×

Purely nonlinear instability of standing waves with minimal energy. (English) Zbl 1072.35165

Based on a general theory by M. G. Grillakis, J. Shatah and W. Strauss about the non-linear stability (with respect to the flow) of solitary waves in abstract evolutionary partial differential equations in presence of symmetries [J. Funct. Anal. 74, 160–197 (1987; Zbl 0656.35122); J. Funct. Anal. 94, 308–348 (1990; Zbl 0711.58013)], in the paper under review it is proved the generic nonlinear stability of the standing waves with minimal energy for Hamiltonian evolutionary partial differential equations admitting a \(U(1)\) symmetry. The authors apply their results to the nonlinear Schrödinger equation with \(U(1)\) symmetry.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Berestycki, Arch Rational Mech Anal 82 pp 313– (1983)
[2] Bona, Proc Roy Soc London Ser A 411 pp 395– (1987)
[3] Buslaev, Math Comput Simulation 56 pp 539– (2001)
[4] Buslaev, Algebra i Analiz 4 pp 63– (1992)
[5] St Petersburg Math J 4 pp 1111– (1993)
[6] On orbital stability of quasistationary solitary waves of minimal energy. Preprint, 2002.
[7] Cuccagna, Comm Pure Appl Math 54 pp 1110– (2001)
[8] Grillakis, Comm Pure Appl Math 41 pp 747– (1988)
[9] Grillakis, J Funct Anal 74 pp 160– (1987)
[10] Grillakis, J Funct Anal 94 pp 308– (1990)
[11] Perturbation theory for linear operators. 2nd ed. Grundlehren der Mathematischen Wissenschaften, 132. Springer, Berlin, 1976.
[12] Pelinovsky, Phys Rev E 53 pp 1940– (1996)
[13] ; Asymptotic methods in soliton stability theory. Nonlinear instability analysis, 245-312. Advances in Fluid Mechanics, 12. Computational Mechanics, Southampton, England, 1997.
[14] Pelinovsky, Phys D 116 pp 121– (1998)
[15] Perelman, Ann Henri Poincar? 2 pp 605– (2001)
[16] ; Methods of modern mathematical physics. IV. Analysis of operators. Academic, New York-London, 1978. · Zbl 0401.47001
[17] Shatah, Comm Math Phys 100 pp 173– (1985)
[18] ; Spectral condition for instability. Nonlinear PDE’s, dynamics and continuum physics (South Hadley, MA, 1998), 189-198. Contemporary Mathematics, 255. American Mathematical Society, Providence, R.I., 2000.
[19] Soffer, J Differential Equations 98 pp 376– (1992)
[20] Soffer, Invent Math 136 pp 9– (1999)
[21] Strauss, Comm Math Phys 55 pp 149– (1977)
[22] Nonlinear wave equations. CBMS Regional Conference Series in Mathematics, 73. Conference Board of the Mathematical Sciences, Washington, D.C.; American Mathematical Society, Providence, R.I., 1989.
[23] ; The nonlinear Schr?dinger equation. Self-focusing and wave collapse. Applied Mathematical Sciences, 139. Springer, New York, 1999.
[24] Partial differential equations. III. Nonlinear equations. Corrected reprint of the 1996 original. Applied Mathematical Sciences, 117. Springer, New York, 1997.
[25] Vakhitov, Radiophys Quantum Electron 16 pp 783– (1973)
[26] Weinstein, SIAM J Math Anal 16 pp 472– (1985)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.