Quadratic bundle and nonlinear equations
Teoretičeskaâ i matematičeskaâ fizika, Tome 44 (1980) no. 3, pp. 342-357
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A class of nonlinear evolution equations solvable by means of the inverse scattering problem method for the quadratic bundle
$$
L_\lambda\psi=\left[i\begin{pmatrix}10\\0-1\end{pmatrix}\frac{d}{dx}+\lambda\begin{pmatrix}0(x)\\p(x)0\end{pmatrix}-\lambda^2\right]\psi(x,\lambda)=0
$$
is described. It is shown that all the equations from this class are completely integrable hamiltonian systems; the corresponding “action-angle” variables are explicitly calculated. For $q=\varepsilon p^*$, $\varepsilon=\pm1$ this class contains such physically interesting equations as the modified nonlinear Schrödinger equation ($iq_t+q_{xx}-i\varepsilon(q^2q^*)_x=0$), the massive Thirring model and others.
@article{TMF_1980_44_3_a4,
author = {V. S. Gerdjikov and M. I. Ivanov and P. P. Kulish},
title = {Quadratic bundle and nonlinear equations},
journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
pages = {342--357},
publisher = {mathdoc},
volume = {44},
number = {3},
year = {1980},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TMF_1980_44_3_a4/}
}
TY - JOUR AU - V. S. Gerdjikov AU - M. I. Ivanov AU - P. P. Kulish TI - Quadratic bundle and nonlinear equations JO - Teoretičeskaâ i matematičeskaâ fizika PY - 1980 SP - 342 EP - 357 VL - 44 IS - 3 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/TMF_1980_44_3_a4/ LA - ru ID - TMF_1980_44_3_a4 ER -
V. S. Gerdjikov; M. I. Ivanov; P. P. Kulish. Quadratic bundle and nonlinear equations. Teoretičeskaâ i matematičeskaâ fizika, Tome 44 (1980) no. 3, pp. 342-357. http://geodesic.mathdoc.fr/item/TMF_1980_44_3_a4/