Abstract
We review some analytic results of the N × N Zakharov-Shabat system dψ/dx = z2Jψ + (zQ+P)ψ, which is a generalization of Beals-Coifman’s results on the first order system dψ/dx = zJψ + Qψ. We also show that for skew-Hermitian generic potentials Q,P, the scattering data has certain symmetric properties. If the scattering data has such symmetric properties, then the inverse problem is solvable. We also give several examples of evolution equations solvable by this inverse scattering transform. The global existence in time of these evolution equations is obtained if the initial data is generic and Skew-Hermitian.
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Beals R and Coifman R R, 1980–81 Scattering, transformation spectrales, et equations d’evolution non lineaires, Seminaire Goulaouic—Meyer—Schwartz exp. 22, Ecole Polytechnique, Palaiseau.
Beals R and Coifman R R, 1984 Scattering and inverse scattering for first order system, Comm. Pure Appl. Math., vol. 37, 39–90.
Beals R and Coifman R R, 1985 Inverse scattering and evolution equations, Comm. Pure Appl. Math., vol. 38, 29–42.
Beals R and Coifman R R, 1987 Scattering and inverse scattering for first—order system: II, Inverse Problem 3, 577–593.
Bullough R K, Caudrey P J (eds), 1980 Solitons, Topics in Current Physics No. 17, Springer—Verlag.
Fordy A P, 1984 Derivative nonlinear Schrodinger equations and Hermitian symmetric spaces, J. Phys. A: Math. Gen. 17, 1235–45.
Gerzhikov V S et al, 1980 Quadratic Bundle and Nonlinear Equations, Theoret. and Math. Phys. 44, No. 3, 784–795.
Kaup D J and Newell A C, 1978 An exact solution for a derivative nonlinear Schrodinger equation, J. Math. Phys. 19 (4), 789–801.
Kaup D J and Newell A C, 1977 On the Coleman correspondence and the solution of the Massive Thirring Model, Lettere Al Nuovo Cimento, Vol. 20, N. 9, 325–331.
Kuznetsov E A and Mikhailov A V, 1977 On the complete integrability of the two—dimensional classical Thirring Model, Theoret. Math. Physics, Vol. 30, No. 3, 193–200.
Lee J H, 1983 Analytic properties of Zakharov—Shabat inverse scattering problem with polynomial spectral dependence of degree 1 in the potential, Ph. D. Dissertation, Yale University.
Lee J H, 1984 A Zakharov—Shabat inverse scattering problem and the associated evolution equations, Chinese J. of Mathematics, Vol. 12, No. 4, 223–233.
Lee J H, 1986 Analytic properties of a Zakharov—Shabat inverse scattering problem (I), Chinese J. of Mathematics, vol. 14, No. 4, 225–248.
Lee J H, 1988 Analytic properties of a Zakharov—Shabat inverse scattering problem II), Chinese J. of Mathematics, Vol. 16, No. 2, 81–110.
Lee J H, 1989 Global solvability of the derivative nonlinear Schrodinger equation, Transations of the American Mathematical Society, Vol. 314, No. 1, 107–118.
Lee J H On the dissipative evolution equations associated with the Zakharov—Shabat system with a quadratic spectral parameter, to appear in the Transactions of the American Mathematical Society.
Mio et al, 1976 Modified nonlinear Schrodinger equation for Alfven waves propagating along the magnetic field in cold plasmas, J. Phys Soc. Japan 41, 265–271.
Mjlhus E, 1976 On the modulational instability of hydromagnetic waves parallel tote magnetic field, J. Plasma Physics, vol. 16, part 3, 321–334.
Morris H C and Dodd R K, 1979 The two component derivative nonlinear Schrodinger equation, physica Scripta 20, 505–508.
Sasaki R, 1982 Canonical structure of soliton equations, II. The Kaup—Newell system, Physica 5D, 66–74.
Zakharov V E and Shabat A B, 1972 A refined theory of two—dimensional self—focussing and one-dimensional self—modulation of waves in non—linear media, Soviet Physics JETP 34, 62–69.
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© 1990 Springer-Verlag Berlin, Heidelberg
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Lee, JH. (1990). A N × N Zakharov-Shabat System with a Quadratic Spectral Parameter. In: Carillo, S., Ragnisco, O. (eds) Nonlinear Evolution Equations and Dynamical Systems. Research Reports in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-84039-5_13
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DOI: https://doi.org/10.1007/978-3-642-84039-5_13
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-51983-6
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