On the variational approach to the stability of standing waves for the nonlinear Schrödinger equation

Advanced Nonlinear Studies

Volumn 4, Issue 4, 2004, Pages 469-501

  Hajaiej, H ^a   Stuart, C A ^b  

^a University of Virginia   (United States)

^b EPFL   (Switzerland)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 10944273602     PISSN: 15361365     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (24)

1. T.B. Benjamin, The stability of solitary waves, Proc. R. Soc. Lond. A 328 (1972), 153-183
2. J. Boussinesq, Essai sur la théorie des eaux courantes, Mém. prés. div. Sav. Acad. Sci. Inst. Fr., 23,(1877)
3. J.C. Bronski, L.D. Carr, B. Deconinck, and J.N. Kutz, Bose-Einstein condensates in standing waves, Phys. Rev. Lettr.86 (2001), 1402-1405.
4. ème cycle, Univ. P. et M. Curie, Paris, 1978.
5. T. Cazenave, An introduction to nonlinear Schrödinger equations, Textos de Métodos Matematicos, Rio de Janeiro, 1996, third edition.
6. T. Cazenave, P.L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys. 85, p 549-561 (1982).
7. M. Colin, Stability of stationary waves for a quasilinear Schrödinger equation in space dimension 2, Adv. Diff. Equat, 8 (2003), 1-28
8. A. Comech and D. Pelinovsky, Purely nonlinear instability of standing waves with minimal energy, Comm. Pure Appl. Math., LVI (2003), 1565-1607
9. G. Fibich, and X.-P. Wang, Stability of solitary waves for nonlinear Schrödinger equations with inhomogeneous nonlinearities, Physica D, 175 (2003), 96-108.
10. M. Grillakis, J. Shatah, and W. Strauss, Stability theory of solitary waves in the presence of symmetry, I., J. Functional Anal. 74, (1987), 160-197.
11. H. Hajaiej, C.A. Stuart, Existence and non-existence of Schwarz symmetric ground states for elliptic eigenvalue problems, to appear in Ann. Mat. Pura Appl.
12. H.P. Heinz, T. Küpper, and C.A. Stuart, Existence and bifurcation of solutions for nonlinear perturbations of the periodic Schrödinger equation, J. Diff. Equat. 100 (1992), 341-354.
13. I.D. Iliev and K.P. Kirchev, Stability and instability of solitary waves for one-dimensional singular Schrödinger equations, J. Diff. Int. Equats, 6, (1993), 685-703
14. O. Kavian, Introduction à la théorie des points critiques, Mathématiques et ses Applications, Springer-Verlag France, Paris 1993.
15. Y.S. Kivshar, and A.A. Sukhorukov, Stability of spatial optical solitons, in Spatial Optical Solitons, editors T. Torruellas and S. Trillo, Springer, New York, 2001.
16. P.L. Lions, The concentration - compactness principle in the calculus of variations. The locally compact case: Part 1, p 109-145, Part 2, p 223-283, Ann. Inst. H. Poincaré, Vol 1, n°4, 1984.
17. J.B. McLeod, C.A. Stuart, W.C. Troy, Stability of standing waves for some nonlinear Schrödinger equations, J. Diff. Int. Equats., 16 (2003), 1025-1035.
18. H.A. Rose, and M.I. Weinstein, On the bound states of the nonlinear Schrödinger equation with a linear potential, Physica D 30 (1988), 207-218.
19. C.A. Stuart, Bifurcation for Dirichlet problems without eigenvalues, Proc. London Math, Soc., 45 (1982), 169-192.
20. C.A. Stuart, Bifurcation into spectral gaps, Bull. Belgian Math. Soc., supplement 1995.
21. A.A. Sukhorukov, and Y.S. Kivshar, Nonlinear guided waves and spatial solitons in a periodic layered medium, J. Opt. Soc. Am. B 19 (2002), 772-781.
22. M.G. Vakhitov and A.A. Kolokolov, Stationary solutions of the wave equation in the medium with nonlinearity saturation, Radiophys. Quantum Electron., 16 (1973), 783
23. M.I. Weinstein, Lyapunov stability of ground states for nonlinear dispersive evolution equations, Comm. Pure Appl. Math., 39 (1986), 51-68
24. P.E. Zhidkov, Korteweg-de Vries and Nonlinear Schrödinger Equations: Qualitative Theory, Lecture Notes in Mathematics No 1756, Springer-Verlag, Berlin 2001