Semiclassical eigenvalue distribution of the Zakharov-Shabat eigenvalue problem

Physica D: Nonlinear Phenomena

Volumn 97, Issue 4, 1996, Pages 376-397

  Bronski, Jared C ^a,b  

^a UNIVERSITY OF MINNESOTA   (United States)

^b STANFORD UNIVERSITY   (United States)

Author keywords

Inverse scattering method; Semiclassical limits; Zakharov Shabat eigenvalue problem

Indexed keywords

APPROXIMATION THEORY; EIGENVALUES AND EIGENFUNCTIONS; NUMERICAL ANALYSIS; NUMERICAL METHODS; PROBLEM SOLVING; SPECTRUM ANALYSIS;

ASYMPTOTIC PREDICTION; EIKONAL EQUATION; INVERSE SCATTERING METHOD; SCHRODINGER EQUATION; SEMICLASSICAL LIMITS; ZAKHAROV-SHABAT EIGENVALUE PROBLEM;

QUANTUM THEORY;

EID: 0030263239     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/0167-2789(95)00311-8     Document Type: Article

References (35)

1. M. Berry, Semiclassically weak reflections above analytic and nonanalytic potential barriers, J. Phys. A: Math. Gen. 15 (1982) 3693-3704.
2. Levine, Segur and Tanveer, eds., Asymptotics Beyond All Orders, NATO/ASI B: Physics, Vol. 284, NATO (Plenum Press, New York, 1991).
3. Ercolani, Serre and Levermore, eds., Singular Limits of Dispersive Waves, NATO/ASI B: Physics, Vol. 320, NATO (Plenum Press, New York, 1994).
4. R. Meyer, Exponential asymptotics, SIAM Rev. 22 (1980) 213-223.
5. R. Meyer, Quasiclassical scattering above barriers in one dimension, J. Math. Phys. 17 (1976).
6. V. Pokrovskii and I. Khalatnikov, On the problem of above-barrier reflection of high energy particles, Sov. Phys. JETP 13 (1961) 1207-1210.
7. S. Jin, C. Levermore and D. McLaughlin, The behavior of the NLS equation in the semiclassical limit, in: Singular Limits of Dispersive Waves, NATO/ASI B: Physics, Vol. 320, NATO (Plenum Press, New York, 1994).
8. O. Wright, Explicit construction of the Lax-Levermore minimizer for the KdV zero dispersion limit, in: Singular Limits of Dispersive Waves, NATO/ASI B: Physics, Vol. 320, NATO (Plenum Press, New York, 1994).
9. F. Tian, On the initial value problem of the Whitham averaged system, in: Singular Limits of Dispersive Waves, NATO/ASI B: Physics, Vol. 320, NATO (Plenum Press, New York, 1994).
10. J. Bronski and D. McLaughlin, Semiclassical behavior in the NLS equation: optical shocks and focusing instabilities, in: Singular Limits of Dispersive Waves, NATO/ASI B: Physics, Vol. 320, NATO (Plenum Press, New York, 1994).
11. V. Geogjaev, The quasiclassical limit of the inverse scattering method, in: Singular Limits of Dispersive Waves, NATO/ASI B: Physics, Vol. 320, NATO (Plenum Press, New York, 1994).
12. J. Gibbons and Y. Kodama, Solving dispersionless Lax equations, in: Singular Limits of Dispersive Waves, NATO/ASI B: Physics, Vol. 320, NATO (Plenum Press, New York, 1994).
13. S. Kamvissis, Critical and subcritical cases of the Toda shock problem, in: Singular Limits of Dispersive Waves, NATO/ASI B: Physics, Vol. 320, NATO (Plenum Press, New York, 1994).
14. P. Deift and X. Zhou, Long-time asymptotics for the autocorrelation function of the transverse Ising chain at the critical magnetic field, in: Singular Limits of Dispersive Waves, NATO/ASI B: Physics, Vol. 320, NATO (Plenum Press, New York, 1994).
15. P. Lax and C. Levermore, The small dispersion limit of the KdV equation I, II, III, Comm. Pure Appl. Math. 36 (1983) 253-290, 571-593, 809-829.
16. S. Venakides, Higher order Lax-Levermore theory, Comm. Pure Appl. Math. 43 (1990) 335-362.
17. S. Venakides, The generation of modulated wavetrains in the solution of the KdV equation, Comm. Pure Appl. Math. 38 (1985) 883-909.
18. S. Venakides, The zero dispersion limit of the KdV equation with periodic initial data, AMS Trans. 301 (1987) 189-225.
19. G. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1974).
20. R. Langer, On the asymptotic solution of ordinary differential equations, Trans. Amer. Math. Soc. 33(23) (1931).
21. P. Deift and X. Zhou, A steepest descent method for oscillatory Riemann-Hilbert problems: Asymptotics for the MKdV equation, Ann. Math. 137 (1993) 295-368.
22. H. Flashka, M. Forest and D. McLaughlin, Multiphase averaging and the inverse spectral solutions of the KdV equation, Comm. Pure. Appl. Math 33 (1980) 739-784.
23. N. Ercolani, S. Jin, C. Levermore and W.D. McEvoy, The zero dispersion limit of the NLS/MKdV hierarchy for the nonselfadjoint ZS operator, Preprint (1993).
24. C. Levermore, The hyperbolic nature of the zero dispersion KdV limit, Comm. P.D.E. 13(4) (1988) 495-514.
25. K. McLaughlin, A continuum limit of the Toda lattice, Ph.D. Thesis, New York Univ., Courant Institute (1994).
26. M.G. Forest and J. Lee, Geometry and modulation theory for periodic nonlinear Schrödinger equations, in: Oscillation Theory, Computation, and Methods of Compensated Compactness, IMA Volumes on Mathematics and its Applications (1986).
27. A. Bishop, M. Forest, D. McLaughlin and E.A. Overman II, A quasi-periodic route to chaos in a near integrable PDE, Physica D 23 (1986) 293.
28. A. Bishop, D. McLaughlin and E.A. Overman II, Coherence and chaos in the driven damped sine Gordon equation, Physica D 19 (1986).
29. D. McLaughlin and E.A. Overman II, Whiskered tori for integrable PDE's; Chaotic behavior in near integrable PDE's, Surv. Appl. Math., to appear.
30. P. Deift, S. Venakides and X. Zhou, personal communications.
31. K. McLaughlin, personal communications.
32. Z. Lewis, Semiclassical solutions of the Zakharov-Shabat scattering problem for phase modulated potentials, Phys. Lett. A 112(3/4) (1985).
33. V. Zakharov and A. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Sov. Phys. JETP 34 (1973) 62-69.
34. V. Zakharov and A. Shabat, Interaction between solitons in a stable medium, Sov. Phys. JETP 37 (1973) 823-828.
35. D. Kaup and B. Malomed, Variational principal for the Zakharov-Shabat equations, Physica D 84 (1995) 319-328.