A N × N Zakharov-Shabat System with a Quadratic Spectral Parameter
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.
Unable to display preview. Download preview PDF.
- 1.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.Google Scholar
- 5.Bullough R K, Caudrey P J (eds), 1980 Solitons, Topics in Current Physics No. 17, Springer—Verlag.Google Scholar
- 7.Gerzhikov V S et al, 1980 Quadratic Bundle and Nonlinear Equations, Theoret. and Math. Phys. 44, No. 3, 784–795.Google Scholar
- 10.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.Google Scholar
- 11.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.Google Scholar
- 16.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.Google Scholar