Smooth branch of rarefaction pulses for the nonlinear Schrödinger equation and the Euler-Korteweg system in 2d. (Branche régulière d’ondes de raréfaction pour l’équation de Schrödinger non linéaire et le système d’Euler-Korteweg en 2d.) (English. French summary) Zbl 1527.35268
Summary: We are interested in the construction of a smooth branch of travelling waves to the Nonlinear Schrödinger Equation and the Euler-Korteweg system for capillary fluids with nonzero condition at infinity. This branch is defined for speeds close to the speed of sound and looks qualitatively, after rescaling, as a rarefaction pulse described by the Kadomtsev-Petviashvili equation. The proof relies on a fixed point theorem based on the nondegeneracy of the lump solitary wave of the Kadomtsev-Petviashvili equation.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 35Q31 | Euler equations |
| 35Q40 | PDEs in connection with quantum mechanics |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35Q41 | Time-dependent Schrödinger equations and Dirac equations |
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 35J60 | Nonlinear elliptic equations |
| 35D30 | Weak solutions to PDEs |
| 35C07 | Traveling wave solutions |
| 35C08 | Soliton solutions |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
Keywords:
travelling waves; nonlinear Schrödinger equation; Euler-Korteweg system; Kadomtsev-Petviashvili equation; lumpReferences:
| [1] | Abid, Malek; Huepe, Cristián; Metens, Stéphane; Nore, Caroline; Pham, Chi-Tuong; Tuckerman, Laurette S.; Brachet, Marc-Etienne, Gross-Pitaevskii dynamics of Bose-Einstein condensates and superfluid turbulence, Fluid Dynamics Research, 33, 5-6, 509-544 (2003) · Zbl 1060.76528 · doi:10.1016/j.fluiddyn.2003.09.001 |
| [2] | Abramowitz, Milton; Stegun, Irene. A., Handbook of mathematical functions with formulas, graphs, and mathematical tables, 55 (1964), U. S. Government Printing Office, Washington, D. C. · Zbl 0171.38503 |
| [3] | Audiard, Corentin, Small energy traveling waves for the Euler-Korteweg system, Nonlinearity, 30, 9, 3362-3399 (2017) · Zbl 1375.35357 · doi:10.1088/1361-6544/aa7cc2 |
| [4] | Benzoni-Gavage, Sylvie, Planar traveling waves in capillary fluids, Differ. Integral Equ., 26, 3-4, 439-485 (2013) · Zbl 1289.76012 |
| [5] | Benzoni-Gavage, Sylvie; Chiron, David, Long wave asymptotics for the Euler-Korteweg system, Rev. Mat. Iberoam., 34, 1, 245-304 (2018) · Zbl 1391.35323 · doi:10.4171/RMI/985 |
| [6] | Béthuel, Fabrice; Gravejat, Philippe; Saut, Jean-Claude, On the KP-I transonic limit of two-dimensional Gross-Pitaevskii travelling waves, Dyn. Partial Differ. Equ., 5, 3, 241-280 (2008) · Zbl 1186.35199 · doi:10.4310/DPDE.2008.v5.n3.a3 |
| [7] | Béthuel, Fabrice; Gravejat, Philippe; Saut, Jean-Claude, Travelling waves for the Gross-Pitaevskii equation. II, Commun. Math. Phys., 285, 2, 567-651 (2009) · Zbl 1190.35196 · doi:10.1007/s00220-008-0614-2 |
| [8] | Béthuel, Fabrice; Orlandi, Giandomenico; Smets, Didier, Vortex rings for the Gross-Pitaevskii equation, J. Eur. Math. Soc., 6, 1, 17-94 (2004) · Zbl 1091.35085 · doi:10.4171/JEMS/2 |
| [9] | Berloff, Natalia G.; Roberts, Paul H., Motions in a Bose condensate: X. New results on stability of axisymmetric solitary waves of the Gross-Pitaevskii equation, J. Phys. A. Math. Gen., 37, 11333-11351 (2004) · Zbl 1062.82055 · doi:10.1088/0305-4470/37/47/003 |
| [10] | Bellazzini, Jacopo; Ruiz, David, Finite energy traveling waves for the Gross-Pitaevskii equation in the subsonic regime, Am. J. Math., 145, 1, 109-149 (2023) · Zbl 1509.35089 · doi:10.1353/ajm.2023.0002 |
