURNAL = {SIAM J. Math. Anal.},
      FJOURNAL = {SIAM Journal on Mathematical Analysis},
      VOLUME = {40},
      YEAR = {2008},
      NUMBER = {3},
      PAGES = {1076--1103},
      ISSN = {0036-1410},
      MRCLASS = {35Q55 (35C07)},
      MRNUMBER = {2452881},
      MRREVIEWER = {Dimitrios Pinotsis},
      DOI = {10.1137/070711189},
      }
  • [stein] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton, N.J.: Princeton Univ. Press, 1970, vol. 30.
    @book {stein, MRKEY = {0290095},
      AUTHOR = {Stein, Elias M.},
      TITLE = {Singular Integrals and Differentiability Properties of Functions},
      SERIES = {Princeton Math. Series},
      VOLUME={30},
      PUBLISHER = {Princeton Univ. Press},
      ADDRESS = {Princeton, N.J.},
      YEAR = {1970},
      PAGES = {xiv+290},
      MRCLASS = {46.38 (26.00)},
      MRNUMBER = {0290095},
      MRREVIEWER = {R. E. Edwards},
      ZBLNUMBER = {0207.13501},
      }
  • [willem] Go to document M. Willem, Minimax Theorems, Boston, MA: Birkhäuser, 1996, vol. 24.
    @book {willem, MRKEY = {1400007},
      AUTHOR = {Willem, Michel},
      TITLE = {Minimax Theorems},
      SERIES = {Progr. Nonlinear Differential Equations Appl.},
      VOLUME={24},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Boston, MA},
      YEAR = {1996},
      PAGES = {x+162},
      ISBN = {0-8176-3913-6},
      MRCLASS = {58E05 (35A15 35J60 58-02)},
      MRNUMBER = {1400007},
      MRREVIEWER = {Andrzej Szulkin},
      DOI = {10.1007/978-1-4612-4146-1},
      ZBLNUMBER = {0856.49001},
      }

Authors

Mihai Maris

Université Paul Sabatier
Institut de Matématiques de Toulouse
31962 Toulouse
France