Nonlinear Problems in Theoretical Physics pp 35-90 | Cite as

# Spectral transform and solvability of nonlinear evolution equations

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Reflection Coefficient Spectral Problem Nonlinear Evolution Equation Discrete Part Discrete Eigenvalue
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## References

- 1.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 (1967).CrossRefGoogle Scholar
- 2.Ablowitz, M.J., Kaup, D.J., Newell, A.C., Segur, H.: The Inverse Scattering transform-Fourier analysis for nonlinear problems. Appl. Math. 53, 249 (1974); hereafter referred to as AKNS.Google Scholar
- 3.Scott, A.C., Chu, F.Y.F., McLaughlin, D.W.: The soliton: a new concept in applied science. Proc. I.E.E.E. 61, 1443 (1973).Google Scholar
- 3.aBollough, R.K.: Solitons. In Interaction of Radiation with Condensed Matter. Vol.I, IAEA-SMR-20/51, Vienna (1977); also lectures delivered at the International Advanced study Institute on Nonlinear Equations in Physics and Mathematics, Istanbul, August 1–13, 1977. Proceedings edited by.O.Barut.Google Scholar
- 4.Calogero, F., Degasperis, A.: Solution by the spectral transform method of a nonlinear evolution equation including as a special case the cylindrical KdV equation. Lett. Nuovo Cimento 23, 150 (1978).Google Scholar
- 5.Well known linear problems are the Schr6dinger and the generalized (non selfadjoint) Zackarov-Shabat spectral problems on the real line, with a potential which vanishes at infinity. The most interesting associated evolution equations are the KdV for the first one and the MKdV (modified KdV), sine-Gordon and nonlinear Schrödinger equations for the second one. A unified treatment of these evolution equations can be obtained by considering the N x N matrix Schrödinger spectral problem: see Ref. 12 and Jaulent, M., Miodek, I.: Connection between Zackarov Shabat and Schr6dinger Type Inverse Scattering Transforms. Preprint PM/77/9, University of Montpellier (1977). Calogero, F., Degasperis, A.: in preparation. The Schrödinger problem with a potential which depends in a simple way on the eigenvalue has been discussed by Jaulent, M. and Miodek, I.: Nonlinear evolution equations associated with “energy-dependent Schrödinger potentials”. Lett. Math. Phys. 1, 243 (1976). The Schrödinger problem with a potential which diverges at infinity has been discussed by Kulish, P.: Inverse scattering problem for Schrödinger equation on a line with potential growing in one direction. Mathematical Notes, Leningrad (1970). Also Calogero, F., Degasperis, A.: Inverse spectral problem for the one-dimensional Schrödinger equation with an additional linear potential. Lett. Nuovo Cimento, 23, 143 (1978. See also Ref. 4.Google Scholar
- 6.Lax, P.D.: Integrals of Nonlinear equations of evolution and solitary waves. Comm. Pure Appl. Math. 21, 467 (1968).Google Scholar
- 7.Calogero, F.: A Method to Generate Solvable Nonlinear Evolution Equations. Lettere Nuovo Cimento, 14, 443 (1975).Google Scholar
- 7.aCalogero, F., Degasperis, A.: Nonlinear Evolution Equations Solvable by the Inverse Spectral Transform. I. Nuovo Cimento 32B, 201 (1976).Google Scholar
- 8.Wahlquist, H.D., Estabrook, F.B.: Prolungation structures of nonlinear evolution equations. J. Math. Phys. 16, 1 (1975).CrossRefGoogle Scholar
- 9.Corones, J., Markovski, B.L., Rizov, V.A.: Bilocal Lie Groups and Solitons. Phys. Lett. 61A, 439 (1977).Google Scholar
- 10.Novikov, S.P.: New applications of algebraic geometry to nonlinear equations and inverse problems. To appear in the Proceedings of a Symposium held at the Accademia Nazionale dei Lincei in Rome in June 1977, Calogero, F. (editor): Nonlinear evolution equations solvable b the spectral transform. Pitman. London (1978).Google Scholar
- 11.Calogero, F., Degasperis, A.: Nonlinear evolution equations solvable by the inverse spectral transform.II. Nuovo Cimento 39B, 1 (1977).Google Scholar
- 12.Wadati, M., Kamijo, T.: On the extension of inverse scattering method. Prog. Theor. Phys. 52, 397 (1974).Google Scholar
- 13.Newton, R.G.: Scattering Theory of Waves and Particles. Mc Graw Hill Book Company, New York (1966).Google Scholar
- 14.It can be easily proved that if Q(x) is hermitian and k is real, then T(k) is not singular.Google Scholar
- 15.Gel'fand, I.M., Levitan, B.M.: On the determination of a differential equation from its spectral function. Amer. Math. Soc. Transl., 1, 253 (1955).Google Scholar
- 15.aAgranovich, Z.S., Marchenko, V.A.: The Inverse Problem of Scattering Theory. (translated from Russian by B.D. Seckler). New York, Gordon and Breach 1963).Google Scholar
- 15.bChadan, K., Sabatier, P.C.: Inverse Problems in Quantum Scattering Theory. Springer Verlag, New York (1977).Google Scholar
- 16.For a detailed treatment (in the case N =1) see the paper by Calogero, F.: Generalized wronskian relations, one-dimensional Schrödinger equation and nonlinear partial differential equations solvable by the inverse-scattering method. Nuovo Cimento 31B, 229 (1976).Google Scholar
