A uniqueness result for travelling waves in the Gross-Pitaevskii equation
Eliot Pacherie1
1 NYUAD Research Institute, New York University Abu Dhabi, PO Box 129188, Abu Dhabi, UAE
Séminaire Laurent Schwartz — EDP et applications (2021-2022), Talk no. 17, 16 p.

This note is a summary of a series of papers [12], [13] and [14], done in collaboration with David Chiron. In them, we establish the uniqueness of the energy minimizer at fixed large momentum for the 2 dimensional Gross-Pitaevskii equation, up to the natural invariances of the problem. The minimizer is a nonradial travelling wave with a small speed, behaving like two well separated vortices. Here, we summarize the key steps of the proof, highlighting the arguments that can be used for similar problems in other equations.

Published online:
DOI: 10.5802/slsedp.148
Eliot Pacherie 1

1 NYUAD Research Institute, New York University Abu Dhabi, PO Box 129188, Abu Dhabi, UAE 
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Eliot Pacherie. A uniqueness result for travelling waves in the Gross-Pitaevskii equation. Séminaire Laurent Schwartz — EDP et applications (2021-2022), Talk no. 17, 16 p. doi : 10.5802/slsedp.148. https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.148/

[1] M. Abid, C. Huepe, S. Metens, C. Nore, C. T. Pham, L. S. Tuckerman, and M. E. Brachet. Gross-Pitaevskii dynamics of Bose-Einstein condensates and superfluid turbulence. Fluid Dynam. Res., 33(5-6):509–544, 2003. | DOI | MR | Zbl

[2] G. Alberti, S. Baldo, and G. Orlandi. Variational convergence for functionals of Ginzburg-Landau type. Indiana Univ. Math. J., 54(5):1411–1472, 2005. | DOI | MR | Zbl

[3] J. Bellazzini and D. Ruiz. Finite energy traveling waves for the Gross-Pitaevskii equation in the subsonic regime. 2019. | arXiv

[4] F. Bethuel, P. Gravejat, and J.-C. Saut. On the KP I transonic limit of two-dimensional Gross-Pitaevskii travelling waves. Dyn. Partial Differ. Equ., 5(3):241–280, 2008. | DOI | MR | Zbl

[5] F. Bethuel, P. Gravejat, and J.-C. Saut. Travelling waves for the Gross-Pitaevskii equation. II. Comm. Math. Phys., 285(2):567–651, 2009. | DOI | MR | Zbl

[6] F. Bethuel, G. Orlandi, and D. Smets. Vortex rings for the Gross-Pitaevskii equation. J. Eur. Math. Soc. (JEMS), 6(1):17–94, 2004. | DOI | Zbl

[7] F. Bethuel and J.-C. Saut. Travelling waves for the Gross-Pitaevskii equation. I. Ann. Inst. H. Poincaré Phys. Théor., 70(2):147–238, 1999. | Zbl

[8] Fabrice Béthuel, Philippe Gravejat, and Jean-Claude Saut. Existence and properties of travelling waves for the Gross-Pitaevskii equation. In Stationary and time dependent Gross-Pitaevskii equations, volume 473 of Contemp. Math., pages 55–103. Amer. Math. Soc., Providence, RI, 2008. | DOI | Zbl

[9] H. Brezis, F. Merle, and T. Rivière. Quantization effects for -Δu=u(1-|u| 2 ) in R 2 . Arch. Rational Mech. Anal., 126(1):35–58, 1994. | DOI | Zbl

[10] X. Chen, C. M. Elliott, and T. Qi. Shooting method for vortex solutions of a complex-valued Ginzburg-Landau equation. Proc. Roy. Soc. Edinburgh Sect. A, 124(6):1075–1088, 1994. | DOI | MR | Zbl

[11] D. Chiron and M. Mariş. Traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity. Arch. Ration. Mech. Anal., 226(1):143–242, 2017. | DOI | Zbl

[12] D. Chiron and E. Pacherie. Smooth branch of travelling waves for the Gross-Pitaevskii equation in ℝ 2 for small speed. Ann. Sc. Norm. Super. Pisa Cl. Sci., 2021. | DOI | Zbl

[13] D. Chiron and E. Pacherie. A uniqueness result for the two vortex travelling wave in the nonlinear Schrödinger equation. Anal. PDE, to appear.

