Symmetry-breaking bifurcation in nonlinear schrödinger/gross- pitaevskii equations

SIAM Journal on Mathematical Analysis

Volumn 40, Issue 2, 2008, Pages 566-604

  Kirr, E W ^a   Kevrekidis, P G ^b   Shlizerman, E ^c   Weinstein, M I ^d  

^a UNIVERSITY OF ILLINOIS AT URBANA CHAMPAIGN   (United States)

^b UNIVERSITY OF MASSACHUSETTS   (United States)

^c WEIZMANN INSTITUTE OF SCIENCE   (Israel)

^d Columbia University ^*  (United States)

Author keywords

Bound state; Gross Pitaevskii; Nonlinear schrodinger; Soliton

Indexed keywords

ANTISYMMETRIC MODES; ASYMMETRIC BRANCHES; BIFURCATION POINTS; BOUND STATE; DOUBLE WELLS; GROSS-PITAEVSKII; LINEAR POTENTIALS; MACROSCOPIC QUANTUM PHENOMENON; MATHEMATICAL MODELING; MIXED MODES; NON-LINEAR OPTICAL; NONLINEAR SCHRODINGER; OPTICAL POWER; PARTICLE NUMBERS; SYMMETRY-BREAKING; SYMMETRY-BREAKING BIFURCATIONS;

BIFURCATION (MATHEMATICS); SOLITONS;

NONLINEAR EQUATIONS;

EID: 54249084092     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/060678427     Document Type: Article

References (35)

