Localization phenomena in nonlinear Schrödinger equations with spatially inhomogeneous nonlinearities: Theory and applications to Bose-Einstein condensates

Physica D: Nonlinear Phenomena

Volumn 238, Issue 15, 2009, Pages 1352-1361

  Pérez García, Víctor M ^a   Pardo, Rosa ^b  

^a E T S de Ingenieros Industriales   (Spain)

^b UNIVERSIDAD COMPLUTENSE DE MADRID   (Spain)

Author keywords

Inhomogeneous interactions; Localization; Nonlinear Schr dinger equations

Indexed keywords

BOSE-EINSTEIN CONDENSATION; CONTROL NONLINEARITIES; GROUND STATE; STATISTICAL MECHANICS;

BOSE-EINSTEIN CONDENSATES; DINGER EQUATION; INHOMOGENEOUS INTERACTIONS; LOCALIZATION; LOCALIZATION PHENOMENA; NONLINEAR INTERACTIONS; SPATIALLY INHOMOGENEOUS; STRONG LOCALIZATIONS;

NONLINEAR EQUATIONS;

EID: 67449141398     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physd.2008.08.020     Document Type: Article

References (53)

1. L.P. Pitaevskii, S. Stringari, Bose-Einstein condensation, Oxford, 2003
2. Dalfovo F., and Stringari S. Bosons in anisotropic traps: Ground state and vortices. Phys. Rev. A 53 (1996) 2477
3. Dalfovo F., Pitaevskii L., and Stringari S. Order parameter at the boundary of a trapped Bose gas. Phys. Rev. A 54 (1996) 4213
4. Hyouguchi T., Seto R., Ueda M., and Adachi S. Divergence-free WKB theory. Ann. Phys. 312 (2004) 177-267
5. Ruprecht P.A., Holland M.J., Burnett K., and Edwards M. Time-dependent solution of the nonlinear Schrödinger equation for Bose-condensed trapped neutral atoms. Phys. Rev. A 51 (1995) 4704
6. Holland M., and Cooper J. Expansion of a Bose-Einstein condensate in a harmonic potential. Phys. Rev. A 53 (1996) R1954
7. Inouye S., Andrews M.R., Stenger J., Miesner H.-J., Stamper-Kurn D.M., and Ketterle W. Observation of Feshbach resonances in a Bose-Einstein condensate. Nature 392 (1998) 151
8. Partridge G.B., Truscott A.G., and Hulet R.G. Formation and propagation of matter wave soliton trains. Nature 417 (2002) 150
9. Khaykovich L., Schreck F., Ferrari G., Bourdel T., Cubizolles J., Carr L.D., Castin Y., and Salomon C. Formation of a matter-wave bright soliton. Science 296 (2002) 1290
10. Cornish S.L., Thompson S.T., and Wieman C.E. Formation of bright matter-wave solitons during the collapse of attractive Bose-Einstein condensates. Phys. Rev. Lett. 96 (2006) 170401
11. Donley E.A., Claussen N.R., Cornish S.L., Roberts J.L., Cornell E.A., and Wieman C.E. Dynamics of collapsing and exploding Bose-Einstein condensates. Nature 412 (2001) 295
12. Abdullaev F.K., Kamchatnov A.M., Konotop V.V., and Brazhnyi V.A. Adiabatic dynamics of periodic waves in Bose-Einstein condensates with time dependent atomic scattering length. Phys. Rev. Lett. 90 (2003) 230402
13. Pérez-García V.M., Konotop V.V., and Brazhnyi V.A. Feshbach resonance induced shock waves in Bose-Einstein condensates. Phys. Rev. Lett. 92 (2004) 220403
14. Abdullaev F., Caputo J.G., Kraenkel R.A., and Malomed B.A. Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length. Phys. Rev. A 67 (2003) 013605
15. Saito H., and Ueda M. Dynamically stabilized bright solitons in a two-dimensional Bose-Einstein condensate. Phys. Rev. Lett. 90 (2003) 040403
16. Montesinos G.D., Pérez-García V.M., and Torres P. Stabilization of solitons of the multidimensional nonlinear Schrödinger equation: matter-wave breathers. Physica D (Amsterdam) 191 (2004) 193
17. Abdullaev F.K., Gammal A., Kamchatnov A., and Tomio L. Dynamics of bright matter wave solitons in a Bose-Einstein condensate. Internat. J. Modern Phys. B 19 (2005) 3415
18. Konotop V.V., and Pacciani P. Collapse of solutions of the nonlinear Schrödinger equation with a time-dependent nonlinearity: Application to Bose-Einstein condensates. Phys. Rev. Lett. 94 (2005) 240405
19. Theis M., Thalhammer G., Winkler K., Hellwig M., Ruff G., Grimm R., and Denschlag J.H. Phys. Rev. Lett. 93 (2004) 123001
20. Rodas-Verde M.I., Michinel H., and Pérez-García V.M. Controllable soliton emission from a Bose-Einstein condensate. Phys. Rev. Lett. 95 (2005) 153903
21. Abdullaev F.K., and Garnier J. Propagation of matter-wave solitons in periodic and random nonlinear potentials. Phys. Rev. A 72 (2005) 061605R
22. Theocharis G., Schmelcher P., Kevrekidis P.G., and Frantzeskakis D.J. Matter-wave solitons of collisionally inhomogeneous condensates. Phys. Rev. A 72 (2005) 033614
23. Sakaguchi H., and Malomed B.A. Matter-wave solitons in nonlinear optical lattices. Phys. Rev. E 72 (2005) 046610
24. Garnier J., and Abdullaev F.K. Transmission of matter-wave solitons through nonlinear traps and barriers. Phys. Rev. A 74 (2006) 013604