[14] D. Chiron and E. Pacherie. Coercivity for travelling waves in the Gross-Pitaevskii equation in ℝ 2 for small speed. Publ. Mat., to appear.

[15] D. Chiron and C. Scheid. Travelling waves for the nonlinear Schrödinger equation with general nonlinearity in dimension two. J. Nonlinear Sci., 26(1):171–231, 2016. | DOI | Zbl

[16] D. Chiron and C. Scheid. Multiple branches of travelling waves for the Gross-Pitaevskii equation. Nonlinearity, 31(6):2809–2853, 2018. | DOI | MR | Zbl

[17] M. del Pino, P. Felmer, and M. Kowalczyk. Minimality and nondegeneracy of degree-one Ginzburg-Landau vortex as a Hardy’s type inequality. Int. Math. Res. Not., (30):1511–1527, 2004. | DOI | Zbl

[18] C. Gallo. The Cauchy problem for defocusing nonlinear Schrödinger equations with non-vanishing initial data at infinity. Comm. Partial Differential Equations, 33(4-6):729–771, 2008. | DOI | Zbl

[19] P. Gérard. The Cauchy problem for the Gross-Pitaevskii equation. Ann. Inst. H. Poincaré Anal. Non Linéaire, 23(5):765–779, 2006. | DOI | MR | Zbl

[20] P. Gérard. The Gross-Pitaevskii equation in the energy space. In Stationary and time dependent Gross-Pitaevskii equations, volume 473 of Contemp. Math., pages 129–148. Amer. Math. Soc., Providence, RI, 2008. | DOI | Zbl

[21] V. L. Ginzburg and L. P. Pitaevskii. On the theory of superfluidity. Sov. Phys. JETP, 7(5):858–861, 1958.

[22] P. Gravejat. Decay for travelling waves in the Gross-Pitaevskii equation. Ann. Inst. H. Poincaré Anal. Non Linéaire, 21(5):591–637, 2004. | DOI | MR | Zbl

[23] P. Gravejat. Asymptotics for the travelling waves in the Gross-Pitaevskii equation. Asymptot. Anal., 45(3-4):227–299, 2005. | Zbl

[24] Philippe Gravejat. A non-existence result for supersonic travelling waves in the Gross-Pitaevskii equation. Comm. Math. Phys., 243(1):93–103, 2003. | DOI | MR | Zbl

[25] Philippe Gravejat. Limit at infinity and nonexistence results for sonic travelling waves in the Gross-Pitaevskii equation. Differential Integral Equations, 17(11-12):1213–1232, 2004. | Zbl

[26] Philippe Gravejat, Eliot Pacherie, and Didier Smets. On the stability of the Ginzburg-Landau vortex. 2021. | arXiv

[27] R.-M. Hervé and M. Hervé. Étude qualitative des solutions réelles d’une équation différentielle liée à l’équation de Ginzburg-Landau. Ann. Inst. H. Poincaré Anal. Non Linéaire, 11(4):427–440, 1994. | DOI | Zbl

[28] R. L. Jerrard and H. M. Soner. The Jacobian and the Ginzburg-Landau energy. Calc. Var. Partial Differential Equations, 14(2):151–191, 2002. | DOI | Zbl

[29] C. A. Jones, J. S. Putterman, and P. H. Roberts. Motions in a Bose condensate. V. Stability of solitary waves solutions of nonlinear Schrödinger equations in two and three dimensions. J. Phys. A: Math. and Gen., 19(15):2991–3011, 1986. | DOI

[30] C. A. Jones and P. H. Roberts. Motions in a Bose condensate. IV. Axisymmetric solitary waves. J. Phys. A: Math. and Gen., 15(8):2599–2619, 1982. | DOI

[31] M. Mariş. Traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity. Ann. of Math. (2), 178(1):107–182, 2013. | DOI | Zbl

[32] P. Mironescu. Les minimiseurs locaux pour l’équation de Ginzburg-Landau sont à symétrie radiale. C. R. Acad. Sci. Paris Sér. I Math., 323(6):593–598, 1996. | Zbl

[33] J. C. Neu. Vortices in complex scalar fields. Phys. D, 43:385–406, 1990. | DOI | MR | Zbl

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

[35] E. Sandier. Lower bounds for the energy of unit vector fields and applications. J. Funct. Anal., 152(2):379–403, 1998. Erratum: Ibid. 171(1):233, 2000.

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