Geodesic
Coulomb Gas Representation for Rational Solutions of the Painlevé Equations
Teoretičeskaâ i matematičeskaâ fizika, Tome 127 (2001) no. 2, pp. 284-303 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We consider rational solutions for a number of dynamic systems of the type of the nonlinear Schrödinger equation, in particular, the Levi system. We derive the equations for the dynamics of poles and Bäcklund transformations for these solutions. We show that these solutions can be reduced to rational solutions of the Painlevé IV equation, with the equations for the pole dynamics becoming the stationary equations for the two-dimensional Coulomb gas in a parabolic potential. The corresponding Coulomb systems are derived for the Painlevé II-VI equations. Using the Hamiltonian formalism, we construct the spin representation of the Painlevé equations. 
@article{TMF_2001_127_2_a3,
     author = {V. G. Marikhin},
     title = {Coulomb {Gas} {Representation} for {Rational} {Solutions} of the {Painlev\'e} {Equations}},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {284--303},
     year = {2001},
     volume = {127},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2001_127_2_a3/}
}
TY  - JOUR
AU  - V. G. Marikhin
TI  - Coulomb Gas Representation for Rational Solutions of the Painlevé Equations
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2001
SP  - 284
EP  - 303
VL  - 127
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_2001_127_2_a3/
LA  - ru
ID  - TMF_2001_127_2_a3
ER  - 
%0 Journal Article
%A V. G. Marikhin
%T Coulomb Gas Representation for Rational Solutions of the Painlevé Equations
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2001
%P 284-303
%V 127
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_2001_127_2_a3/
%G ru
%F TMF_2001_127_2_a3
V. G. Marikhin. Coulomb Gas Representation for Rational Solutions of the Painlevé Equations. Teoretičeskaâ i matematičeskaâ fizika, Tome 127 (2001) no. 2, pp. 284-303. http://geodesic.mathdoc.fr/item/TMF_2001_127_2_a3/

[1] E. L. Ains, Obyknovennye differentsialnye uravneniya, DNTVU, Kharkov, 1939 | MR

[2] M. Boiti, F. Pempinelli, Nuovo Cimento B, 59 (1980), 40–58 | DOI | MR

[3] V. G. Marikhin, A. B. Shabat, M. Boiti, F. Pempinelli, ZhETF, 117:3 (2000), 634–643 | MR

[4] M. Ablovits, Kh. Sigur, Solitony i metod obratnoi zadachi, Mir, M., 1987 | MR

[5] V. G. Marikhin, A. B. Shabat, TMF, 118:2 (1999), 217–228 | DOI | MR | Zbl

[6] V. E. Adler, A. B. Shabat, R. I. Yamilov, TMF, 125:3 (2000), 355–424 | DOI | MR | Zbl

[7] V. E. Zakharov, S. V. Manakov, S. P. Novikov, L. P. Pitaevskii, Teoriya solitonov. Metod obratnoi zadachi, Nauka, M., 1980 | MR | Zbl

[8] K. Okamoto, Proc. Japan Acad. Ser. A, 56:6 (1980), 264–268 | DOI | MR | Zbl

[9] V. E. Adler, Physica D, 73 (1994), 335–351 | DOI | MR | Zbl