Geodesic
Nonlocal reductions of the~multicomponent nonlinear Schr\"odinger equation on symmetric spaces
Teoretičeskaâ i matematičeskaâ fizika, Tome 197 (2018) no. 1, pp. 45-67

Voir la notice de l'article provenant de la source Math-Net.Ru

Our aim is to develop the inverse scattering transform for multicomponent generalizations of nonlocal reductions of the nonlinear Schrödinger (NLS) equation with $\mathcal{PT}$ symmetry related to symmetric spaces. This includes the spectral properties of the associated Lax operator, the Jost function, the scattering matrix, the minimum set of scattering data, and the fundamental analytic solutions. As main examples, we use the Manakov vector Schrödinger equation (related to A.III-symmetric spaces) and the multicomponent NLS (MNLS) equations of Kullish–Sklyanin type (related to BD.I-symmetric spaces). Furthermore, we obtain one- and two-soliton solutions using an appropriate modification of the Zakharov–Shabat dressing method. We show that the MNLS equations of these types admit both regular and singular soliton configurations. Finally, we present different examples of one- and two-soliton solutions for both types of models, subject to different reductions.
Keywords: integrable system, multicomponent nonlinear Schrödinger equation, Lax representation, Zakharov–Shabat system, $\mathcal{PT}$ symmetry, inverse scattering transform, Riemann–Hilbert problem, dressing method.
Mots-clés : spectral decompositions 
@article{TMF_2018_197_1_a2,
     author = {G. G. Grahovski and A. J. Mustafa and H. Susanto},
     title = {Nonlocal reductions of the~multicomponent nonlinear {Schr\"odinger} equation on symmetric spaces},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {45--67},
     publisher = {mathdoc},
     volume = {197},
     number = {1},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2018_197_1_a2/}
}
TY  - JOUR
AU  - G. G. Grahovski
AU  - A. J. Mustafa
AU  - H. Susanto
TI  - Nonlocal reductions of the~multicomponent nonlinear Schr\"odinger equation on symmetric spaces
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2018
SP  - 45
EP  - 67
VL  - 197
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TMF_2018_197_1_a2/
LA  - ru
ID  - TMF_2018_197_1_a2
ER  - 
%0 Journal Article
%A G. G. Grahovski
%A A. J. Mustafa
%A H. Susanto
%T Nonlocal reductions of the~multicomponent nonlinear Schr\"odinger equation on symmetric spaces
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2018
%P 45-67
%V 197
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TMF_2018_197_1_a2/
%G ru
%F TMF_2018_197_1_a2
G. G. Grahovski; A. J. Mustafa; H. Susanto. Nonlocal reductions of the~multicomponent nonlinear Schr\"odinger equation on symmetric spaces. Teoretičeskaâ i matematičeskaâ fizika, Tome 197 (2018) no. 1, pp. 45-67. http://geodesic.mathdoc.fr/item/TMF_2018_197_1_a2/