Nonlinearity-defect interaction: symmetry breaking bifurcation in a NLS with a 𝛿′ impurity
Abstract
We illustrate some new results and comment on perspectives of a recent research line, focused on the stability of stationary states of nonlinear NLS with point interactions. We describe in detail the case of a “𝛿′” interaction, that provides a rich model endowed with a pitchfork bifurcation with symmetry breaking in the family of ground states. Finally, we give a direct proof of the stability of the ground states in the cases of a subcritical and critical (in the sense of the blow-up) nonlinearity power.
About the Authors
R. Adami
Universit`a di Milano Bicocca
Italy
Italy
Riccardo Adami – Dipartimento di Matematica e Applicazioni, Assistant Professor
D. Noja
Universit`a di Milano Bicocca
Italy
Italy
Diego Noja – Dipartimento di Matematica e Applicazioni, Assistant Professor
References
1. Adami R., Bardos C., Golse. F, Teta A. Towards a rigorous derivation of the cubic NLSE in dimension one // Asymp. An. — 2004. — V. 40(2). — P. 93-108.
2. Adami R., Golse, F, Teta A.: Rigorous derivation of the cubic NLS in dimension one // J. Stat. Phys. — 2007. — V. 127. — P. 1193-1220.
3. Adami R., Noja D. Existence of dynamics for a 1-d NLS equation in dimension one // J. Phys. A Math. Theor. — 2009. — V. 42. — 495302, 19 p.
4. Adami R., Cacciapuoti C., Finco D., Noja D. Fast Solitons on Star Graphs // Rev. Math. Phys. — 2011. — V. 23(4).
5. Adami R., Cacciapuoti C., Finco D., Noja D. Stationary states of NLS on star graphs. — arXiv: 1104.3839, 2011.
6. Adami R., Noja D. Stability, instability and symmetry breaking bifurcation for the ground states of a NLS with a 𝛿′ interaction // arXiv.1112.1318, 2011.
7. Adami R., Noja D., Sacchetti A. On the mathematical description of the effective behaviour of one-dimensional Bose-Einstein condensates with defects // In: Bose-Einstein Condensates: Theory, Characteristics, and Current Research. — New York: Nova Publishing, 2010.
8. Adami R., Sacchetti A. The transition from diffusion to blow-up for a NLS equation in dimension one // J. Phys. A Math. Gen. — 2005. — V. 38. — P. 8379-8392.
9. Albeverio S., Brze´zniak Z., Dabrowski L. Fundamental solutions of the Heat and Schr¨odinger Equations with point interaction // Journal of Functional Analysis. — 1995. — V. 130. — P. 220-254.
10. Albeverio S., Gesztesy F., Hoegh-Krohn R., Holden H. Solvable Models in Quantum Mechanics: Second Edition, With an Appendix by Pavel Exner. — AMS Chelsea Publishing, Providence, 2005.
11. Albeverio S., Kurasov P. Singular Perturbations of Differential Operators. — Cambridge University Press, 2000.
12. Ammari Z., Breteaux S. Propagation of chaos for many-boson systems in one dimension with a point pairinteraction. — arXiv: 0906.3047, 2009.
13. Brazhnyi V., Konotop V., Perez-Garc´ıa V. Driving defect modes of Bose Einstein condensates in optical lattices // Phys. Rev.Lett. — 2006. — 96. 060403.
14. Brazhnyi V., Konotop V., Perez-Garc´ıa V. Defect modes of a Bose Einstein condensate in an optical lattice with a localized impurity // Phys. Rev.A. — 2006. — V. 74. — 023614.
15. Brezis H., Lieb E.H. A relation between pointwise convergence of functions and convergence of functionals // Proc. Amer. Math. Soc. — 1983. — V. 88. — P. 486–490.
16. Burioni R., Cassi D., Sodano P., Trombettoni A., Vezzani A. Soliton propagation on chains with simple nonlocal defects // Physica D –. 2006. — V. 216. — P. 71–76.
17. Cazenave T. Semilinear Schr¨odinger Equations. — Courant Lecture Notes, 2003.
18. Cheon T., Shigehara T. Realizing discontinuous wave functions with renormalized short-range potentials // Phys. Lett. A. — 1998. — V. 243(22). — P. 111–116.
19. Datchev K., Holmer J. Fast soliton scattering by attractive delta impurities // Comm. Part. Diff. Eq. — 2009. — V. 34. — P. 1074-1113.
20. Erd˝os L., Schlein B., Yau H-T. Derivation of the Gross-Pitaevskii hierarchy for the dynamics of the Bose Einstein condensate // Comm. Pure Appl. Math. — 2006. — V. 59. — P. 1659–1741.
21. Erd˝os L., Schlein B., Yau H-T. Derivation of the cubic nonlinear Schr¨odinger equation from quantum dynamics of many body systems // Invent. Math. — 2007. — V. 167. — P. 515–614.
22. Erd˝os L., Schlein B., Yau H-T. Derivation of the Gross-Pitaevskii equation for the dynamics of the Bose Einstein condensate. // Ann. Math. — 2010. — V. 172(1). — P. 291-370.
23. Exner P., Grosse P. Some properties of the one-dimensional generalized point interactions (a torso). — mp-arc 99-390, math-ph/9910029, 1999.
24. Exner P. Neidhart H. Zagrebnov V.A. Potential approximations to 𝛿′: an inverse Klauder phenomenon with norm-resolvent convergence // Comm. Math. Phys. — 2001. — V. 224. — P. 593–612.
