Lectures on the orbital stability of standing waves and application to the nonlinear Schrödinger equation

Milan Journal of Mathematics

Volumn 76, Issue 1, 2008, Pages 329-399

  Stuart, C A ^a  

^a EPFL   (Switzerland)

Author keywords

Hamiltonian system; Nonlinear Schr dinger equation; Orbital stability

Indexed keywords

EID: 58149173394     PISSN: 14249286     EISSN: 14249294     Source Type: Journal    
DOI: 10.1007/s00032-008-0089-9     Document Type: Article

References (36)

1. A. Ambrosetti and G. Prodi. A Primer of Nonlinear Analysis, Cambridge University Press, Cambridge, 1993
2. T. B. Benjamin, The stability theory of solitary waves, Proc. Royal Soc. London, 328 (1972), 153-183
3. J. Bona, On the stability theory of solitary waves, Proc. Royal Soc. London, 344 (1975), 363-374
4. T. Cazenave, Semilinear Schrödinger equations, Courant Lecture notes in Mathematics, AMS 2003
5. T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561
6. A. Comech and D. Pelinovsky, Purely nonlinear instability of standing waves with minimal energy, Comm. Pure Appl. Math., LVI (2003), 1565-1607
7. M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Anal., 8 (1971), 321-340
8. S. Cuccagna, Stability of standing waves for NLS with perturbed Lamé potential, J. Differntial Equations, 223 (2006), 112-160
9. A. de Bouard, and R. Fukuizumi, Stability of standing waves for nonlinear Schrödinger equations with inhomogeneous nonlinearities, Ann. Henri Poincaré 6 (2005), 1157-1177.
10. N, J. Diff. Equat., 124 (1996), 279-301
11. M. Esteban and W. Strauss, Nonlinear bound states outside an insulated sphere, Comm. Partial Differential Equations, 19 (1994), 177-197
12. R. Fukuizumi, Stability and instability of standing waves for nonlinear Schrödinger equations, Tohoku Mathematical Publications, No 25, Tohoku University, Japan, 2003
13. F. Genoud and C. A. Stuart, Schrödinger equation with a spatially decaying nonlinearity, Existence and stability of standing waves, Discrete & Continuous Dynamical Systems, 21 2008, 137-186
14. M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry, I, J. Functional Anal., 74 (1987), 160-190
15. M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry, II, J. Functional Anal., 90 1990, 308-348
16. M. Grillakis, M. Linearized instability for nonlinear Schrödinger and KLein-Gordon equations, Comm. Pure Appl. Math., XLI (1988), 747-774
17. M. Grillakis, Analysis of the linearization around a critical point of an infinite dimensional dynamical system, Comm. Pure Appl. Math., XLIII (1990), 299-333
18. H. Hajaiej and C. A. Stuart, On the variational approach to the stability of standing waves for the nonlinear Schrödinger equation, Adv. Nonlinear Studies, 4 (2004), 469-501
19. H. Jeanjean and C. A. Stuart, Nonlinear eigenvalue problems having an unbounded branch of symmetric bound states, Adv. Differential Equations, 4 (1999), 639-670
20. L. Jeanjean and S. Le Coz, An existence and stability result for standing waves of nonlinear Schrödinger equations, Adv. Differential Equations, 11 (2006), 813-840.
21. T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin 1967
22. Y. S. Kivshar and A. A. Sukkhorukov, Stability of spatial optical solitons, in Spatial Optical Solitons, editors T. Torruellas and S. Trillo, Springer, New York 2001
23. J. H. Maddocks and R. L. Sachs, On the stability of KdV multi-solitons, Comm. Pure Appl. Math., XLVI (1993), 867-901
24. J. B. McLeod, C. A. Stuart and W. C. Troy, Stability of standing waves for some nonlinear Schrödinger equations J. Differential Integral Equations, 16 (2003), 1025-1038
25. T. Mizumachi, A remark on linearly unstable standing wave solutions to NLS, Nonlinear Anal. TMA, 64 (2006), 657-676
26. D. E. Pelinovsky, A. A. Sukhorukov and Y. S. Kivshar, Bifurcations and stability of gap solitons in periodic potentials, Phys. Rev. E, 70 (2004),
27. 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
28. A. Sofer and M. I. Weinstein, Selection of ground states for nonlinear Schrödinger equations, Rev. Math. Phys., 16 (2004), 977-1071
29. N, in Nonlinear Functional Analysis and Applications to Differential Equations, editors A. Ambrosetti, K.-C. Chang and I. Ekeland, World Scientific, Singapore, 1998
30. C. A. Stuart, Uniqueness and stability of ground states for some nonlinear Schrödinger equations, J. Eur. Math. Soc., 8 (2006), 399-414
31. C. A. Stuart, Existence and stability of TE modes in a stratified non-linear dielectric, IMA J. of Applied Math. , 72 (2007), 659-679
32. M. G. Vakhitov and A. A. Kolokolov, Stationary solutions of the wave equation in a medium with nonlinearity saturation, Radiophys. Quantum Electron., 16 (1973), 783-785
33. T.-P. Tsai and H.-T. Yau, Asymptotic dynamics of nonlinear Schrödinger equations: Resonance dominated and radiation dominated solutions, Comm. Pure Appl. Math., LV 2002, 153-216
34. M. I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math., XXXIX (1986), 51-68
35. V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Soviet Physics JETP, 34 (1972), 62-69
36. P. E. Zhidkov, Korteweg de Vries and Nonlinear Schrödinger Equations: Qualitative Theory, Lecture Notes in Mathematics No 1756, Springer-Verlag, Berlin, 2001