Geodesic
Model equation of the theory of solitons
Teoretičeskaâ i matematičeskaâ fizika, Tome 153 (2007) no. 1, pp. 29-45

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We consider the hierarchy of integrable $(1+2)$-dimensional equations related to the Lie algebra of vector fields on the line. We construct solutions in quadratures that contain $n$ arbitrary functions of a single argument. A simple equation for the generating function of the hierarchy, which determines the dynamics in negative times and finds applications to second-order spectral problems, is of main interest. Considering its polynomial solutions under the condition that the corresponding potential is regular allows developing a rather general theory of integrable $(1+1)$-dimensional equations.
Keywords: hierarchy of commuting vector fields
Mots-clés : Riemann invariant, Dubrovin equations. 
V. E. Adler; A. B. Shabat. Model equation of the theory of solitons. Teoretičeskaâ i matematičeskaâ fizika, Tome 153 (2007) no. 1, pp. 29-45. http://geodesic.mathdoc.fr/item/TMF_2007_153_1_a2/
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[1] L. Martínez Alonso, A. B. Shabat, Phys. Lett. A, 300:1 (2002), 58–64 ; J. Nonlinear Math. Phys., 10:2 (2003), 229–242 ; Л. Мартинес Алонсо, А. Б. Шабат, ТМФ, 140:2 (2004), 216–229 ; A. B. Shabat, L. Martínez Alonso, “On the prolongation of a hierarchy of hydrodynamic chains”, New Trends in Integrability and Partial Solvability (Cadiz, Spain, 2002), NATO Sci. Ser. II. Math. Phys. Chem., 132, eds. A. B. Shabat et al., Kluwer, Dordrecht, 2004, 263–280 | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | MR | Zbl

[2] E. V. Ferapontov, K. R. Khusnutdinova, M. V. Pavlov, TMF, 144:1 (2005), 35–43 ; E. V. Ferapontov, K. R. Khusnutdinova, S. P. Tsarev, Commun. Math. Phys., 261:1 (2006), 225–243 | DOI | MR | Zbl | DOI | MR | Zbl

[3] E. V. Ferapontov, Phys. Lett. A, 158:3–4 (1991), 112–118 | DOI | MR

[4] A. B. Shabat, J. Nonlinear Math. Phys., 12: Suppl. 1 (2005), 614–624 | DOI | MR

[5] N. Kh. Ibragimov, A. B. Shabat, DAN SSSR, 244:1 (1979), 57–61 | MR | Zbl

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

[7] A. Hone, V. S. Novikov, C. Verhoeven, Inverse Problems, 22:6 (2006), 2001–2020 | DOI | MR | Zbl