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Nonlinear wave and Schrödinger equations. I: Instability of periodic and quasiperiodic solutions. (English) Zbl 0780.35106

Stability of periodic and quasiperiodic solutions of linear wave and Schrödinger equations under nonlinear perturbations is investigated. Using the Mourre estimate and the Fermi Golden rule for non-elliptic and nonlinear equations the author shows that for the wave equation such solutions are unstable for generic perturbations and gives spectral conditions which guarantee instability. In this case of the Schrödinger equation periodic solutions are stable while quasiperiodic ones are not.
These results are extended to periodic solutions of nonlinear equations. Some illustrative examples are considered.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35K55 Nonlinear parabolic equations
35B35 Stability in context of PDEs
35B10 Periodic solutions to PDEs
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[1] [AF] Albanese, C., Fröhlich, J.: Periodic solutions ... I. Math. Phys.116, 475–502 (1988) · Zbl 0696.35185
[2] [AHS] Agmon, S., Herbst, I., Skibsted, E.: Perturbation of embedded eigenvalues in generalizedN-body problem. Commun. Math. Phys.122, 411–438 (1989) · Zbl 0668.35078
[3] [CFKS] Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger operators. Berlin, Heidelberg, New York: Springer-Verlag, 1987
[4] [FH1] Froese, R., Herbst, I.: A new proof of the Mourre estimate. Duke Math. J.49, 1075–1985 (1982) · Zbl 0514.35025
[5] [FH2] Froese, R., Herbst, I.: Exponential bounds and absence of positive eigenvalues forN-body Schrödinger operators. Commun. Math. Phys.87, 429–447 (1982) · Zbl 0509.35061
[6] [FHi] Froese, R., Hislop, P.: Spectral analysis of second order elliptic operators on noncompact manifolds. Duke Math. J.58, 103–128 (1989) · Zbl 0687.35060
[7] [FS] Froese, R., Sigal, I.M.: Lectures on scattering theory. Preprint, Toronto, 1987.
[8] [How] Howland, J.S.: Perturbation of embedded eigenvalues. Bull. A.M.S.78, 280–283 (1972) · Zbl 0259.47013
[9] [HoS] Horwitz, L.P., Sigal, I.M.: On a mathematical model for non-stationary physical systems. Helvetica Physica Acta.51, 685–715 (1978)
[10] [Kato] Kato, T.: Perturbation theory for linear operators. Berlin, Heidelberg, New York: Springer 1966 · Zbl 0148.12601
[11] [Mo] Mourre, E.: Absence of singular continuous spectrum for certain self-adjoint operators. Commun. Math. Phys.78, 391–408 (1981) · Zbl 0489.47010
[12] [PSS] Perry, P., Sigal, I.M., Simon, B.: Spectral analysis ofN-body Schrödinger operators. Ann. Math.114, 519–567 (1981) · Zbl 0477.35069
[13] [Sig] Sigal, I.M.: Scattering theory for many-body quantum mechanical systems. Springer Lect. Notes in Math., No. 1011, 1983 · Zbl 0522.47006
[14] [Sim] Simon, B.: Resonances inn-body quantum systems with dilation analytic potentials and the foundations of time-dependent perturbation theory. Ann. Math.97, 247–274 (1973) · Zbl 0252.47009
[15] [Yaj] Yajima, K.: Resonances for the AC Stark effec. Commun. Math. Phys.87, 331–352 (1982) · Zbl 0538.47010
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