1. M. ALBIEZ, R. GATI, J. FOLLING, S. HUNSMANN, M. CRISTIANI, AND M. K. OBERTHALER, Direct observation of tunneling and nonlinear self-trapping in a single bosonic Josephson junction, Phys. Rev. Lett., 95 (2005), p. 010402.
2. G. L. ALFIMOV, P. G. KEVREKIDIS, V. V. KONOTOP, AND M. SALERNO, Wannier functions analysis of the nonlinear Schrödinger equation with a periodic potential, Phys. Rev. E, 66 (2002), p. 046608.
3. W. H. AHBACHER, J. FRÖHLICH, G. M. GRAF, K. SCHNEE, AND M. TROVER, Symmetry breaking regime in the nonlinear Hartree equation, J. Math. Phys., 43 (2002), pp. 3879-3891.
4. C. CAMBOURNAC, T. SYLVESTRE, H. MAILLOTTE, B. VANDERLINDEN, P. KOCKAERT, PH. EMPLIT, AND M. HAELTERMAN, Symmetry-breaking instability of multimode vector solitons, Phys. Rev. Lett., 89 (2002), p. 083901.
5. S. CUCCAGNA, D. PELINOVSKY, AND V. VOUGALTER, Spectra of positive and negative energies in the linearized NLS problem, Comm. Pure Appl, Math., 58 (2005), pp. 1-29.
6. B. DECONINCK AND J. NATHAN KUTZ, Singular instability of exact stationary solutions of the nonlocal Gross-Pitaevskii equation, Phys. Lett. A, 319 (2003), pp. 97-103.
7. J. FLEISCHER, G. BARTAL, O. COHEN, T. SCHWARTZ, O. MANELA, B. FREEDMAN, M. SEGEV, H. BULJAN, AND N. EFREMIDIS, Spatial photonics in nonlinear waveguide arrays, Optics Express, 13 (2005), pp. 1780-1796.
8. M. GOLUBITSKY AND D. G. SCHAEFFER, Singularities and Groups in Bifurcation Theory, Vol. 1-2, Appl. Math. Sci. 51, Springer, New York, 1985.
9. M. G. GRILLAKIS, Linearized instability for nonlinear Schrodinger and Klein-Gordon equations, Comm. Pure Appl. Math., 41 (1988), pp. 747-774.
10. M. G. GRILLAKIS. J. SHATAH, AND W. A. STRAUSS, Stability theory of solitary waves in the presence of symmetry I, J. Funct. Anal., 74 (1987), pp. 160-197.
11. J. GUCKENHEIMER AND P. HOLMES, Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields, Appl. Math. Sci. 42, Springer, New York, 1983.
12. E. M. HARRELL, Double wells, Commdie;. Math. Phys., 75 (1980), pp, 239-261.
13. T. HEIL, I. FISCHER, AND W. ELSÄSSER, Chaos synchronization and spontaneous symmetry-breaking in symmetrically delay-coupled semiconductor lasers, Phys. Rev. Lett., 86 (2001), pp. 795-798.
14. R. K. JACKSON AND M. I. WEINSTEIN, Geometric analysis of bifurcation and symmetry breaking in a Gross-Pitaevskii equation, J. Statist. Phys., 116 (2004), pp. 881-905.
15. C. K. R. T. JONES, An instability mechanism for radially symmetric standing waves of a nonlinear Schrödinger equation, J. Differential Equations, 71 (1988), pp. 34-62.
16. T. KAPITULA, Stability of waves in perturbed Hamiltonian systems, Phys. D, 156 (2001), pp. 186-200.
17. T. KAPITULA AND P. KEVREKIDIS, Bose-Einstein condensates in the presence of a magnetic trap and optical lattice: Two-mode approximation, Nonlinearity, 18 (2005), pp. 2491-2512.
18. T. KAPITULA, P. G. KEVREKIDIS, AND Z. CHEN, Three is a crowd: Solitary waves in photore fractive media with three potential wells, SIAM J. Appl. Dyn. Syst., 5 (2006), pp. 598-633.
19. T. KAPITULA, P. G. KEVREKIDIS, AND B. SANDSTEDE, Counting eigenvalues via the Krein signature in infinite-dimensional Hamiltonian systems, Phys. D, 195 (2004), pp. 263-282.
20. P. G. KEVREKIDIS, Z. CHEN, B. A. MALOMED, D. J. FRANTZESKAKIS, AND M. I. WEINSTEIN, Spontaneous symmetry breaking in photonic lattices: Theory and experiment, Phys. Lett. A, 340 (2005), pp. 275-280.
21. E. KIRR AND A. ZARNESCU, On the asymptotic stability of bound states in 2D cubic Schrödinger equation, Comm. Math. Phys., 272 (2007), pp. 443-468.
22. W. KROLIKOWSKI, O. BANG, N. I. NIKOLOV. D. NESHEV, J. WYLLER, J. J. RASMUSSEN, AND D. EDMUNDSON, Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media, J. Opt. B Quantum Semiclass. Opt., 6 (2004), pp. S288-S294.
23. K. W. MAHMUD, J. N. KUTZ, AND W. P. REINHARDT, Bose-Einstein condensates in a one dimensional double square well: Analytical solutions of nonlinear Schrödinger equation, Phys. Rev. A, 66 (2002), p. 063607.
24. L. NIRENBERG, Lectures on Nonlinear Functional Analysis, Courant Institute, New York, 1974.
25. C.-A. PILLET AND C. E. WAYNE, Invariant manifolds for a class of dispersive, Hamiltonian partial differential equations, J. Differential Equations, 141 (1997), pp. 310-326.
26. H. A. ROSE AND M. I. WEINSTEIN, On the bound states of the nonlinear Schrödinger equation with a linear potential, Phys. D, 30 (1988), pp. 207-218.
27. A. SACCHETTI, Nonlinear time-dependent one-dimensional Schrödinger equation with double- well potential, SIAM J. Math. Anal., 35 (2003), pp. 1160-1176.
28. S. SAWAI, Y. MAEDA, AND Y. SAWADA, Spontaneous symmetry breaking turing-type pattern formation in a confined dictyostelium cell mass, Phys. Rev. Lett., 85 2000), pp. 2212-2215.
29. A. SOFFER AND M. I. WEINSTEIN, Selection of the ground state for nonlinear Schröoinger equations, Rev. Math. Phys., 16 (2004), pp. 977-1071.
30. A. SOFFER AND M. I. WEINSTEIN, Theory of nonlinear dispersive waves and selection of the ground state, Phys. Rev. Lett., 95 (2005), p. 213905.
31. A. G. VANAKARAS, D. J. FOTINOS, AND E. T. SAMULSKI, Tilt, polarity, and spontaneous symmetry breaking in liquid crystals, Phys. Rev. E, 57 (1998), pp. R4875-R4878.
32. M. I. WEINSTEIN, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16 (1985), pp. 472-491.
33. M. I. WEINSTEIN, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math., 39 (1986), pp. 51-68.
34. C. YANNOULEAS AND U. LANDMAN, Spontaneous symmetry breaking in single and molecular quantum dots, Phys. Rev. Lett., 82 (1999), pp. 5325-5328.
35. G. ZHOU, Perturbation expansion and Nth order Fermi golden rule of the nonlinear Schrödinger equations with potential, J. Math. Phys.., 48 (2007), p. 053509.