Dark solitons in external potentials

Zeitschrift fur Angewandte Mathematik und Physik

Volumn 59, Issue 4, 2008, Pages 559-599

  Pelinovsky, Dmitry E ^a   Kevrekidis, Panayotis G ^b  

^a MCMASTER UNIVERSITY   (Canada)

^b UNIVERSITY OF MASSACHUSETTS   (United States)

Author keywords

Dark solitions; Eigenvalues; Gross Pitaevskii equation; Persistence and stability

Indexed keywords

CHLORINE COMPOUNDS; COMPUTER SIMULATION; LINEAR EQUATIONS; NONLINEAR EQUATIONS; TURBULENT FLOW;

COMPLEX EIGENVALUES; DARK SOLITIONS; DARK SOLITONS; EFFECTIVE POTENTIALS; EIGENVALUES; EXTERNAL-; GP EQUATIONS; GROSS-PITAEVSKII EQUATION; GROSS-PITAEVSKII EQUATIONS; MONOTONIC INSTABILITY; NUMERICA L RESULTS; NUMERICAL APPROXIMATIONS; NUMERICAL SIMULATIONS; OSCILLATORY INSTABILITIES; PERSISTENCE AND STABILITY; POSITIVE REAL; ZERO SPEED;

SOLITONS;

EID: 47049104873     PISSN: 00442275     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00033-007-6120-0     Document Type: Article

References (45)

