Abstract
Over the past fifteen years the theory of solitons and the related theory of integrable nonlinear evolution equations in two space-time dimensions has attracted a large number of research workers of different orientations ranging from algebraic geometry to applied hydrodynamics. Modern mathematical physics has witnessed the development of a vast new area of research devoted to this theory and called the inverse scattering method of solving nonlinear equations (other names are: the inverse spectral transform, the method of isospectral deformations and, more colloquially, the L-A pair method).
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References
Ablowitz, M. J., Segur, H.: Solitons and the Inverse Scattering Transform. SIAM, Philadelphia 1981
Bullough, R. K., Caudrey, P. J. (ed.): Solitons, Topics in Current Physics 17, Berlin-Heidelberg-New York, Springer 1980
Calogero, F. (ed.): Nonlinear Evolution Equations Solvable by the Spectral Transform. Research Notes in Math. 26, London, Pitman 1978
Calogero, F., Degasperis, A.: Spectral Transform and Solitons. Vol. 1. Amsterdam, North-Holland 1982
Dodd, R. K., Eilbeck, J. C., Gibbon, J. D., Morris, H. C.: Solitons and Nonlinear Waves. New York, Academic Press 1982
Eilenberger, G.: Solitons. Mathematical Method for Physicists. Berlin, Springer 1981
Faddeev, L. D.: A Hamiltonian interpretation of the inverse scattering method. In: Solitons, edited by Bullough R. K., Caudrey P. J., Topics in Current Physics 17, 339–354, Berlin-Heidelberg-New York, Springer 1980
Faddeev, L. D.: Quantum completely integrable models in field theory. In: Mathematical Physics Review. Sect. C.: Math. Phys. Rev. 1, 107–155, Harwood Academic 1980
Faddeev, L. D.: Two-dimensional integrable models in quantum field theory. Physica Scripta 24, 832–835 (1981)
Faddeev, L. D.: Recent development of QST. In: Recent development in gauge theory and integrable systems. Kyoto, Kyoto Univ. Research Inst. for Math. Sci., 53–71, 1982
Faddeev, L. D.: Integrable models in 1+ 1-dimensional quantum field theory. In: Les Houches, Session XXXIX, 1982, Recent Advances in Field Theory and Statistical Mechanics, Zuber, J.-B., Stora, R. (editors), 563608. Elsevier Science Publishers 1984
Faddeev, L. D.: Quantum scattering transformation. In: Structural Elements in Particle Physics and Statistical Mechanics. (Freiburg Summer Inst. on Theor. Physics 1981) vol. 82, 93–114, New York-London, Plenum Press 1983
Faddeev, L. D., Korepin, V. E.: Quantum Theory of Solitons. Physics Reports 42C (1), 1–87 (1978)
Gardner, C. S., Greene, J. M., Kruskal, M. D., Miura, R. M.: Method for solving the Korteweg-de Vries equation. Phys. Rev. Lett. 19, 1095–1097 (1967)
Izergin, A. G., Korepin, V. E.: The quantum inverse scattering method. Physics of elementary particles and atomic nuclei, v. 13, N 3, 501–541 (1982)
Izergin, A. G., Korepin, V. E.: English trans!. in Soviet J. Particles and Nuclei 13 (3), 207–223 (1982)
KF 1977] Korepin, V. E., Faddeev, L. D.: Quantization of solitons. In: Physics of elementary particles (Proceedings of the XII winter school of the Leningrad institute of nuclear physics), 130–146, Leningrad 1977 [Russian]
Kulish, P. P., Sklyanin, E. K.: Solutions of the Yang-Baxter equation. In: Differential geometry, Lie groups and mechanics. III. Zapiski Nauchn. Semin. LOMI 95, 129–160 (1980) [Russian]; English transi. in J. Soy. Math. 19 (5), 1596–1620 (1982)
Kulish, P. P., Sklyanin, E. K.: Quantum spectral transform method. Recent Developments. Lecture Notes in Physics, vol. 151, 61–119, Berlin-Heidelberg-New York, Springer 1982
Lax, P. D.: Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure Appl. Math. 21, 467–490 (1968)
Lamb, G. L., Jr.: Elements of Soliton Theory. New York, Wiley 1980
Lonngren, K., Scott, A. (eds.): Solitons in Action. New York, Academic Press 1978
Miura, R. (ed.): Bäcklund transformations. Lecture Notes in Mathematics, vol. 515, Berlin-Heidelberg-New York, Springer 1976
Manakov, S. L., Zakharov, V. E. (eds.): Soliton Theory. Proceedings of the Soviet-American Symposium on Soliton Theory. Physica D, 3 D, no. 1+ 2 (1981)
Takhtajan, L. A.: Integrable models in classical and quantum field theory. In: Proceedings of the International Congress of Mathematicians 1983, 1331–1340, Warszawa, North-Holland 1984
Takhtajan, L. A., Faddeev, L. D.: The quantum inverse problem method and the XYZ Heisenberg model. Uspekhi Mat. Nauk 34 (5), 13–63 (1979)
Takhtajan, L. A., Faddeev, L. D.: English trans!. in Russian Math. Surveys 34 (5), 2–68 (1979)
Zakharov, V. E., Faddeev, L. D.: Korteweg-de Vries equation, a completely integrable Hamiltonian system. Funk. Anal. Priloz. 5 (4) 18–27 (1971) [Russian]: English transi. in Funct. Anal. Appl. 5, 280–287 (1971)
Zakharov, V. E., Manakov, S. V., Novikov, S. P., Pitaievski, L. P.: Theory of Solitons. The Inverse Problem Method. Moscow, Nauka 1980
Zakharov, V. E., Manakov, S. V., Novikov, S. P., Pitaievski, L. P.: English transi.: New York, Plenum 1984
Zakharov, V. E., Shabat, A. B.: Exact theory of two-dimensional selffocusing and one-dimensional self-modulation of waves in non-linear media. Zh. Exp. Teor. Fiz. 61, 118–134 (1971)
Zakharov, V. E., Shabat, A. B.: English transi. in Soviet Phys. JETP 34, 62–69 (1972)
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Faddeev, L.D., Takhtajan, L.A. (2007). Introduction. In: Hamiltonian Methods in the Theory of Solitons. Springer Series in Soviet Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69969-9_1
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