| [11] | Béthuel, Fabrice; Saut, Jean-Claude, Travelling waves for the Gross-Pitaevskii equation. I, Ann. Inst. Henri Poincaré, Phys. Théor., 70, 2, 147-238 (1999) · Zbl 0933.35177 |
| [12] | Chiron, David, Travelling waves for the Gross-Pitaevskii equation in dimension larger than two, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods, 58, 1-2, 175-204 (2004) · Zbl 1054.35091 · doi:10.1016/j.na.2003.10.028 |
| [13] | Chiron, David, Travelling waves for the Nonlinear Schrödinger Equation with general nonlinearity in dimension one, Nonlinearity, 25, 813-850 (2012) · Zbl 1278.35226 · doi:10.1088/0951-7715/25/3/813 |
| [14] | Chiron, David, Error bounds for the (KdV)/(KP-I) and (gKdV)/(gKP-I) asymptotic regime for nonlinear Schrödinger type equations, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 31, 6, 1175-1230 (2014) · Zbl 1307.35274 · doi:10.1016/j.anihpc.2013.08.007 |
| [15] | Chiron, David; Mariş, Mihai, Rarefaction pulses for the Nonlinear Schrödinger Equation in the transonic limit, Commun. Math. Phys., 326, 2, 329-392 (2014) · Zbl 1292.35274 · doi:10.1007/s00220-013-1879-7 |
| [16] | Chiron, David; Mariş, Mihai, Traveling Waves for Nonlinear Schrödinger Equations with Nonzero Conditions at Infinity, Arch. Ration. Mech. Anal., 226, 1, 143-242 (2017) · Zbl 1391.35351 · doi:10.1007/s00205-017-1131-2 |
| [17] | Chiron, David; Pacherie, Eliot, Smooth branch of travelling waves for the Gross-Pitaevskii equation in \(\mathbb{R}^2\) for small speed, Ann. Sc. Norm. Super. Pisa, Cl. Sci., 22, 4, 1937-2038 (2021) · Zbl 1484.35110 |
| [18] | Chiron, David; Pacherie, Eliot, Coercivity for travelling waves in the Gross-Pitaevskii equation in \(\mathbb{R}^2\) for small speed, Publ. Mat., Barc., 67, 1, 277-410 (2023) · Zbl 1509.35091 · doi:10.5565/PUBLMAT6712307 |
| [19] | Chiron, David; Rousset, Frédéric, The KdV/KP-I limit of the nonlinear Schrödinger equation, SIAM J. Math. Anal., 42, 1, 64-96 (2010) · Zbl 1210.35229 · doi:10.1137/080738994 |
| [20] | Chiron, David; Scheid, Claire, Travelling waves for the Nonlinear Schrödinger Equation with general nonlinearity in dimension two, J. Nonlinear Sci., 26, 2, 171-231 (2016) · Zbl 1336.35318 · doi:10.1007/s00332-015-9273-6 |
| [21] | Chiron, David; Scheid, Claire, Multiple branches of travelling waves for the Gross-Pitaevskii equation, Nonlinearity, 31, 6, 2809-2853 (2018) · Zbl 1393.35216 · doi:10.1088/1361-6544/aab4cc |
| [22] | del Pino, Manuel; Felmer, Patricio; Kowalczyk, Michał, Minimality and nondegeneracy of degree-one Ginzburg-Landau vortex as a Hardy’s type inequality, Int. Math. Res. Not., 30, 1511-1527 (2004) · Zbl 1112.35055 · doi:10.1155/S1073792804133588 |
| [23] | Dávila, Juan; del Pino, Manuel; Medina, Maria; Rodiac, Rémy, Interacting helical traveling waves for the Gross-Pitaevskii equation, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 1319-1367 (2021) · Zbl 1510.35295 |
| [24] | Gravejat, Philippe, Decay for travelling waves in the Gross-Pitaevskii equation, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 21, 5, 591-637 (2004) · Zbl 1057.35060 · doi:10.1016/j.anihpc.2003.09.001 |
| [25] | Gravejat, Philippe, Asymptotics of the solitary waves for the generalized Kadomtsev-Petviashvili equations, Discrete Contin. Dyn. Syst., 21, 3, 835-882 (2008) · Zbl 1155.35086 · doi:10.3934/dcds.2008.21.835 |
| [26] | Gilbarg, D.; Trudinger, N. S., Elliptic partial differential equations of second order (2001), Springer · Zbl 1042.35002 · doi:10.1007/978-3-642-61798-0 |
| [27] | Howie, John M., Complex analysis (2003), Springer · Zbl 1017.00001 · doi:10.1007/978-1-4471-0027-0 |
| [28] | Iordanskii, S. V.; Smirnov, A. V., Three-dimensional solitons in He II, JETP Lett., 27, 10, 535-538 (1978) |
