Integrability of linear and nonlinear evolution equations and the associated nonlinear Fourier transforms. (English) Zbl 0807.35138

Summary: The inverse spectral method is a nonlinear Fourier transform method for solving certain equations. Here, we emphasize that such transforms should be considered in their own right. We also elucidate further the connection between the Fourier transform and inverse spectral methods by establishing that linear equations can also be solved through the inverse spectral method.


35Q58 Other completely integrable PDE (MSC2000)
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
58J72 Correspondences and other transformation methods (e.g., Lie-Bäcklund) for PDEs on manifolds
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[1] Gardner, C. S., Greene, J. M., Kruskal, M. D., and Miura, R. M.,Phys. Rev. Lett. 19, 1095 (1967);Comm. Pure Appl. Math. 27, 97 (1974). For recent developments see Fokas, A. S. and Zakharov, V. E. (eds),Important Developments in Soliton Theory, Springer-Verlag, New York, 1993. · Zbl 1103.35360
[2] Lax, P. D.,Comm. Pure Appl. Math. 21, 467 (1968). · Zbl 0162.41103
[3] Ablowitz, M. J., Kaup, D. J., Newell, A. C., and Segur, H.,Phys. Rev. Lett. 30, 1262 (1973a);31, 125 (1973b).
[4] Fokas, A. S. and Its, A. R.,Phys. Rev. Lett. 68, 3117-3120 (1992). · Zbl 0969.35537
[5] Gelfand, I. M. and Levitan, B. M.,Izv. Akad. Nauk SSR, Ser. Math. 15, 309 (1951) (in Russian); English transl. inAmer. Math. Soc. Trans. Ser. 2,1, 253 (1955).
[6] Zakharov, V. E. and Shabat, A. B.,Soviet Phys. JETP 34, 62 (1972).
[7] Zhou, X.,SIAM J. Math. Anal. 20, 966-980 (1989). · Zbl 0685.34021
[8] Fokas, A. S. and Sung, L. Y.,Inverse Problems 8, 673-708 (1992). · Zbl 0768.35069
[9] Fokas, A. S. and Zakharov, V.,J. Nonlinear Sci. 2, 109-134 (1992). · Zbl 0872.58032
[10] Fokas, A. S.,Phys. Rev. Lett. 51, 3 (1983).
[11] Beals, R. and Coifman, R. R.,Comm. Pure Appl. Math. 37, 39 (1984). · Zbl 0523.34020
[12] Beals, R. and Coifman, R. R.,Proc. Symp. Pure Math. 43, 45 (1985).
[13] Beals, R. and Coifman, R. R., The spectral problem for the Davey-Stewartson and Ishimori Hierarchies,Proc. Conf. Nonlinear Evolution Equations: Integrability and Spectral Methods, Como, University of Manchester. · Zbl 0725.35096
[14] Sung, L. Y., An inverse scattering transform for the DS II equations, to appear inJ. Math. Anal. Appl.
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