Spectral instability of the semiclassical Zakharov-Shabat eigenvalue problem

Physica D: Nonlinear Phenomena

Volumn 152-153, Issue , 2001, Pages 163-170

  Bronski, J C ^a  

^a UNIVERSITY OF ILLINOIS AT URBANA CHAMPAIGN   (United States)

Author keywords

NLS equation; Spectral instability; Zakharov Shabat eigenvalue

Indexed keywords

EID: 4244193319     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0167-2789(01)00167-1     Document Type: Article

References (18)

1. M.G. Forest, J.E. Lee, Geometry and modulation theory for the periodic nonlinear Schrodinger equation, in: C. Dafermos, et. al. (Eds.), Oscillation Theory, Computation, and Methods of Compensated Compactness, Vol. 2, IMA, Springer, New York, 1986.
2. Zakharov V.E., Shabat A.B. Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP. 34:1972;62-69.
3. Deift P., Zhou X. A steepest descent method for oscillatory Riemann-Hilbert problems. Bull. Am. Math. Soc. 26:1992;119-123.
4. Deift P., Venakides S., Zhou X. New results in small dispersion KdV by an extension of the steepest descent method for Riemann-Hilbert problems. Int. Math. Res. Notices. 6:1997;286-299.
5. P. Deift, S. Venakides, X. Zhou, unpublished.
6. P.D. Miller, Some remarks on a WKB method for the non-self-adjoint Zakharov-Shabat eigenvalue problem with analytic potentials and fast phase, in press.
7. Bronski J.C., Kutz J.N. Numerical simulation of the semi-classical limit of the focusing nonlinear Schrödinger equation. Phys. Lett. A. 254:1999;326-336.
8. Davies E.B. Semiclassical states for non-self-adjoint Schrödinger operators. Commun. Math. Phys. 200:1999;35-41.
9. A. Aslanyan, E.B. Davies, Spectral instability for some Schrodinger operators, Electronic Preprint arXiv.math.SP/9810063.
10. Trefethen L.N. Pseudospectra of linear operators. SIAM Rev. 39:1997;383-406.
11. Bronski J.C. Semiclassical eigenvalue distribution of the Zakharov-Shabat eigenvalue problem. Physica D. 97:1996;376-397.
12. S. Jin, C.D. Levermore, D.W. McLaughlin, The behavior of solutions of the NLS equation in the semi-classical limit, Singular Limits of Dispersive Waves, NATO Adv. Sci. Inst. Ser. B, Vol. 320, Plenum Press, New York, 1994.
13. S. Jin, C.D. Levermore, D.W. McLaughlin, The semi-classical limit of the NLS hierarchy, Commun. Pure Appl. Math. (1997), submitted for publication.
14. Miller P.D., Kamvissis S. On the semi-classical limit of the focusing nonlinear Schrödinger equation. Phys. Lett. A. 247(1/2):1998;75-86.
15. Krylov D., Leng L., Bergman K., Bronski J.C., Kutz J.N. Observation of the breakup of a pre-chirped N-soliton in an optical fiber. Opt. Lett. 24:1999;1191-1193.
16. N.M Ercolani, S. Jin, C.D. Levermore, W. MacEvoy Jr., The zero dispersion limit of the NLS/mKdV hierarchy for the non-self-adjoint ZS operator, Preprint, 1993.
17. J.C. Bronski, D.W. McLaughlin, Semi-classical behavior in the NLS equation: optical shocks - focusing instabilities, Singular Limits of Dispersive Waves, NATO Adv. Sci. Inst. Ser. B, Vol. 320, Plenum Press, New York, 1994.
18. K.T.R. McLaughlin, P.D. Miller, S. Kamvissis, Preprint.