| [29] | Jones, C.; Putterman, Seth J.; Roberts, Paul H., 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, 2991-3011 (1986) · doi:10.1088/0305-4470/19/15/023 |
| [30] | Jones, C.; Roberts, Paul H., Motion in a Bose condensate IV. Axisymmetric solitary waves, J. Phys. A. Math. Gen., 15, 2599-2619 (1982) · doi:10.1088/0305-4470/15/8/036 |
| [31] | Kivshar, Yuri S.; Luther-Davies, Barry, Dark optical solitons: physics and applications, Phys. Rep., 298, 81-197 (1998) · doi:10.1016/S0370-1573(97)00073-2 |
| [32] | Kivshar, Yuri S.; Pelinovsky, Dmitry E., Self-focusing and transverse instabilities of solitary waves, Phys. Rep., 331, 4, 117-195 (2000) · doi:10.1016/S0370-1573(99)00106-4 |
| [33] | Lang, Serge, Complex analysis, 103 (1999), Springer · Zbl 0933.30001 · doi:10.1007/978-1-4757-3083-8 |
| [34] | Lannes, David, Consistency of the KP approximation, Discrete Contin. Dyn. Syst., suppl., 517-525 (2003) · Zbl 1066.35017 |
| [35] | Lizorkin, Pëtr I., Multipliers of Fourier integrals in the spaces \({L}_{p,\,\theta }\), Tr. Mat. Inst. Steklova, 89, 231-248 (1967) · Zbl 0159.17402 |
| [36] | Liu, Yong; Wei, Juncheng, Nondegeneracy, Morse index and orbital stability of the KP-I lump solution, Arch. Ration. Mech. Anal., 234, 3, 1335-1389 (2019) · Zbl 1428.35458 |
| [37] | Liu, Yong; Wei, Juncheng, Multivortex traveling waves for the Gross-Pitaevskii equation and the Adler-Moser polynomials, SIAM J. Math. Anal., 52, 4, 3546-3579 (2020) · Zbl 1445.35106 |
| [38] | Liu, Yong; Wang, Zhengping; Wei, Juncheng; Yang, Wen, From KP-I lump solution to travelling waves of Gross-Pitaevskii equation (2021) |
| [39] | Mariş, Mihai, Traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity, Ann. Math., 178, 107-182 (2013) · Zbl 1315.35207 · doi:10.4007/annals.2013.178.1.2 |
| [40] | Muscalu, Camil; Pipher, Jill; Tao, Terence; Thiele, Christoph, A short proof of the Coifman-Meyer multilinear theorem |
| [41] | Manakov, S.; Zakharov, V.; Bordag, L.; Its, A.; Matveev, V., Two-dimensional solitons of the Kadomtsev-Petviashvili equation and their interaction, Phys. Lett., A, 63, 205-206 (1977) · doi:10.1016/0375-9601(77)90875-1 |
| [42] | Pismen, Len M., Vortices in Nonlinear Fields: From Liquid Crystals to Superfluids, From Non-Equilibrium Patterns to Cosmic Strings, 100 (1999), Oxford University Press · Zbl 0987.76001 |
| [43] | del Pino, Manuel; Kowalczyk, Michał; Musso, Monica, Variational reduction for Ginzburg-Landau vortices, J. Funct. Anal., 239, 2, 497-541 (2006) · Zbl 1387.35561 · doi:10.1016/j.jfa.2006.07.006 |
| [44] | Roberts, Paul H.; Berloff, Natalia G.; Barenghi, C. F.; Donnelly, R. J.; Vinen, W. F., Quantized Vortex Dynamics and Superfluid Turbulence, 571, Chapter V. The nonlinear Schrödinger equation as a model of superfluid helium, 233-257 (2001), Springer · Zbl 0994.82105 |
| [45] | de Bouard, Anne; Saut, Jean-Claude, Mathematical problems in the theory of water waves (Luminy, 1995), 200, Remarks on the stability of generalized KP solitary waves, 75-84 (1996), American Mathematical Society · Zbl 0863.35095 · doi:10.1090/conm/200/02510 |
| [46] | de Bouard, Anne; Saut, Jean-Claude, Solitary waves of generalized Kadomtsev-Petviashvili equations, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 14, 2, 211-236 (1997) · Zbl 0883.35103 · doi:10.1016/s0294-1449(97)80145-x |
| [47] | Vassenet, Marc-Antoine, Transonic limit of traveling waves of the Euler-Korteweg system (2022) |
| [48] | Wang, Zhi-Qiang; Willem, Michel, A multiplicity result for the generalized Kadomtsev-Petviashvili equation, Topol. Methods Nonlinear Anal., 7, 2, 261-270 (1996) · Zbl 0902.35100 · doi:10.12775/TMNA.1996.012 |
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