On the Whitham equations for the semiclassical limit of the defocusing nonlinear Schrödinger equation

Communications on Pure and Applied Mathematics

Volumn 52, Issue 6, 1999, Pages 655-692

  Tian, Fei Ran ^a   Ye, Jian ^a  

^a OHIO STATE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0033435348     PISSN: 00103640     EISSN: None     Source Type: Journal    
DOI: 10.1002/(SICI)1097-0312(199906)52:6<655::AID-CPA1>3.0.CO;2-A     Document Type: Article

References (28)

1. Deift, P.; McLaughlin, K. T-R. A continuum limit of the Toda lattice. Mem. Amer. Math. Soc. 131 (1998), no. 624.
2. Dubrovin, B. A.; Novikov, S. P. Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory. (Russian) Uspekhi Mat. Nauk 44 (1989), no. 6(270), 29-98, 203; translation in Russian Math. Surveys 44 (1989), no. 6, 35-124.
3. Dubrovin, B. A.; Novikov, S. P. Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory. (Russian) Uspekhi Mat. Nauk 44 (1989), no. 6(270), 29-98, 203; translation in Russian Math. Surveys 44 (1989), no. 6, 35-124.
4. Flaschka, H.; Forest, M. G.; McLaughlin, D. W. Multiphase averaging and the inverse spectral solution of the Korteweg-de Vries equation. Comm. Pure Appl. Math. 33 (1980), no. 6, 739-784.
5. Forest, M. G.; Lee, J. E. Geometry and modulation theory for periodic nonlinear Schrödinger equation. Oscillation theory, computation, and methods of compensated compactness (Minneapolis, Minn., 1985), 35-69, IMA Vol. Math. Appl., 2. Springer, New York, 1986.
6. Forest, M. G.; McLaughlin, K. T-R. Onset of oscillations in nonsoliton pulses in nonlinear dispersive fibers. J. Nonlinear Sci. 8 (1998), no. 1, 43-62.
7. Geogdzhaev, V. V. Solution of the Benney equations by the inverse scattering method. Teoret. Mat. Fiz. 73 (1987), no. 2, 255-263.
8. Gurevich, A. V.; Krylov, A. L. Dissipationless shock waves in media with positive dispersion. Sov. Phys. JETP. 65 (1987), 944-953.
9. Jin, S. The semiclassical limit of the defocusing nonlinear Schrödinger flows. Doctoral dissertation, University of Arizona, 1991.
10. Jin, S.; Levermore, C. D.; McLaughlin, D. W. The behavior of solutions of the NLS equation in the semiclassical limit. Singular limits of dispersive waves (Lyon, 1991), 235-255. Edited by N. M. Ercolani, I. R. Gabitov, C. D. Levermore, and D. Serre. NATO Adv. Sci. Inst. Ser. B Phys., 320. Plenum, New York, 1994.
11. Kodama, Y. The Whitham equations for optical communications: Mathematical theory of NRZ. Preprint, 1998.
12. Kodama, Y.; Gibbons, J. Integrability of the dispersionless KP hierarchy. Nonlinear world, Vol. 1 (Kiev, 1989), 166-180. World Sci. Publishing, River Edge, N.J., 1990.
13. Krichever, I. M. The averaging method for two-dimensional "integrable" equations. (Russian) Functional Anal. Appl. 22 (1988), no. 3, 200-213.
14. Kudashev, V. R.; Sharapov, S. E. Hydrodynamic symmetries for the Whitham equations for the nonlinear Schrödinger equation (NSE). Phys. Lett. A 154 (1991), no. 9, 445-448.
15. Lax, P. D. Hyperbolic systems of conservation laws and the mathematical theory of shock waves. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 11. Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973.
16. Lax, P. D.; Levermore, C. D. The small dispersion limit of the Korteweg-de Vries equation. I, II, III. Comm. Pure Appl. Math. 36 (1983), no. 3, 253-290; no. 5, 571-593; no. 6, 809-830.
17. Lax, P. D.; Levermore, C. D. The small dispersion limit of the Korteweg-de Vries equation. I, II, III. Comm. Pure Appl. Math. 36 (1983), no. 3, 253-290; no. 5, 571-593; no. 6, 809-830.
18. Lax, P. D.; Levermore, C. D. The small dispersion limit of the Korteweg-de Vries equation. I, II, III. Comm. Pure Appl. Math. 36 (1983), no. 3, 253-290; no. 5, 571-593; no. 6, 809-830.
19. McLaughlin, K. T-R. A continuum limit of the Toda lattice. Doctoral dissertation, New York University, 1994.
20. Pavlov, M. V. The nonlinear Schrödinger equation and Bogolyubov-Whitham averaging method. (Russian) Teoret. Mat. Fiz. 71 (1987), no. 3, 351-356.
21. Tian, F-R. Oscillations of the zero dispersion limit of the Korteweg-de Vries equation. Comm. Pure Appl. Math. 46 (1993), no. 8, 1093-1129.
22. Tian, F.-R. The initial value problem for the Whitham averaged system. Comm. Math. Phys. 166 (1994), no. 1, 79-115.
23. Tian, F.-R. The Whitham-type equations and linear overdetermined systems of Euler-Poisson-Darboux type. Duke Math. Jour. 74 (1994), no. 1, 203-221.
24. Tian, F.-R.; Ye, J. Development of singularities of solutions of nonlinear hyperbolic partial differential equations with phase transition. OSMRI preprint #97-8, 1997.
25. Tsarev, S. P. Poisson brackets and one-dimensional Hamiltonian systems of hydrodynamic type. Sov. Math. Dokl. 31 (1985), 488-491.
26. Venakides, S. The zero dispersion limit of the Korteweg-de Vries equation for initial potential with nontrival reflection coefficient. Comm. Pure Appl. Math. 38 (1985), no. 2, 125-155.
27. Whitham, G. B. Non-linear dispersive waves. Proc. Roy. Soc. Ser. A 283 (1965), 238-261.
28. Wright, O. C. Korteweg-de Vries zero dispersion limit: through first breaking for cubic-like analytic initial data. Comm. Pure Appl. Math. 46 (1993), no. 3, 423-440.