Smooth branch of rarefaction pulses for the Nonlinear Schrödinger Equation and the Euler–Korteweg system in 2d
Annales Henri Lebesgue, Volume 6 (2023), pp. 767-845.

Metadata

Keywords Travelling waves, Nonlinear Schrödinger Equation, Euler–Korteweg system, Kadomtsev–Petviashvili equation, lump

Abstract

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.


References

[AHM + 03] 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, Volume 33 (2003) no. 5-6, pp. 509-544 | DOI | MR | Zbl

[AS64] Abramowitz, Milton; Stegun, Irene. A. Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, 55, U. S. Government Printing Office, Washington, D. C., 1964 (For sale by the Superintendent of Documents) | Zbl

[Aud17] Audiard, Corentin Small energy traveling waves for the Euler–Korteweg system, Nonlinearity, Volume 30 (2017) no. 9, pp. 3362-3399 | DOI | Zbl

[BG13] Benzoni-Gavage, Sylvie Planar traveling waves in capillary fluids, Differ. Integral Equ., Volume 26 (2013) no. 3-4, pp. 439-485 | MR | Zbl

[BGC18] Benzoni-Gavage, Sylvie; Chiron, David Long wave asymptotics for the Euler–Korteweg system, Rev. Mat. Iberoam., Volume 34 (2018) no. 1, pp. 245-304 | DOI | MR | Zbl

[BGS08] Béthuel, Fabrice; Gravejat, Philippe; Saut, Jean-Claude On the KP-I transonic limit of two-dimensional Gross–Pitaevskii travelling waves, Dyn. Partial Differ. Equ., Volume 5 (2008) no. 3, pp. 241-280 | DOI | MR | Zbl

[BGS09] Béthuel, Fabrice; Gravejat, Philippe; Saut, Jean-Claude Travelling waves for the Gross–Pitaevskii equation. II, Commun. Math. Phys., Volume 285 (2009) no. 2, pp. 567-651 | DOI | MR | Zbl

[BOS04] Béthuel, Fabrice; Orlandi, Giandomenico; Smets, Didier Vortex rings for the Gross–Pitaevskii equation, J. Eur. Math. Soc., Volume 6 (2004) no. 1, pp. 17-94 | DOI | MR | Zbl

[BR04] 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., Volume 37 (2004), pp. 11333-11351 | DOI | MR | Zbl

[BR23] Bellazzini, Jacopo; Ruiz, David Finite energy traveling waves for the Gross–Pitaevskii equation in the subsonic regime, Am. J. Math., Volume 145 (2023) no. 1, pp. 109-149 | DOI | MR | Zbl

[BS99] Béthuel, Fabrice; Saut, Jean-Claude Travelling waves for the Gross-Pitaevskii equation. I, Ann. Inst. Henri Poincaré, Phys. Théor., Volume 70 (1999) no. 2, pp. 147-238 | Numdam | MR | Zbl

[Chi04] Chiron, David Travelling waves for the Gross–Pitaevskii equation in dimension larger than two, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods, Volume 58 (2004) no. 1-2, pp. 175-204 | DOI | MR | Zbl

[Chi12] Chiron, David Travelling waves for the Nonlinear Schrödinger Equation with general nonlinearity in dimension one, Nonlinearity, Volume 25 (2012), pp. 813-850 | DOI | MR | Zbl

[Chi14] 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, Volume 31 (2014) no. 6, pp. 1175-1230 | DOI | MR | Zbl

[CM14] Chiron, David; Mariş, Mihai Rarefaction pulses for the Nonlinear Schrödinger Equation in the transonic limit, Commun. Math. Phys., Volume 326 (2014) no. 2, pp. 329-392 | DOI | Zbl

[CM17] Chiron, David; Mariş, Mihai Traveling Waves for Nonlinear Schrödinger Equations with Nonzero Conditions at Infinity, Arch. Ration. Mech. Anal., Volume 226 (2017) no. 1, pp. 143-242 | DOI | Zbl

[CP21] Chiron, David; Pacherie, Eliot Smooth branch of travelling waves for the Gross–Pitaevskii equation in 2 for small speed, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 22 (2021) no. 4, pp. 1937-2038 | MR | Zbl

[CP23] Chiron, David; Pacherie, Eliot Coercivity for travelling waves in the Gross–Pitaevskii equation in 2 for small speed, Publ. Mat., Barc., Volume 67 (2023) no. 1, pp. 277-410 | DOI | MR | Zbl

[CR10] Chiron, David; Rousset, Frédéric The KdV/KP-I limit of the nonlinear Schrödinger equation, SIAM J. Math. Anal., Volume 42 (2010) no. 1, pp. 64-96 | DOI | Zbl

[CS16] Chiron, David; Scheid, Claire Travelling waves for the Nonlinear Schrödinger Equation with general nonlinearity in dimension two, J. Nonlinear Sci., Volume 26 (2016) no. 2, pp. 171-231 | DOI | Zbl

[CS18] Chiron, David; Scheid, Claire Multiple branches of travelling waves for the Gross–Pitaevskii equation, Nonlinearity, Volume 31 (2018) no. 6, pp. 2809-2853 | DOI | MR | Zbl

[dPFK04] 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. (2004) no. 30, pp. 1511-1527 | DOI | MR | Zbl

[DPMMR21] 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 (2021), pp. 1319-1367 | Zbl

[Gra04] Gravejat, Philippe Decay for travelling waves in the Gross–Pitaevskii equation, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 21 (2004) no. 5, pp. 591-637 | DOI | Numdam | MR | Zbl

