Strong instability of solitary waves for nonlinear Klein-Gordon equations and generalized Boussinesq equations

Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

Volumn 24, Issue 4, 2007, Pages 539-548

  Liu, Yue ^a   Ohta, Masahito ^b   Todorova, Grozdena ^c  

^a UNIVERSITY OF TEXAS AT ARLINGTON   (United States)

^b SAITAMA UNIVERSITY   (Japan)

^c UNIVERSITY OF TENNESSEE   (United States)

Author keywords

Blowup; Boussinesq equation; Nonlinear Klein Gordon equation; Solitary wave; Standing wave; Strong instability

Indexed keywords

FINITE DIFFERENCE TIME DOMAIN METHOD; NONLINEAR EQUATIONS; PROBLEM SOLVING; SYSTEM STABILITY; WAVE PROPAGATION;

BLOWUP; BOUSSINESQ EQUATION; NONLINEAR KLEIN-GORDON EQUATION; STANDING WAVES; STRONG INSTABILITY;

SOLITONS;

EID: 34250208422     PISSN: 02941449     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.anihpc.2006.03.005     Document Type: Article

References (30)

1. Berestycki H., and Cazenave T. Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires. C. R. Acad. Sci. Paris 293 (1981) 489-492
2. Berestycki H., Gallouët T., and Kavian O. Équations de champs scalaires euclidiens non linéaires dans le plan. C. R. Acad. Sci. Paris 297 (1983) 307-310
3. Berestycki H., and Lions P.L. Nonlinear scalar field equations. Arch. Rat. Mech. Anal. 82 (1983) 313-345
4. Bona J., and Sachs R. Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation. Comm. Math. Phys. 118 (1988) 15-29
5. Boussinesq J. Théorie des ondes et de remous qui se propagent.... J. Math. Pures Appl. 17 (1872) 55-108
6. Brezis H., and Lieb E. A relation between pointwise convergence of functions and convergence of functionals. Proc. Amer. Math. Soc. 88 (1983) 486-490
7. R.E. Caflisch, Shallow water waves, Lecture notes, New York University, New York
8. Cazenave T. Semilinear Schrödinger Equations. Courant Lecture Notes in Mathematics vol. 10 (2003), New York University, Courant Institute of Mathematical Sciences, American Mathematical Society, Providence, RI
9. Cazenave T., and Lions P.L. Orbital stability of standing waves for some nonlinear Schrödinger equations. Comm. Math. Phys. 85 (1982) 549-561
10. Fröhlich J., Lieb E.H., and Loss M. Stability of Coulomb systems with magnetic fields I. The one-electron atom. Comm. Math. Phys. 104 (1986) 251-270
11. Ginibre J., and Velo G. The global Cauchy problem for the non linear Klein-Gordon equation. Math. Z. 189 (1985) 487-505
12. Grillakis M., Shatah J., and Strauss W. Stability theory of solitary waves in the presence of symmetry, I. J. Funct. Anal. 74 (1987) 160-197
13. Grillakis M., Shatah J., and Strauss W. Stability theory of solitary waves in the presence of symmetry, II. J. Funct. Anal. 94 (1990) 308-348
14. n. Arch. Rational Mech. Anal. 105 (1989) 234-266
15. t t = - A u + F (u). Trans. Amer. Math. Soc. 192 (1974) 1-21
16. Lieb E. On the lowest eigenvalue of the Laplacian for the intersection of two domains. Invent. Math. 74 (1983) 441-448
17. Liu Y. Instability of solitary waves for generalized Boussinesq equations. J. Dynam. Differential Equations 5 (1993) 537-558
18. Liu Y. Instability and blow-up of solutions to a generalized Boussinesq equation. SIAM J. Math. Anal. 26 (1995) 1527-1546
19. Liu Y. Strong instability of solitary-wave solutions of a generalized Boussinesq equation. J. Differential Equations 164 (2000) 223-239
20. Liu Y. Blow up and instability of solitary-wave solutions to a generalized Kadomtsev-Petviashvili equation. Trans. Amer. Math. Soc. 353 (2000) 191-208
21. Liu Y. Strong instability of solitary-wave solutions to a Kadomtsev-Petviashvili equation in three dimensions. J. Differential Equations 180 (2002) 153-170
22. Ohta M., and Todorova G. Strong instability of standing waves for nonlinear Klein-Gordon equations. Discrete Contin. Dynam. Syst. 12 (2005) 315-322
23. M. Ohta, G. Todorova, Strong instability of standing waves for nonlinear Klein-Gordon equation and Klein-Gordon-Zakharov system, Preprint
24. Payne L.E., and Sattinger D.H. Saddle points and instability of nonlinear hyperbolic equations. Israel J. Math. 22 (1975) 273-303
25. Shatah J. Stable standing waves of nonlinear Klein-Gordon equations. Comm. Math. Phys. 91 (1983) 313-327
26. Shatah J. Unstable ground state of nonlinear Klein-Gordon equations. Trans. Amer. Math. Soc. 290 (1985) 701-710
27. Shatah J., and Strauss W. Instability of nonlinear bound states. Comm. Math. Phys. 100 (1985) 173-190
28. Strauss W. Existence of solitary waves in higher dimensions. Comm. Math. Phys. 55 (1977) 149-162
29. Weinstein M.I. Nonlinear Schrödinger equations and sharp interpolation estimates. Comm. Math. Phys. 87 (1983) 567-576
30. Weinstein M.I. Lyapunov stability of ground states of nonlinear dispersive evolution equations. Comm. Pure Appl. Math. 39 (1986) 51-68