- 17.Indeed this result applies to all SNEE's associated to the generalized Zacharov-Shabat spectral problem since they are a subclass of the present class of SNEE (see below and the references reported in footnote 5 ). Nonlinear evo lution equations which are not isospectral flows but can be still investigated by the ST method associated to the single-channel Schrödinger problem and the generalized Zackarov-Shabat problem, have been discussed by Newell, A.: The general structure of integrable evolution equations, to appear in Proc. Roy. Soc., 1978; and by Calogero, F., Degasperis, A.: Extension of the spectral transform method for solving nonlinear evolution equations. I & II. Lett Nuovo Cimento 22, 131 and 263 (1978).Google Scholar
- 18.Since the evolution equation (2.28) is invariant under time-translations, there is no loss of generality in considering t = 0 as the initial time.Google Scholar
- 19.Calogero, F., Degasperis, A.: Coupled nonlinear evolution equations solvable via the inverse spectral transform and solitons that come back: the boomeron. Lettere Nuovo Cimento 16, 425 (1976).Google Scholar
- 19.aCalogero, F., Degasperis, A.: Bäcklund transformations, nonlinear superposition principle, multisoliton solutions and conserved quantities for the “boomeron” nonlinear evolution equation. Lettere Nuovo Cimento 16, 434 (1976).Google Scholar
- 20.A special class of non hermitian potentials has been discussed by: Bruschi,M. Levi, D., Ragnisco, O.: Evolution equations associated to the triangular matrix Schrödinger problem solvable by the inverse spectral transform. Nuovo Cimento 45A, 225 (1978).Google Scholar
- 21.Scott-Russel, J.: Report on waves. Report of the Fourteenth Meeting on the British Association for the Advancement of Science, London, 1845, pp.311–390.Google Scholar
- 22.Zabusky, N.J. Kruskal, M.D.: Interaction of solitons in a collisionless plas ma and the recurrence of initial states. Phys. Rev. Lett. 15, 240 (1965)Google Scholar
- 22.aZabusky, N.J.: A synergetic approach to problems of nonlinear dispersive wave propagation and interaction. In Nonlinear Partial Differential Equations Ames, W. (editor). New York: Academic Press (1967), pp.223–258.Google Scholar
- 23.Kruskal, M.D.: The birth of the soliton. Proceedings of the Symposium held at the Accademia Nazionale dei Lincei in Rome in June 1977. Calogero, F. (editor): Nonlinear evolution equations solvable by the spectral transform. Pitman. London (1978).Google Scholar
- 24.Zackarov, V.E.: Kinetic equation for solitons. Soviet Phys. JETP, 33, 538 (1971).Google Scholar
- 24.aWadati, M., Toda, M.: The exact N-soliton solution of the Korteweg-de Vries equation. J.Phys. Soc. Japan 32, 1403 (1972).Google Scholar
- 24.bTanaka, S.: Publ. Res. Inst. Math. Sci. Kyoto University 8, 419 (1972/73).Google Scholar
- 24.cSee also: Gardner, C.S., Greene, J.M., Kruskal, M.D., Miura, R.M.: Kortewegde Vries equation and generalization. VI. Methods for exact solution. Comm. Pure Appl. Math. 27, 97 (1974).Google Scholar
- 25.Condrey, P.J., Eilbeck, J.C., Gibbon, J.D.: The Sine-Gordon equation as a model classical field theory. Nuovo Cimento 25B, 497 (1975).Google Scholar
- 26.In the special case a \(\mathop a\limits^ \to \Lambda \mathop b\limits^ \to = 0\) the nonlinear evolution equation (3.34a) coincides with the Landau-Lifshitz equation describing the magnetization of a ferromagnetic substance.Google Scholar
- 27.Calogero, F., Degasperis, A.: Transformations between solutions of different nonlinear evolution equations solvable via the same inverse spectral transform, generalized resolvent formulas and nonlinear operator identities. Lettere Nuovo Cimento 16, 181 (1976).Google Scholar
- 28.See, for instance: Lamb Jr., G.L.: Bäcklund transformations for certain nonlinear evolution equations.J. Math. Phys. 15, 2157 (1974)CrossRefGoogle Scholar
- 28.aChen, H.H.: General derivation of Bäcklund transformations from inverse scatte ring problems. Phys. Rev. Lett. 33, 925 (1974).Google Scholar
- 28.bMiura, R.M. (editor) Bäcklund transformations. Lectures Notes in Mathematics, 515, Berlin, Heidelberg, New York: Springer Verlag (1976). See also Ref. X30.Google Scholar
- 29.Case, K.M., Chu, S.C.: Some remarks on the wronskian technique and the inverse scattering transform. J. Math. Phys. 18, 2044 (1977).Google Scholar
- 30.Wahlquist, H.D., Estabrook, F.B.: Bäcklund transformations for solutions of the Korteweg-de Vries Equation. Phys. Rev. Lett. 31, 1386 (1973).CrossRefGoogle Scholar
- 31.See, for instance: Kruskal, M.D.: Nonlinear wave equations, in Dynamical Systems, Theory and Applications. Moser, J. (editor). Lectures Notes in Physics, 38, Berlin, Heidelberg, New York: Springer Verlag (1975).Google Scholar

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