Abstract
The main ideas and some recent results on (classical) solitons are tersely surveyed.
Invited lecture presented at the International Conference on the Mathematical Problems in Theoretical Physics, Rome University, June 6–15, 1977.
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References
C.S.Gardner, J.M.Greene, M.D.Kruskal and R.M.Miura, “Method for Solving the Korteweg-de Vries Equation”, Phys.Rev.Lett. 19, 1095–1097 (1967).
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V.E.Zakharov and A.B.Shabat, “A Scheme for Integrating the Nonlinear Equations of Mathematical Physics by the Method of the Inverse Scattering Problem, I”, Func.Anal.Appl. 8, 226–235 (1974) [Rus sian original: Funk.Anal.Pril. 8, 43–53 (1974)].
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See, for instance: P.D.Lax, “Integrals of Nonlinear Equations of Evolution and Solitary Waves”, Comm.Pure Appl.Math. 21, 467–490 (1968); R.M.Miura, C.S.Gardner and M.D.Kruskal, “Korteweg-de Uries equation and generalizations. II. Existence of conservation laws and constants of motion”, J.Math.Phys. 9, 1204–1209 (1968).
See, for instance: H.D.Wahlquist and F.B.Estabrook, “BAcklund Tran sformation for Solutions of the Korteweg-de Uries Equation”, Phys. Rev.Lett. 31, 1386–1390 (1973); G.L.Lamb jr., “BAcklund Transfor mations for Certain Nonlinear Evolution Equations”, J.Math.Phys. 15, 2157–2165 (1974); H.H.Chen, “General Derivation of BAcklund Transformations from Inverse Scattering Problems”, Phys.Rev.Lett. 33, 925–928 (1974).
F.Calogero, “BAcklund Transformations and Functional Relation forn Solutions of Nonlinear Partial Differential Equations Solvable via the Inverse Scattering Method”, Lett. Nuovo Cimento 14, 537–543 (1975); F.Calogero and A.Degasperis, “Transformations between Solu tions of Different Nonlinear Evolution Equations Solvable via the same Inverse Spectral Transform, Generalized Resolvent Formulas and Nonlinear Operator Identities”, Lett. Nuovo Cimento 16, 181–186 (1976).
See, for instance: S.V.Manakov, Sov.Phys. JETP 38, 693 (1974); H. Segur and M.J.Ablowitz, “Asymptotic Solutions and Conservation Laws for the Nonlinear Schroedinger Equation”, J.Math.Phys. 17, 710 (1976); M.J.,Ablowitz and H.Segur, “Asymptotic Solutions of the Kor teweg-de Vries Equation”, Stud.Appl.Math. 1977 (in press); V.E.Zakharov and S.V.Manakov, “Asymptotic Behavior of Nonlinear Wave Systems Solvable by the Method of the Inverse Scattering Transform”, Zh.Eksp.Teor.Fiz. 71, 203 (1976).
V.E.Zakharov and L.D.Faddeev, “The Korteweg-de Vries Equation: a Completely Integrable Hamiltonian System”, Func.Anal.Appls. 5, 28–287 (1971) [Russian original: Funk.Anal.Pril. 5, 1818-227 (1971)].
P.D.Lax, “Integrals of Nonlinear Equations of Evolution and Solitary Waves”, Comm.Pure Appl.Math. 21, 467–490 (1968).
Recently there have been very interesting developments on this last topic: H.Airault, H.P.McKean and J.Moser, “Rational and Elliptic Solutions of the Korteweg-de Vries Equation and a Related Many-Bo dy Problem”, (NYU preprint, to be published); D.V.Choodnovsky and G.V.Choodnovsky, “Pole Expansion of Nonlinear Partial Differential Equations”, Nuovo Cimento B (in press); F.Calogero, “Motion of poles and zeros of special solutions of nonlinear and linear partial differential equations and related “solvable” many-body problems”, Nuovo Cimento B (in press).
See, for instance: M.J.Ablowitz, D.J.Kaup, A.C.Newell and H.Segur, “Method for Solving the Sine-Gordon Equation”, Phys.Rev.Lett. 30, 1262–1264 (1973); L.D.Faddeev and L.A. Takhtajan, “Essentially Nonlinear One-Dimensional Model of Classical Field Theory”, Commun. JINR Dubna E2-7998 (1974).
A.C.Scott, F.Y.F.Chu and D.W.McLaughlin, “The Soliton: a New Concept in Applied Science”, Proc.IEEE 61, 1443–1483 (1973); B.A. Dubrovin, V.B.Matveev and S.P.Novikov, “Nonlinear Equations of Korte weg-de Vries Type, Finite-Zone Linear Operators and Abelian Varie ties”, Uspekhy Mat.Nauk 31, 55–136 (1976); R.Rajaraman, “Some Non-Perturbative Semi-Classical Methods in Quantum Field Theory (A Pedagogical Review)”, Physics Reports 21, 227–313 (1975).
Nonlinear Wave Motion (A.C.Newell, ed.), Lect.Appl.Math. 15, AMS, Providence, R.I., 1974; Dynamical Systems, Theory and Applications (J.Moser, ed.), Lect.Notes in Physics 38, Springer, 1975; Bäcklund Transformations (R.M.Miura, ed.), Lect.Notes in Math. 515, Springer, 1976. Two other books now in preparation are going to be published soon, one by Springer under the editorship of R.K.Bullough, and one by Pitman under the editorship of F.Calogero.
F.Calogero, “Generalized Wronskian Relations, One-Dimensional Schroedinger Equation and Nonlinear Partial Differential Equations Solvable by the inverse Scattering Method”, Nuovo Cimento 31B, 229–249 (1976). See also the papers of Ref.[71.
N.J.Zabusky and M.D.Kruskal, “Interactions of “solitons” in a collisionless plasma and the recurrence of initial states”, Phys.Rev. Lett. 15, 240–243 (1966).
F.Calogero and A.Degasperis, “Coupled Nonlinear Evolution Equations Solvable Via the Inverse Spectral Transform, and Solitons that Come Back: the Boomeron”, Lett.Nuovo Cimento 16, 425–433 (1976); “Bä cklund Transformations, Nonlinear Superposition Principle, Multiso liton Solutions and Conserved Quantities for the “Boomeron” Nonlinear Evolution Equation”, Lett.Nuovo Cimento 16, 434–438 (1976).
F.Calogero and A.Degasperis, “Special Solution of Coupled Nonlinear Evolution Equations with Bumps that Behave as Interacting Particles”, Lett.Nuovo Cimento 19,525–533 (1977).
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Calogero, F. (1978). Nonlinear evolution equations solvable by the inverse spectral transform. In: Dell'Antonio, G., Doplicher, S., Jona-Lasinio, G. (eds) Mathematical Problems in Theoretical Physics. Lecture Notes in Physics, vol 80. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-08853-9_21
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