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Instability of standing waves for non-linear Schrödinger-type equations

Published online by Cambridge University Press:  10 December 2009

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Abstract

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A theorem is proved giving a condition under which certain standing wave solutions of non-linear Schrödinger-type equations are linearly unstable. The eigenvalue equations for the linearized operator at the standing wave can be analysed by dynamical systems methods. A positive eigenvalue is then shown to exist by means of a shooting argument in the space of Lagrangian planes. The theorem is applied to a situation arising in optical waveguides.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 8 , Volume 8: Charles Conley Memorial Issue , December 1988 , pp. 119 - 138
Copyright
Copyright © Cambridge University Press 1988

References

REFERENCES

Akhmediev, N. N.. Novel class of nonlinear surface waves: asymmetric modes in a symmetric layered structure. Sov. Phys. JETP 56 (1982).Google Scholar
Arnol'd, V. I.. Characteristic class entering in quantization conditions. Funct. Anal. Appl. 1 (1967), 1–14.Google Scholar
Arnol'd, V. I.. The Sturm theorem and symplectic geometry. Funct. Anal. Appl. 19 (1985), 251–259.10.1007/BF01077289Google Scholar
Berestycki, H. & Lions, P.-L.. Théorie des points critiques et instabilité des ordres stationnaires pour des équations de Schrödinger non linéaires. C. R. Acad. Sci., Paris 300, série 1, # 10 (1985), 319–322.Google Scholar
Bishop, R. L. & Crittenden, R. J.. Geometry of Manifolds. Academic Press, New York (1964).Google Scholar
Bott, R.. On the iterations of closed geodesics and the Sturm intersection theory. Commun. Pure Appl. Math. 9 (1956), 171–206.Google Scholar
Cazenave, T. & Lions, P.-L.. Orbital stability of standing waves for some nonlinear Schrödinger equations. Commun. Math. Phys. 85 (1982), 549–561.Google Scholar
Conley, C.. An oscillation theorem for linear systems with more than one degree of freedom. IBM Technical Report # 18004. IBM Watson Research Center, Yorktown Heights, New York (1972).Google Scholar
Conley, C.. Isolated Invariant Sets and the Morse Index. CBMS Regional Conference Series in Mathematics 38. AMS, Providence, RI (1978).Google Scholar
Griffiths, & Harris, J.. Principles of Algebraic Geometry. Wiley, New York (1978).Google Scholar
Jones, C. & Moloney, J.. Instability of nonlinear waveguide modes. Phys. Lett. A 117 (1986).Google Scholar
Kato, T.. Perturbation Theory for Linear Operators. Springer-Verlag, New York (1966).Google Scholar
Newell, A.. Solitons in Mathematics and Physics. SIAM, Philadelphia (1985).Google Scholar
Shatah, J. & Strauss, W.. Instability of nonlinear bound states. Commun. Math. Phys. 100 (1985), 173–190.Google Scholar
Stegeman, G. I. & Seaton, C. T.. Nonlinear surface polaritons. Opt. Lett. 9 (1984).Google Scholar
Strauss, W.. The nonlinear Schrödinger equation. In Contemporary Developments in Continuous Mechanics and Partial Differential Equations, ed. de la Penha, G. M. & Medeiros, L. A.. North-Holland, Amsterdam (1978).Google Scholar
Weinstein, M.. Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Appl. Math. 16 (1985).Google Scholar