Linearized instability for nonlinear Schrödinger and Klein-Gordon equations. (English) Zbl 0632.70015

In this paper I am introducing a new technique for proving instability of bound states for Hamilton systems. There are already two disparate types of instability results in the literature. The approach developed by Strauss-Shatah gave an instability criterion coming from the variational structure of the problem; on the other hand Jones’ approach produced a complementary criterion related to the difference between the number of negative eigenvalues of two selfadjoint operators using quite different techniques. It turns out that with the methods developed in this paper these two criteria can be derived within a single framework that also leads to a generalization of the previous results. Finally in order to demonstrate how this method works I apply the instability criterion in some specific examples.


70H99 Hamiltonian and Lagrangian mechanics
Full Text: DOI


[1] Saddle point analysis for an ordinary differential equation, and an application to dynamical buckling of a beam, Proc. Symp. Nonlinear Elasticity (ed. ) Acad. Press. 1973, pp. 93–160.
[2] and , Invariant manifolds for semilinear partial differential equations, to appear. · Zbl 0674.58024
[3] Berestycki, C. R. Acad. Sci. 293 pp 489– (1981)
[4] and , Theorie de Point Critique et Instabilité des Ondes Stationaires pour des Equations de Schrödinger Nonlineaire, C.R. Acad. Sci., 1985,.
[5] Berestycki, Arch. Rat. Mech. Anal. 82 pp 313– (1983)
[6] Cazenave, Comm. Math. Phys. 85 pp 549– (1982)
[7] Coffman, Arch. Rat. Mech. Anal. 46 pp 81– (1981)
[8] and , Non-spreading wave packets for the cubic Schrödinger equation with a bounded potential, to appear.
[9] Friedberg, Physical Review D. 13 pp 2739– (1976)
[10] Grillakis, J. Fun. Anal. 74 pp 160– (1987)
[11] Instability of standing waves for nonlinear Schrödinger equations, to appear. · Zbl 0636.35017
[12] Jones, SIAM J. Math. Anal. 17 pp 803– (1986)
[13] Keller, J. Diff. Eqns. 50 pp 330– (1983)
[14] Maddocks, SIAM J. Math. Anal. 16 pp 47– (1985)
[15] Makhankov, Phys. Reports 35 pp 1– (1978)
[16] Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. · Zbl 0516.47023
[17] and , Method of Modern Mathematical Physics, Vol, I, II, III, IV, Academic Press, 1979.
[18] Shatah, Trans. AMS 290 pp 701– (1985)
[19] Shatah, Comm. Math. Phys. 91 pp 313– (1983)
[20] and , Instability of bound states for nonlinear Klein-Gordon and Schrödinger equations, preprint.
[21] Sternberg, Trans. AMS 296 pp 315–
[22] Strauss, Anais Acad. Brasil Cienc. 42 pp 645– (1970)
[23] Strauss, Comm. Math. Phys. 55 pp 149– (1977)
[24] Strauss, Contemp. Math. 17 pp 429– (1983) · Zbl 0588.35027
[25] Weinstein, Com. Pure Appl. Math. 39 pp 51– (1986)
[26] Weinstein, SIAM J. Appl. Math. 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.