# Spectral transform and solvability of nonlinear evolution equations

• A. Degasperis
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 98)

## Keywords

Reflection Coefficient Spectral Problem Nonlinear Evolution Equation Discrete Part Discrete Eigenvalue
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

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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
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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
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## Authors and Affiliations

• A. Degasperis
• 1
• 2
• 3
1. 1.Istituto di FisicaUniversita di RomaRomaItaly
2. 2.Istituto di FisicaUniversita di LecceItaly
3. 3.Istituto Nazionale di Fisica NucleareSezione di RomaItaly