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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
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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
[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
[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
[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
[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
[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
[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
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