Transverse Nonlinear Instability for Two-Dimensional Dispersive Models

Rousset, F.; Tzvetkov, N.

Annales de l'I.H.P. Analyse non linéaire, Tome 26 (2009) no. 2, pp. 477-496.

[1] C Alexander J., Pego R. L., Sachs R. L., On the Transverse Instability of Solitary Waves in the Kadomtsev-Petviashvili Equation, Phys. Lett. A 226 (1997) 187-192. | MR | Zbl

[2] Benjamin T., The Stability of Solitary Waves, Proc. London Math. Soc. (3) 328 (1972) 153-183. | MR

[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., Sachs R., Global Existence of Smooth Solutions and Stability of Solitary Waves for a Generalized Boussinesq Equation, Comm. Math. Phys. 118 (1988) 15-29. | MR | Zbl

[5] Burq N., Gérard P., Tzvetkov N., Two Singular Dynamics of the Nonlinear Schrödinger Equation on a Plane Domain, Geom. Funct. Anal. 13 (2003) 1-19. | MR | Zbl

[6] Cazenave T., Lions P. L., Orbital Stability of Standing Waves for Some Nonlinear Schrödinger Equations, Comm. Math. Phys. 85 (1982) 549-561. | MR | Zbl

[7] Coppel W. A., Dichotomies in Stability Theory, Lecture Notes in Mathematics, vol. 629, Springer-Verlag, Berlin, 1978. | MR | Zbl

[8] Friedlander S., Strauss W., Vishik M., Nonlinear Instability in an Ideal Fluid, Ann. Inst. H. Poincaré 14 (1997) 187-209. | Numdam | MR | Zbl

[9] Friedlander S., Vishik M., Nonlinear Instability in Two-Dimensional Ideal Fluids: the Case of a Dominant Eigenvalue, Comm. Math. Phys. 243 (2003) 261-273. | MR | Zbl

[10] Grenier E., On the Nonlinear Instability of Euler and Prandtl Equations, Comm. Pure Appl. Math. 53 (2000) 1067-1091. | MR | Zbl

[11] Guo Y., Strauss W. A., Instability of Periodic BGK Equilibria, Comm. Pure Appl. Math. 48 (1995) 861-894. | MR | Zbl

[12] Henry D., Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin, 1981. | MR | Zbl

[13] A. Ionescu, C. Kenig, Local and global well-posedness of periodic KP-I equations, Preprint, 2005. | MR

[14] Janssen P., Rasmussen J., Nonlinear Evolution of the Transverse Instability of Plane Envelope Solitons, Phys. Fluids 26 (1983) 1279-1287. | MR | Zbl

[15] Kadomtsev B. B., Petviashvili V. I., On the Stability of Solitary Waves in Weakly Dispersive Media, Soviet Phys. Dokl. 15 (1970) 539-541. | Zbl

[16] Kato T., Perturbation Theory for Linear Operators, Classics in Mathematics, Reprint of the 1980 edition, Springer-Verlag, Berlin, 1995. | MR | Zbl

[17] Kenig C., Ponce G., 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. | MR | Zbl

[18] Koch H., Tzvetkov N., On Finite Energy Solutions for the KP-I Equation, Math. Z. 256 (2008) 55-68. | MR

[19] Liu Y., Strong Instability of Solitary Wave Solutions to a Kadomtsev-Petviashvili Equation in Three Dimensions, J. Differential Equations (2002) 153-170. | MR | Zbl

[20] Merle F., Vega L., $L^2$ Stability of Solitons for KdV Equation, Int. Math. Res. Notices 13 (2003) 735-753. | MR | Zbl

[21] Pego R., Weinstein M., Eigenvalues, and Instabilities of Solitary Waves, Philos. Trans. Roy. Soc. London A 340 (1992) 47-97. | MR | Zbl

[22] Saut J.-C., Remarks on the Generalized Kadomtsev-Petviashvili Equations, Indiana Univ. Math. J. 42 (1993) 1011-1026. | MR | Zbl

[23] Takaoka H., Tzvetkov N., On 2D Nonlinear Schrödinger Equations With Data on $R\times T$, J. Funct. Anal. 182 (2001) 427-442. | MR | Zbl

[24] Titchmarch E. C., Eigenfunction Expansions Associated to Second Order Differential Equations, Clarendon Press, Oxford, 1946. | MR | Zbl

[25] Weinstein M., Modulational Stability of Ground States of Nonlinear Schrödinger Equations, SIAM J. Math. Anal. 16 (1985) 472-491. | MR | Zbl

[26] Zakharov V. E., Instability and Nonlinear Oscillations of Solitons, JETP Lett. 22 (1975) 172-173.