Abstract
The purpose of this paper is to provide a rigorous mathematical proof of the existence of travelling wave solutions to the Gross-Pitaevskii equation in dimensions two and three. Our arguments, based on minimization under constraints, yield a full branch of solutions, and extend earlier results (see [3,4,8]) where only a part of the branch was built. In dimension three, we also show that there are no travelling wave solutions of small energy.
Similar content being viewed by others
References
Almeida L.: Topological sectors for Ginzburg-Landau energy. Rev. Mat. Iber. 15(3), 487–546 (1999)
Berloff, N.: Quantum vortices, travelling coherent structures and superfluid turbulence. Preprint
Béthuel F., Orlandi G., Smets D.: Vortex rings for the Gross-Pitaevskii equation. J. Eur. Math. Soc. 6(1), 17–94 (2004)
Béthuel F., Saut J.-C.: Travelling waves for the Gross-Pitaevskii equation I. Ann. Inst. Henri Poincaré, Physique Théorique. 70(2), 147–238 (1999)
Béthuel, F., Saut, J.-C.: Vortices and sound waves for the Gross-Pitaevskii equation. In: Nonlinear PDE’s in Condensed Matter and Reactive Flows, Volume 569 of NATO Science Series C. Mathematical and Physical Sciences, Dordrecht, Kluwer Academic Publishers, 2002, pp. 339–354
Bona J.L., Li Y.A.: Analyticity of solitary-wave solutions of model equations for long waves. SIAM J. Math. Anal. 27(3), 725–737 (1996)
Bona J.L., Li Y.A.: Decay and analyticity of solitary waves. J. Math. Pures Appl. 76(5), 377–430 (1997)
Chiron D.: Travelling waves for the Gross-Pitaevskii equation in dimension larger than two. Nonlinear Anal. 58(1-2), 175–204 (2004)
de Bouard, A., Saut, J.-C.: Remarks on the stability of generalized KP solitary waves. In: Mathematical problems in the theory of water waves (Luminy, 1995), Volume 200 of Contemp. Math., Providence, RI: Amer. Math. Soc., 1996, pp. 75–84
de Bouard A., Saut J.-C.: Solitary waves of generalized Kadomtsev-Petviashvili equations. Ann. Inst. Henri Poincaré, Analyse Non Linéaire. 14(2), 211–236 (1997)
de Bouard A., Saut J.-C.: Symmetries and decay of the generalized Kadomtsev-Petviashvili solitary waves. SIAM J. Math. Anal. 28(5), 1064–1085 (1997)
Ekeland I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)
Farina A.: From Ginzburg-Landau to Gross-Pitaevskii. Monatsh. Math. 139, 265–269 (2003)
Gallo, C.: The Cauchy problem for defocusing nonlinear Schrödinger equations with non-vanishing initial data at infinity. Preprint
Gérard P.: The Cauchy problem for the Gross-Pitaevskii equation. Ann. Inst. Henri Poincaré, Analyse Non Linéaire. 23(5), 765–779 (2006)
Gérard, P.: The Gross-Pitaevskii equation in the energy space. Preprint
Goubet O.: Two remarks on solutions of Gross-Pitaevskii equations on Zhidkov spaces. Monatsh. Math. 151(1), 39–44 (2007)
Gravejat P.: Limit at infinity for travelling waves in the Gross-Pitaevskii equation. C. R. Math. Acad. Sci. Paris. 336(2), 147–152 (2003)
Gravejat P.: A non-existence result for supersonic travelling waves in the Gross-Pitaevskii equation. Commun. Math. Phys. 243(1), 93–103 (2003)
Gravejat P.: Decay for travelling waves in the Gross-Pitaevskii equation. Ann. Inst. Henri Poincaré, Analyse Non Linéaire. 21(5), 591–637 (2004)
Gravejat P.: Limit at infinity and nonexistence results for sonic travelling waves in the Gross-Pitaevskii equation. Differ. Int. Eqs. 17(11–12), 1213–1232 (2004)
Gravejat P.: Asymptotics for the travelling waves in the Gross-Pitaevskii equation. Asymptot. Anal. 45(3–4), 227–299 (2005)
Gravejat P.: First order asymptotics for the travelling waves in the Gross-Pitaevskii equation. Adv. Differ. Eqs. 11(3), 259–280 (2006)
Gravejat P.: Asymptotics of the solitary waves for the generalised Kadomtsev-Petviashvili equations. Disc. Cont. Dynam. Syst. 21(3), 835–882 (2008)
Gross E.P.: Hydrodynamics of a superfluid condensate. J. Math. Phys. 4(2), 195–207 (1963)
Gustafson, S., Nakanishi, K., Tsai, T.-P.: Scattering theory for the Gross-Pitaevskii equation in three dimensions. Preprint, available at http://arxiv.org/abs/0803.3208vi[math.AP], 2008
Gustafson S., Nakanishi K., Tsai T.-P.: Scattering for the Gross-Pitaevskii equation. Math. Res. Lett. 13(2), 273–285 (2006)
Gustafson S., Nakanishi K., Tsai T.-P.: Global dispersive solutions for the Gross-Pitaevskii equation in two and three dimensions. Ann. Henri Poincaré. 8(7), 1303–1331 (2007)
Iordanskii S.V., Smirnov A.V.: Three-dimensional solitons in He II. JETP Lett. 27(10), 535–538 (1978)
Jones C.A., Putterman S.J., Roberts P.H.: Motions in a Bose condensate V. Stability of solitary wave solutions of nonlinear Schrödinger equations in two and three dimensions. J. Phys. A, Math. Gen. 19, 2991–3011 (1986)
Jones C.A., Roberts P.H.: Motions in a Bose condensate IV. Axisymmetric solitary waves. J. Phys. A, Math. Gen. 5, 2599–2619 (1982)
Kato K., Pipolo P.-N.: Analyticity of solitary wave solutions to generalized Kadomtsev-Petviashvili equations. Proc. Roy. Soc. Edinb. A. 131(2), 391–424 (2001)
Lizorkin P.I.: On multipliers of Fourier integrals in the spaces L p,θ . Proc. Steklov Inst. Math. 89, 269–290 (1967)
Lopes O.: A constrained minimization problem with integrals on the entire space. Bol. Soc. Bras. Mat. 25(1), 77–92 (1994)
Maris M.: Analyticity and decay properties of the solitary waves to the Benney-Luke equation. Differ. Int. Eqs. 14(3), 361–384 (2001)
Maris M.: On the existence, regularity and decay of solitary waves to a generalized Benjamin-Ono equation. Nonlinear Anal. 51(6), 1073–1085 (2002)
Nakanishi, K.: Scattering theory for the Gross-Pitaevskii equation. Preprint
Pitaevskii L.P.: Vortex lines in an imperfect Bose gas. Sov. Phys. JETP. 13(2), 451–454 (1961)
Stein, E.M.: Harmonic analysis : Real-variable methods, orthogonality, and oscillatory integrals. Volume 43 of Princeton Mathematical Series. Monographs in Harmonic Analysis. Princeton, NJ: Princeton Univ. Press, 1993 (With the assistance of T.S. Murphy)
Tarquini É.: A lower bound on the energy of travelling waves of fixed speed for the Gross-Pitaevskii equation. Monatsh. Math. 151(4), 333–339 (2007)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by P. Constantin
Rights and permissions
About this article
Cite this article
Béthuel, F., Gravejat, P. & Saut, JC. Travelling Waves for the Gross-Pitaevskii Equation II. Commun. Math. Phys. 285, 567–651 (2009). https://doi.org/10.1007/s00220-008-0614-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-008-0614-2