1. I. V. Barashenkov, Stability criterion for dark solitons, Phys. Rev. Lett. 77 (1996), 1193-1197.
2. I. V. Barashenkov and E. Y. Panova, Stability nd evolution of the quiescient and travelling solitonic bubbles, Physica D 69 (1993), 114-134.
3. M. D. Barrett, J. A. Sauer and M. S. Chapman, All-optical formation of an atomic Bose-Einstein condensate, Phys. Rev. Lett. 87 (2001), 010404.
4. A. de Bouard, Instability of stationary bubbles, SIAM J. Math. Anal. 26 (1995), 566-582.
5. V. A. Brazhnyi and V. V. Konotop, Evolution of a dark soliton in a parabolic potential: application to Bose-Einstein condensates, Phys. Rev. A 68 (2003), 043613.
6. J. C. Bronski and R. L. Jerrard, Soliton dynamics in a potential, Math. Res. Lett. 7 (2000), 329-342.
7. L. D. Carr, M. J. Holland and B. A. Malomed, Macroscopic quantum tunnelling of Bose-Einstein condensates in a finite potential well, J. Phys. B: At. Mol. Opt. Phys. 38 (2005), 3217-3231.
8. X. J. Chen, Z. D. Chen, and N. N. Huang, A direct perturbation theory for dark solitons based on a complete set of the squared Jost solutions, J. Phys. A: Math. Gen. 31 (1998), 6929-6947.
9. M. Chugunova and D. Pelinovsky, Count of eigenvalues in the generalized eigenvalue problem, math. DS/0602386
10. E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations. McGraw-Hill, New York 1955.
11. S. Cuccagna and D. Pelinovsky, Bifurcations from the end points of the essential spectrum in the linearized NLS problem, J. Math. Phys. 46 (2005), 053520.
12. B. Deconinck, D. Pelinovsky, and J. D. Carter, Transverse instabilities of deep-water solitary waves, Proc. Roy. Soc. A 462 (2006), 2039-2061.
13. J. Fröhlich, S. Gustafson, B. L. G. Jonsson, and I. M. Sigal, Solitary wave dynamics in an external potential, Comm. Math. Phys. 250 (2004), 613-642.
14. P. D. Hislop and I. M. Sigal, Introduction to Spectral Theory with Applications to Schrödinger Operators, Springer-Verlag, 1996.
15. N. N. Huang, S. Chi, and X. J. Chen, Foundations of direct perturbation method for dark solitons, J. Phys. A: Math. Gen. 32 (1999), 3939-3945.
16. B. L. G. Jonsson, J. Fröhlich, S. Gustafson and I.M. Sigal, Long time motion of NLS solitary waves in a confining potential, Ann. Henri Poincare 7 (2006), 621-660.
17. M. Johansson and Yu. S. Kivshar, Discreteness-induced oscillatory instabilities of dark solitons, Phys. Rev. Lett. 82 (1999), 85-88.
18. T. Kapitula, Stability of waves in perturbed Hamiltonian systems, Physica D 156 (2001), 186-200.
19. T. Kapitula and J. Rubin, Existence and stability of standing hole solutions to complex Ginzburg-Landau equations, Nonlinearity 13 (2000), 77-112.
20. T. Kapitula and B. Sandstede, Eigenvalues and resonances using the Evans function, Discr. Cont. Dyn. Syst. B 10 (2004), 857-869.
21. P. G. Kevrekidis and D. J. Frantzeskakis, Pattern forming dynamical instabilities of Bose-Einstein Condensates, Mod. Phys. Lett. B 18 (2004), 173-202.
22. Yu. S. Kivshar and B. Luther-Davies, Dark optical solitons: physics and applications, Phys. Rep. 298 (1988), 81-197.
23. Yu. S. Kivshar and X. Yang, Perturbation-induced dynamics of dark solitons, Phys. Rev. E 49 (1994), 1657-1670.
24. V. V. Konotop and V. E. Vekslerchik, Direct perturbation theory for dark solitons, Phys. Rev. E 49 (1994), 2397-2407.
25. E. A. Kuznetsov and S. A. Turitsyn, Instabilities and collapse of solitons in the medium with defocusing nonlinearity, Sov. Phys. JETP 67 (1988), 1583-1588.
26. V. M. Lashkin, Perturbation theory for dark solitons: Inverse scattering transform approach and radiative effects, Phys. Rev. E 70 (2004), 066620.
27. J. Lega and S. Fauve, Traveling hole solutions to the complex Ginzburg-Landau equation as perturbations of nonlinear Schrödinger dark solitons, Physica D 102 (1997), 234-252.
28. Z. Lin, Stability and instability of traveling solitonic bubbles, Adv. Diff. Eqs. 7 (2002), 897-918.
29. M. Maris, Stationary solutions to a nonlinear Schrödinger equation with potential in one dimension, Proc. Roy. Soc. Edinburgh A 133 (2003), 409-437.
30. M. Maris, Global branches of travelling waves to a Gross-Pitaevskii- Schrödinger system in one dimension, SIAM J. Math. Anal. 37 (2006), 1535-1559.
31. L. Di Menza and C. Gallo, The black solitons of one-dimensional NLS equations, Nonlinearity 20 (2007), 461-496.
32. R. Onofrio, C. Raman, J. M. Vogels, J. R. Abo-Shaeer, A. P. Chikkatur and W. Ketterle, Observation of superfluid flow in a Bose-Einstein gas, Phys. Rev. Lett. 85 (2000), 2228-2231.
33. D. E. Pelinovsky and R. H. J. Grimshaw, Asymptotic methods in soliton stability theory, in Advances in Fluid Mechanics Series, 12: Nonlinear Instability Analysis, edited by L. Debnath and S. R.Choudhury, Computational Mechanics Publications, Southampton, Boston 1997, pp. 245-312.
34. D. E. Pelinovsky, D. Frantzeskakis, and P. G. Kevrekidis, Oscillations of dark solitons in trapped Bose-Einstein condensates, Phys. Rev. E 72 (2005), 016615.
35. D. E. Pelinovsky, Yu. S. Kivshar, and V. V. Afanasjev, Instability-induced dynamics of dark solitons, Phys. Rev. E 54 (1996), 2015-2032.
36. D. E. Pelinovsky, Yu. A. Stepanyants and Yu. S. Kivshar, Self-focusing of plane dark solitons in nonlinear defocusing media, Phys.Rev. E 51 (1995), 5016-5026.
37. D. E. Pelinovsky, A. A. Sukhorukov, and Yu. S. Kivshar, Bifurcations and stability of gap solitons in periodic potentials, Phys. Rev. E 70 (2004), 036618.
38. D. Pelinovsky and J. Yang, Stability analysis of embedded solitons in the generalized third-order NLS equation, Chaos 15 (2005), 037115.
39. C. J. Pethick and H. Smith, Bose-Einstein condensation in dilute gases, Cambridge University Press, Cambridge 2002.
40. L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensation, Oxford University Press, Oxford 2003.
41. B. Prinari, M. J. Ablowitz, and G. Biondini, Inverse scattering transform for the vector nonlinear Schrödinger equation with non-vanishing boundary conditions, J. Math. Phys. 47 (2006), 063508.
42. N. P. Proukakis, N. G. Parker, D. J. Frantzeskakis, and C.S. Adams, Analogies between dark solitons in atomic Bose-Einstein condensates and optical systems, J. Opt. B: Quantum Semiclass. Opt. 6 (2004), S380-S391.
43. B. Simon, Resonances in one dimension and Fredholm determinants, J. Funct. Anal. 178 (2000), 396-420.
44. J. D. Wright and A. Scheel, Solitary waves and their linear stability in weakly coupled KdV equations, Z. angew. Math. Phys. doi 10.1007/s00033-006-6076-5
45. P. E. Zhidkov, Existence of solutions to the Cauchy problem and stability of kink-solutions of the nonlinear Schrödinger equation, Siberian. Math. J. 33 (1992), 239-246.