25. Fibich G., Wang X. P. Stability for solitary waves for nonlinear Schr¨odinger equations with inhomogenous nonlinearities // Physica D. — 2003. — V. 175. — P. 96-108.
26. Fukuizumi R., Jeanjean L.: Stability of standing waves for a nonlinear Schr¨odinger equation with a repulsive Dirac delta potential // Disc. Cont. Dyn. Syst. (A). — 2008. — 21. — P. 129-144.
27. Fukuizumi R., Ohta M, Ozawa T. Nonlinear Schr¨odinger equation with a point defect // Ann. I.H.Poincar´e, AN. — 2008. — 25. — P. 837-845.
28. Fukuizumi R., Sacchetti A. Bifurcation and stability for Nonlinear Schroedinger equations with double well potential in the semiclassical limit, preprint arXiv:1104.1511 (2011).
29. Goodman R. H., Holmes P. J., Weinstein M. I. Strong NLS soliton-defect interactions // Physica D. — 2004. — 192. — P. 215-248.
30. Grillakis M. Shatah J. Strauss W. Stability theory of solitary waves in the presence of symmetry I. // J. Func. An. — 1987. — V. 74. — P. 160-197.
31. Grillakis M. Shatah J. Strauss W. Stability theory of solitary waves in the presence of symmetry II // J. Func. An. — 1990. — V. 94. — P. 308-348.
32. Holmer J., Marzuola J., Zworski M. Fast soliton scattering by delta impurities // Comm. Math. Phys. — 2007. — V. 274. — P. 187-216.
33. Jackson R.K, Weinstein M.I. Geometric analysis of bifurcation and symmetry breaking in a Gross-Pitaevskii equation // J. Stat. Phys. — 2004. — V. 116. — P. 881-905.
34. Kirkpatrick K., Staffilani G., Schlein B. Derivation of the two-dimensional nonlinear Schr¨odinger equation from many body quantum dynamics // Am. J. of Math. — 2011. — V. 133(1). — P. 91-130.
35. Kirr E.W., Kevrekidis P.G., Pelinovsky D. E. Symmetry-breaking bifurcation in the nonlinear Schrodinger equation with symmetric potentials. — arXiv:1012.3921, 2010.
36. Kirr E.W., Kevrekidis R.G., Shlizerman E., Weinstein M.I. Symmetry-breaking bifurcation in nonlinear Schr¨odinger/Gross–Pitaevskii equations // SIAM J. Math. Anal. — 2008. — V. 40. — P. 566-604.
37. Kostrykin V., Schrader R. Kirchhoff’s rule for quantum wires // J. Phys. A: Math. Gen. — 1999. — V. 32(4). — P. 595-630.
38. Kuchment P. Quantum graphs. I. Some basic structures // Waves Random Media. — 2004. — V. 14(1). — S107–S128.
39. Kuchment P.: Quantum graphs. II. Some spectral properties of quantum and combinatorial graphs // J. Phys. A: Math. Gen. — 2005. — V. 38(22). — P. 4887-4900.
40. Le Coz S., Fukuizumi ., Fibich G., Ksherim B., Sivan Y.: Instability of bound states of a nonlinear Schr¨odinger equation with a Dirac potential // Phys. D. — 2008. — V. 237(8). — P. 1103–1128.
41. Lions, P.L. The concentration-compactness principle in the calculus of variations. The locally compact case. I // Ann. Inst. H. Poincar´e Anal. Non Lin´eaire. — 1984. — 1. — P. 109-145.
42. Lions, P.L. The concentration-compactness principle in the calculus of variations. The locally compact case. II // Ann. Inst. H. Poincar´e Anal. Non Lin´eaire. — 1984. — V. 1. — P. 223-283.
43. Pelinovsky D.E., Phan T. Normal form for the symmetry-breaking bifurcation in the nonlinear Schr¨odinger equation. — arXiv: 1101.5402v1, 2011.
44. Perelman G., A remark on soliton-potential interaction for nonlinear Schr¨odinger equations // Math. Res. Lett. — 2009. — V. 16(3). — P. 477-486.
45. Pitaevskii L., Stringari S. Bose-Einstein condensation. — Oxford University Press, 2003.
46. Sacchetti A. Universal Critical Power for Nonlinear Schr¨odinger Equations with a Symmetric Double Well Potential // Phys. Rev. Lett. — 2009. — V. 103. — 194101.
47. M. G. Vakhitov, A. A. Kolokolov: Stationary solutions of the wave equation in a medium with nonlinearity saturation // Radiophys. Quantum Electron. — 1973. — V. 16. — P. 783–789.
48. Weinstein M.: Nonlinear Schr¨odinger equations and sharp interpolation estimates // Comm. Math. Phys. — 1983. — V. 87. — P. 567-576.
49. Weinstein M. Modulational stability of ground states of nonlinear Schr¨odinger equations // SIAM J. Math. Anal. — 1985. — V. 16. — P. 472-491.
50. Weinstein M.: Lyapunov stability of ground states of nonlinear dispersive evolution equations // Comm. Pure Appl. Math. — 1986. — V. 39. — P. 51–68.
Review
For citations:
Adami R., Noja D. Nonlinearity-defect interaction: symmetry breaking bifurcation in a NLS with a 𝛿′ impurity. Nanosystems: Physics, Chemistry, Mathematics. 2011;2(4):5-19.
Views:
9
Email this article 