Integrable and nonintegrable cases of the Lax equations with a source
Teoretičeskaâ i matematičeskaâ fizika, Tome 99 (1994) no. 3, pp. 471-477
Cet article a éte moissonné depuis la source Math-Net.Ru
The Korteweg–de Vries equation with a source given as a Fourier integral over eigenfunctions of the so-called generating operator is considered. It is shown that depending on the choice of a basis of eigenfunctions we have the following three possibilities: 1) evolution equations for the scattering data are nonintegrable; 2) evolution equations for the scattering data are integrable but the solution of the Cauchy problem for the Korteweg–de Vries equation with a source at some $t'>t_0$ leaves the considered class of functions decreasing rapidly enough as $x\to \pm \infty$; 3) evolution equations for the scattering data are integrable and the solution of the Cauchy problem for the Korteweg–de Vries equation with a source exists at all $t>t_0$. All these possibilities are widespread and occur in other Lax equations with a source.
@article{TMF_1994_99_3_a15,
author = {V. K. Mel'nikov},
title = {Integrable and nonintegrable cases of the {Lax} equations with a source},
journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
pages = {471--477},
year = {1994},
volume = {99},
number = {3},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TMF_1994_99_3_a15/}
}
V. K. Mel'nikov. Integrable and nonintegrable cases of the Lax equations with a source. Teoretičeskaâ i matematičeskaâ fizika, Tome 99 (1994) no. 3, pp. 471-477. http://geodesic.mathdoc.fr/item/TMF_1994_99_3_a15/
[1] V. K. Mel'nikov, “Integration of the Korteweg–de Vries equation with asource”, Inverse Problems, 6:2 (1990), 233–246 | DOI | MR | Zbl
[2] V. K. Mel'nikov, “Creation and annihilation of solitons in the system described by the Korteweg–de Vries equations with a self-consistent source”, Inverse Problems, 6:5 (1990), 809–823 | DOI | MR | Zbl