[Gra08] Gravejat, Philippe Asymptotics of the solitary waves for the generalized Kadomtsev–Petviashvili equations, Discrete Contin. Dyn. Syst., Volume 21 (2008) no. 3, pp. 835-882 | DOI | MR | Zbl

[GT01] Gilbarg, D.; Trudinger, N. S. Elliptic partial differential equations of second order, Classics in Mathematics, Springer, 2001 (reprint of the 1998 edition) | DOI | Zbl

[How03] Howie, John M. Complex analysis, Springer Undergraduate Mathematics Series, Springer, 2003 | DOI | Zbl

[IS78] Iordanskii, S. V.; Smirnov, A. V. Three-dimensional solitons in He II, JETP Lett., Volume 27 (1978) no. 10, pp. 535-538

[JPR86] 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., Volume 19 (1986), pp. 2991-3011 | DOI

[JR82] Jones, C.; Roberts, Paul H. Motion in a Bose condensate IV. Axisymmetric solitary waves, J. Phys. A. Math. Gen., Volume 15 (1982), pp. 2599-2619 | DOI

[KLD98] Kivshar, Yuri S.; Luther-Davies, Barry Dark optical solitons: physics and applications, Phys. Rep., Volume 298 (1998), pp. 81-197 | DOI

[KP00] Kivshar, Yuri S.; Pelinovsky, Dmitry E. Self-focusing and transverse instabilities of solitary waves, Phys. Rep., Volume 331 (2000) no. 4, pp. 117-195 | DOI | MR

[Lan99] Lang, Serge Complex analysis, Graduate Texts in Mathematics, 103, Springer, 1999 | DOI | Zbl

[Lan03] Lannes, David Consistency of the KP approximation, Discrete Contin. Dyn. Syst., Volume suppl. (2003), pp. 517-525 Dynamical systems and differential equations (Wilmington, NC, 2002) | MR | Zbl

[Liz67] Lizorkin, Pëtr I. Multipliers of Fourier integrals in the spaces L p,θ , Tr. Mat. Inst. Steklova, Volume 89 (1967), pp. 231-248 | MR | Zbl

[LW19] Liu, Yong; Wei, Juncheng Nondegeneracy, Morse index and orbital stability of the KP-I lump solution, Arch. Ration. Mech. Anal., Volume 234 (2019) no. 3, pp. 1335-1389 | MR | Zbl

[LW20] Liu, Yong; Wei, Juncheng Multivortex traveling waves for the Gross–Pitaevskii equation and the Adler–Moser polynomials, SIAM J. Math. Anal., Volume 52 (2020) no. 4, pp. 3546-3579 | MR | Zbl

[LWWY21] Liu, Yong; Wang, Zhengping; Wei, Juncheng; Yang, Wen From KP-I lump solution to travelling waves of Gross–Pitaevskii equation (2021) (preprint, https://arxiv.org/abs/2110.15472)

[Mar13] Mariş, Mihai Traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity, Ann. Math., Volume 178 (2013), pp. 107-182 | DOI | MR | Zbl

[MPTT] Muscalu, Camil; Pipher, Jill; Tao, Terence; Thiele, Christoph A short proof of the Coifman-Meyer multilinear theorem

[MZB + 77] Manakov, S.; Zakharov, V.; Bordag, L.; Its, A.; Matveev, V. Two-dimensional solitons of the Kadomtsev–Petviashvili equation and their interaction, Phys. Lett., A, Volume 63 (1977), pp. 205-206 | DOI

[Pis99] Pismen, Len M. Vortices in Nonlinear Fields: From Liquid Crystals to Superfluids, From Non-Equilibrium Patterns to Cosmic Strings, International Series of Monographs on Physics, 100, Oxford University Press, 1999 | Zbl

[PMK06] del Pino, Manuel; Kowalczyk, Michał; Musso, Monica Variational reduction for Ginzburg–Landau vortices, J. Funct. Anal., Volume 239 (2006) no. 2, pp. 497-541 | DOI | MR | Zbl

[RB01] Roberts, Paul H.; Berloff, Natalia G. Chapter V. The nonlinear Schrödinger equation as a model of superfluid helium, Quantized Vortex Dynamics and Superfluid Turbulence (Barenghi, C. F.; Donnelly, R. J.; Vinen, W.F., eds.) (Lecture Notes in Physics), Volume 571, Springer, 2001, pp. 233-257 | Zbl

[S96] de Bouard, Anne; Saut, Jean-Claude Remarks on the stability of generalized KP solitary waves, Mathematical problems in the theory of water waves (Luminy, 1995) (Contemporary Mathematics), Volume 200, American Mathematical Society, 1996, pp. 75-84 | DOI | MR | Zbl

[S97] de Bouard, Anne; Saut, Jean-Claude Solitary waves of generalized Kadomtsev–Petviashvili equations, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 14 (1997) no. 2, pp. 211-236 | DOI | Numdam | MR | Zbl

[Vas22] Vassenet, Marc-Antoine Transonic limit of traveling waves of the Euler-Korteweg system (2022) (preprint, https://arxiv.org/abs/2212.02819)

[WW96] Wang, Zhi-Qiang; Willem, Michel A multiplicity result for the generalized Kadomtsev–Petviashvili equation, Topol. Methods Nonlinear Anal., Volume 7 (1996) no. 2, pp. 261-270 | DOI | MR | Zbl