×

Travelling waves for the Gross-Pitaevskii equation. I. (English) Zbl 0933.35177

The authors consider the nonlinear Schrödinger equation \[ -i \frac{\partial v}{\partial t} + \Delta v + v(1-|v|^2) = 0 \] on \(\mathbb R^2 \times \mathbb R \) for complex valued functions. The authors are only interested in finite energy solutions \[ E(v) = \tfrac{1}{2} \int_{\mathbb R^2} |\Delta v|^2 + \frac{1}{4} \int _{\mathbb R^2} (1-|v|^2)<\infty. \]
In view of the form of \(E\) and the potential \(V(v)=(1-|v|^2)^2,\) as natural boundary condition, one takes \(v(x) \to 1\) as \(x \to \infty.\)
The authors investigate the existence of travelling wave solutions of the form \(v(x,t)=\widetilde v(x_1 - ct,x_2); \;\;x=(x_1,x_2)\), where \(c > 0\) represents the speed of the travelling wave and \(\widetilde v\) is a solution to the equation \[ -ic \frac{\partial \widetilde v}{\partial x_1} = \Delta \widetilde v + \widetilde v (1- \widetilde v)^2 \] on \(\mathbb R^2\).

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
82D55 Statistical mechanics of superconductors
PDFBibTeX XMLCite
Full Text: Numdam EuDML

References:

[1] L. Almeida and F. Bethuel , Topological methods for the Ginzburg-Landau equation , J. Math. Pures Appl. , 77 , 1998 , pp. 1 - 49 . MR 1617594 | Zbl 0904.35023 · Zbl 0904.35023 · doi:10.1016/S0021-7824(98)80064-0
[2] F. Bethuel , H. Brézis and F. Hélein , Ginzburg-Landau vortices , Birkhaüser 1994 . MR 1269538 | Zbl 0802.35142 · Zbl 0802.35142
[3] H. Brézis , J.-M. Coron and E. Lieb , Harmonic maps with defects , Comm. Math. Phys. , 107 , 1986 , pp. 649 - 705 . Article | MR 868739 | Zbl 0608.58016 · Zbl 0608.58016 · doi:10.1007/BF01205490
[4] H. Brézis , F. Merle and T. Rivière , Quantization effects for -\Delta v = v(1 - |v|2) in R2 , Arch. Rational Mech. Anal. , 126 , 1994 , pp. 35 - 58 . MR 1268048 | Zbl 0809.35019 · Zbl 0809.35019 · doi:10.1007/BF00375695
[5] F. Bethuel and T. Rivière , Vortices for a minimization problem related to superconductivity , Annales IHP, Analyse Non Linéaire , 12 , 1995 , pp. 243 - 303 . Numdam | MR 1340265 | Zbl 0842.35119 · Zbl 0842.35119
[6] F. Bethuel and J.-C. Saut , Travelling waves for the Gross-Pitaevskii equation II , in preparation. · Zbl 0933.35177
[7] J.E. Colliander and R.L. Jerrard , Vortex dynamics for the Ginzburg-Landau-Schrödinger equation , Preprint, 1997 . arXiv | MR 1623410 · Zbl 0914.35128
[8] R. Coifman , P.-L. Lions , Y. Meyer and S. Semmes , Compensated compactness and Hardy spaces , J. Math. Pures Appl. , 72 , 1993 , pp. 247 - 286 . MR 1225511 | Zbl 0864.42009 · Zbl 0864.42009
[9] I. Ekeland , Convexity methods in Hamiltonian Mechanics , Springer-Verlag , 1990 . MR 1051888 | Zbl 0707.70003 · Zbl 0707.70003
[10] 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. , 7 , 2 , 1974 , pp. 260 - 279 .
[11] N. Ghoussoub and D. Preiss , A general mountain pass principle for locating and classifying critical points , Annales IHP , Analyse non Linéaire , 6 , 1989 , pp. 321 - 330 . Numdam | MR 1030853 | Zbl 0711.58008 · Zbl 0711.58008
[12] F. Helein , Applications harmoniques, lois de conservation et repères mobiles , Diderot éd. , Paris , 1996 . · Zbl 0920.58022
[13] C.A. Jones S . J. Putterman and P.H. Roberts , 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. , 15 , 1982 , pp. 2599 - 2619 .
[14] C.A. Jones and P.H. Roberts , Motions in a Bose condensate IV. Axisymmetric solitary waves J. Phys. A. Math. Gen. , 15 , 1982 , pp. 2599 - 2619 .
[15] T. Kato , On nonlinear Schrödinger equations , Ann. Inst. Henri Poincaré , Physique Théorique , 46 , 1 , 1987 , pp. 113 - 129 . Numdam | MR 877998 | Zbl 0632.35038 · Zbl 0632.35038
[16] T. Kato , Nonlinear Schrödinger equations , Lecture Notes in Physics , vol. 345 , Springer Verlag , Berlin , 1989 , pp. 218 - 263 . MR 1037322 | Zbl 0698.35131 · Zbl 0698.35131
[17] E.A. Kuznetsov and J.J. Rasmussen , Instability of two-dimensional solitons and vortices in defocusing media , Phys. Rev. E , 51 , 5 , 1995 , pp. 4479 - 4484 .
[18] E.A. Kuznetsov and J.J. Rasmussen , Self-focusing Instability of two-dimensional solitons and vortices , JETP, Lett. , Vol. 62 , 2 , 1995 , pp. 105 - 112 .
[19] F.H. Lin , Solutions of Ginzburg equations and critical points of the renormalized energy , Annales IHP , Analyse Non Linéaire , 12 , 1995 , pp. 599 - 622 Numdam | MR 1353261 | Zbl 0845.35052 · Zbl 0845.35052
[20] F.H. Lin , Some dynamical properties of Ginzburg-Landau vortices , Comm. Pure Appl. Math.. MR 1376654 | Zbl 0853.35058 · Zbl 0853.35058 · doi:10.1002/(SICI)1097-0312(199604)49:4<323::AID-CPA1>3.0.CO;2-E
[21] K.M. Pismen and A.A. Nepomnyashchy , Stability of vortex maps in a model of superflow , Physica , D 69 , 1993 , pp. 163 - 175 . Zbl 0791.35128 · Zbl 0791.35128 · doi:10.1016/0167-2789(93)90187-6
[22] L.M. Pismen and J. Rubinstein , Motion of vortex lines in the Ginzburg-Landau model , Physica D 47 , 1991 , pp. 353 - 360 . MR 1098255 | Zbl 0728.35090 · Zbl 0728.35090 · doi:10.1016/0167-2789(91)90035-8
[23] M. Struwe , On the asymptotic behavior of minimizers of the Ginzburg-Landau model in 2 dimensions , J. Diff. Int. Eq. , 7 , 1994 , pp. 1617 - 1624 ; Erratum J. Diff. Int. Eq. , 8 , 1995 , p. 224 . MR 1269674 | Zbl 0809.35031 · Zbl 0809.35031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.