Perturbation theory for the nonlinear Schr\"odinger equation in the solitonless sector
Teoretičeskaâ i matematičeskaâ fizika, Tome 51 (1982) no. 1, pp. 34-43
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The nonlinear Schrödinger equation with a perturbation of polynomial type is considered.
A perturbation theory in the solitonless situation is developed on the basis of the perturbed
equations of motion for canonical variables constructed from scattering data. It is shown
that closed equations of the perturbation theory containing only canonical variables can be
obtained in the case of a “quasi-asymptotic” initial condition. The cases in which these
equations can be solved iteratively are established. Also considered is the ease of an initial condition with spectrum cut off in the “infrared” region. In this case, averaging
over the rapid unperturbed motions makes it possible to reduce the equations of the
perturbation theory to a closed form as well. A solution to these equations is obtained
in implicit form.
@article{TMF_1982_51_1_a3,
author = {B. A. Malomed},
title = {Perturbation theory for the nonlinear {Schr\"odinger} equation in the solitonless sector},
journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
pages = {34--43},
publisher = {mathdoc},
volume = {51},
number = {1},
year = {1982},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TMF_1982_51_1_a3/}
}
TY - JOUR AU - B. A. Malomed TI - Perturbation theory for the nonlinear Schr\"odinger equation in the solitonless sector JO - Teoretičeskaâ i matematičeskaâ fizika PY - 1982 SP - 34 EP - 43 VL - 51 IS - 1 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/TMF_1982_51_1_a3/ LA - ru ID - TMF_1982_51_1_a3 ER -
B. A. Malomed. Perturbation theory for the nonlinear Schr\"odinger equation in the solitonless sector. Teoretičeskaâ i matematičeskaâ fizika, Tome 51 (1982) no. 1, pp. 34-43. http://geodesic.mathdoc.fr/item/TMF_1982_51_1_a3/