Transverse nonlinear instability for two-dimensional dispersive models

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

Volumn 26, Issue 2, 2009, Pages 477-496

  Rousset, F ^a   Tzvetkov, N ^b  

^a UNIVERSITÉ CÔTE D'AZUR   (France)

^b UNIV LILLE   (France)

Author keywords

Dispersive equations; Nonlinear instability; Solitary waves

Indexed keywords

LIQUID WAVES; STABILITY; WAVES;

DINGER EQUATIONS; DISPERSIVE EQUATIONS; DISPERSIVE MODELS; NONLINEAR INSTABILITY; SOLITARY WAVES; TRANSVERSE NONLINEAR; TRANSVERSE PERTURBATIONS; TWO-DIMENSIONAL;

NONLINEAR EQUATIONS;

EID: 61549142438     PISSN: 02941449     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.anihpc.2007.09.006     Document Type: Article

References (26)

1. C Alexander J., Pego R.L., and Sachs R.L. On the transverse instability of solitary waves in the Kadomtsev-Petviashvili equation. Phys. Lett. A 226 (1997) 187-192
2. Benjamin T. The stability of solitary waves. Proc. London Math. Soc. (3) 328 (1972) 153-183
3. K. Blyuss, T. Bridges, G. Derks, Transverse instability and its long-term development for solitary waves of the (2 + 1)-Boussinesq equation, Preprint, 2002
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. Burq N., Gérard P., and Tzvetkov N. Two singular dynamics of the nonlinear Schrödinger equation on a plane domain. Geom. Funct. Anal. 13 (2003) 1-19
6. Cazenave T., and Lions P.L. Orbital stability of standing waves for some nonlinear Schrödinger equations. Comm. Math. Phys. 85 (1982) 549-561
7. Coppel W.A. Dichotomies in Stability Theory. Lecture Notes in Mathematics vol. 629 (1978), Springer-Verlag, Berlin
8. Friedlander S., Strauss W., and Vishik M. Nonlinear instability in an ideal fluid. Ann. Inst. H. Poincaré 14 (1997) 187-209
9. Friedlander S., and Vishik M. Nonlinear instability in two-dimensional ideal fluids: The case of a dominant eigenvalue. Comm. Math. Phys. 243 (2003) 261-273
10. Grenier E. On the nonlinear instability of Euler and Prandtl equations. Comm. Pure Appl. Math. 53 (2000) 1067-1091
11. Guo Y., and Strauss W.A. Instability of periodic BGK equilibria. Comm. Pure Appl. Math. 48 (1995) 861-894
12. Henry D. Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics vol. 840 (1981), Springer-Verlag, Berlin
13. A. Ionescu, C. Kenig, Local and global well-posedness of periodic KP-I equations, Preprint, 2005
14. Janssen P., and Rasmussen J. Nonlinear evolution of the transverse instability of plane envelope solitons. Phys. Fluids 26 (1983) 1279-1287
15. Kadomtsev B.B., and Petviashvili V.I. On the stability of solitary waves in weakly dispersive media. Soviet Phys. Dokl. 15 (1970) 539-541
16. Kato T. Perturbation Theory for Linear Operators. Classics in Mathematics. Reprint of the 1980 edition (1995), Springer-Verlag, Berlin
17. Kenig C., Ponce G., and Vega L. Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle. Comm. Pure Appl. Math. 46 (1993) 527-629
18. Koch H., and Tzvetkov N. On finite energy solutions for the KP-I equation. Math. Z. 256 (2008) 55-68
19. Liu Y. Strong instability of solitary wave solutions to a Kadomtsev-Petviashvili equation in three dimensions. J. Differential Equations (2002) 153-170
20. 2 stability of solitons for KdV equation. Int. Math. Res. Notices 13 (2003) 735-753
21. Pego R., and Weinstein M. Eigenvalues, and instabilities of solitary waves. Philos. Trans. Roy. Soc. London A 340 (1992) 47-97
22. Saut J.-C. Remarks on the generalized Kadomtsev-Petviashvili equations. Indiana Univ. Math. J. 42 (1993) 1011-1026
23. Takaoka H., and Tzvetkov N. On 2D nonlinear Schrödinger equations with data on R × T. J. Funct. Anal. 182 (2001) 427-442
24. Titchmarch E.C. Eigenfunction Expansions Associated to Second Order Differential Equations (1946), Clarendon Press, Oxford
25. Weinstein M. Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Math. Anal. 16 (1985) 472-491
26. Zakharov V.E. Instability and nonlinear oscillations of solitons. JETP Lett. 22 (1975) 172-173