Geodesic
An integrable class of differential equations with nonlocal nonlinearity on Lie groups
Teoretičeskaâ i matematičeskaâ fizika, Tome 161 (2009) no. 3, pp. 332-345 Cet article a éte moissonné depuis la source Math-Net.Ru

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We construct the general and $N$-soliton solutions of an integro-differential Schrödinger equation with a nonlocal nonlinearity. We consider integrable nonlinear integro-differential equations on the manifold of an arbitrary connected unimodular Lie group. To reduce the equations on the group to equations with a smaller number of independent variables, we use the method of orbits in the coadjoint representation and the generalized harmonic analysis based on it. We demonstrate the capacities of the algorithm with the example of the $SO(3)$ group.
Keywords: nonlinear integro-differential equation, harmonic analysis.
Mots-clés : soliton, Lie group, coadjoint representation 
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     title = {An~integrable class of differential equations with nonlocal nonlinearity {on~Lie} groups},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
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M. M. Goncharovskiy; I. V. Shirokov. An integrable class of differential equations with nonlocal nonlinearity on Lie groups. Teoretičeskaâ i matematičeskaâ fizika, Tome 161 (2009) no. 3, pp. 332-345. http://geodesic.mathdoc.fr/item/TMF_2009_161_3_a3/

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

[2] A. A. Kirillov, Elementy teorii predstavlenii, Nauka, M., 1978 | MR | MR | Zbl

[3] I. V. Shirokov, $K$-orbity, garmonicheskii analiz na odnorodnykh prostranstvakh i integrirovanie differentsialnykh uravnenii, OmGU, Omsk, 1998

[4] A. V. Shapovalov, I. V. Shirokov, TMF, 104:2 (1995), 195–213 | DOI | MR | Zbl

[5] S. P. Baranovskii, V. V. Mikheev, I. V. Shirokov, TMF, 129:1 (2001), 3–13 | DOI | MR | Zbl

[6] V. V. Belov, A. Yu. Trifonov, A. V. Shapovalov, TMF, 130:3 (2002), 460–492 | DOI | MR | Zbl

[7] I. Bryuning, S. Yu. Dobrokhotov, R. V. Nekrasov, A. I. Shafarevich, TMF, 155:2 (2008), 215–235 | DOI | MR | Zbl

[8] Zh. Diksme, Universalnye obertyvayuschie algebry, Mir, M., 1978 | MR | MR | Zbl

[9] I. V. Shirokov, TMF, 123:3 (2000), 407–423 | DOI | MR | Zbl