25. Carpentier A.V., Michinel H., Rodas-Verde M.I., and Perez-Garcia V.M. Analysis of an atom soliton laser based on the spatial control of the scattering length. Phys. Rev. A 74 (2006) 013619
26. Bludov Y.V., and Konotop V.V. Localized modes in arrays of boson-fermion mixtures. Phys. Rev. A 74 (2006) 043616
27. Sakaguchi H., and Malomed B. Two-dimensional solitons in the Gross-Pitaevskii equation with spatially modulated nonlinearity. Phys. Rev. E 73 (2006) 026601
28. Belmonte-Beitia J., Pérez-García V.M., Vekslerchik V., and Torres P.J. Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities. Phys. Rev. Lett. 98 (2007) 064102
29. Niarchou P., Theocharis G., Kevrekidis P.G., Schmelcher P., and Frantzeskakis D.J. Soliton oscillations in collisionally inhomogeneous attractive Bose-Einstein condensates. Phys. Rev. A 76 (2007) 023615
30. Torres P. Guided waves in a multi-layered optical structure. Nonlinearity 19 (2006) 2103-2113
31. Fibich G., Sivan Y., and Weinstein M.I. Bound states of nonlinear Schrodinger equations with a periodic nonlinear microstructure. Physica D 217 (2006) 31-57
32. Sivan Y., Fibich G., and Weinstein M.I. Waves in nonlinear lattices: Ultrashort optical pulses and Bose-Einstein condensates. Phys. Rev. Lett. 97 (2006) 193902
33. Porter M.A., Kevrekidis P.G., Malomed B.A., and Frantzeskakis D.J. Modulated amplitude waves in collisionally inhomogenous Bose-Einstein condensates. Physica D 229 (2007) 104-115
34. Belmonte-Beitia J., Pérez-García V.M., Vekslerchik V., and Torres P.J. Lie Symmetries, qualitative analysis and exact solutions of Nonlinear Schrödinger equations with inhomogeneous nonlinearities. Discrete Contin. Dyn. Syst. B 9 (2008) 221-233
35. Belmonte-Beitia J., Pérez-García V.M., Vekslerchik V., and Konotop V.V. Localized nonlinear waves in systems with time- and space-modulated nonlinearities. Phys. Rev. Lett. 100 (2008) 164102
36. Rodrigues A.S., Kevrekidis P.G., Porter M.A., Frantzeskakis D.J., Schmelcher P., and Bishop A.R. Matter-wave solitons with a periodic, piecewise-constant scattering length. Phys. Rev. A 78 (2008) 013611
37. Pérez-García V.M., Michinel H., and Herrero H. Bose-Einstein solitons in highly asymmetric traps. Phys. Rev. A 57 (1998) 3837
38. Murray J.D. Mathematical Biology (1993), Springer-Verlag, Berlin
39. García-Melián J., Gomez-Reñasco R., Lopez-Gomez J., and Sabina De Lis J.C. Point-wise growth and uniqueness of positive solutions for a class of sublinear elliptic problems where bifurcation from infinity occurs. Arch. Ration. Mech. Anal 145 (1998) 261
40. Fraile J.M., Koch M., Medina P., López-Gómez J., and Merino S. Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation. J. Differential Equations 127 (1996) 295-319
41. López-Gómez J., and Sabina de Lis J.C. First variations of principal eigenvalues with respect to the domain and point-wise growth of positive solutions for problems where bifurcation from infinity occurs. J. Differential Equations 148 (1998) 47-64
42. Bao W., and Du Q. Computing the ground state solution of Bose-Einstein condensates by normalized gradient flow. SIAM J. Sci. Comput. 25 (2004) 1674-1697
43. Welsh W. Monotonicity for eigenvalues of the Schrödinger operator on unbounded domains. Arch. Ration. Mech. Anal. 49 (1972/73) 129-136
44. Krein M.G., and Rutman M.A. Linear operators leaving invariant a cone in a Banach space. Uspekhi Mat. Nauk. 3 1 (1948) 3-95 (in Russian); English transl. in Amer. Math. Soc. Transl. 26 (1950)
45. M.A. Krasnosel'skij, J.A. Lifshits, A.V. Sobolev, Positive Linear Systems, In: The Method of Positive Operators, Heldermann, Berlin, 1989
46. Brezis H. Analyse Fonctionnelle (1983), Masson, Paris
47. Crandall M.G., and Rabinowitz P.H. Bifurcation, perturbation of simple eigenvalues and linearized stability. Arch. Ration. Mech. Anal. 52 (1973) 161-180
48. Rabinowitz P.H. Some global results for nonlinear eigenvalue problems. J. Funct. Anal. 7 (1971) 487-513
49. Rabinowitz P.H. On bifurcation from infinity. J. Differential Equations 14 (1973) 462-475
50. Amann H., and López-Gómez J. J. Differential Equations 146 (1998) 336-374
51. Chang S.-M., Lin C.-S., Lin T.-C., and Lin W.-W. Segregated nodal domains of two-dimensional multispecies Bose-Einstein condensates. Physica D 196 (2004) 341-361
52. Field J.E., Hahn K.H., and Harris S.E. Phys. Rev. Lett. 67 (1991) 3062-3065
53. Michinel H., Paz-Alonso M.I., and Pérez-García V.M. Turning light into a liquid via atomic coherence. Phys. Rev. Lett. 96 (2006) 023903