On the construction and integration of a hierarchy for the periodic Toda lattice with a self-consistent source

B. A. Babajanov   ; M. M. Ruzmetov

The Bulletin of Irkutsk State University. Series Mathematics, Tome 38 (2021), pp. 3-18

Résumé

In this paper, it is derived a rich hierarchy for the Toda lattice with a self-consistent source in the class of periodic functions. We discuss the complete integrability of the constructed systems that is based on the transformation to the spectral data of an associated discrete Hill`s equation with periodic coefficients. In particular, Dubrovin-type equations are derived for the time-evolution of the spectral data corresponding to the solutions of any system in the hierarchy. At the end of the paper, we illustrate our theory on concrete example with analytical and numerical results.

Détail

Keywords: periodic Toda lattice hierarchy, Hill's equation, self-consistent source, inverse spectral problem, trace formulas.

Bibliographie

[1] Babajanov B. A., Feckan M., Urazbaev G. U., “On the periodic Toda Lattice hierarchy with an integral source”, Commun. Sci. Numer. Simul., 52 (2017), 110–123 | DOI | Zbl

[2] Babajanov B. A., Feckan M., Urazbaev G. U., “On the periodic Toda Lattice with self-consistent source”, Commun. Sci. Numer. Simul., 22 (2015), 379–388 | DOI

[3] Babajanov B. A., “Integration of the Toda-Type Chain with a special self-consistent source”, Algebra, complex analysis and pluripotential theory, v. 4, Springer, 2018, 1–13 | DOI

[4] Babajanov B. A., Khasanov A. B., “Periodic Toda chain an integral source”, Theoret. and Math. Phys., 184 (2015), 1114–1128 | DOI | Zbl

[5] Bulla W., Gesztesy F., Holden H., Teschl G., “Algebro-Geometric Quasi-Periodic Finite-Gap Solutions of the Toda and Kac-van Moerbeke Hierarchies”, Memoirs of the Amer. Math. Soc., 1998, 135–641 | DOI

[6] Cabada A., Urazboev G. U., “Integration of the Toda lattice with an integral-type source”, Inverse Problems, 26 (2010), 085004 | DOI | Zbl

[7] Claude C., Latifi A., Leon J., “Nonliear resonant scattering and plasma instability: an integrable model”, J. Math. Phys., 32 (1991), 3321–3330 | DOI | Zbl

[8] Chvartatskyi O., Dimakis A., Muller-Hoissen F., “Self-Consistent Sources for Integrable Equations Via Deformations of Binary Darboux Transformations”, Lett. Math. Phys., 106 (2016), 1139–1179 | DOI | Zbl

[9] Date E., Tanaka S., “Analog of inverse scattering theory for discrete Hill's equation and exact solutions for the periodic Toda lattice”, Progress Theor. Physics, 55 (1976), 217–222 | DOI

[10] David C., Niels G. J., Bishop A. R., Findikoglu A. T., Reago D., “A perturbed Toda lattice model for low loss nonlinear transmission lines”, Phys. D: Nonlinear Phenom., 123 (1998), 291–300 | DOI

[11] Dubrovin B. A., Matveev V. B., Novikov S. P., “Nonlinear equations of Korteweg-de Vries type, finite-zone linear operators, and Abelian varieties”, Uspekhi Mat. Nauk, 31:1 (1976), 55–136 | DOI | Zbl

[12] Flaschka H., “On the Toda lattice. II”, Progress Theor. Physics, 51 (1974), 703–716 | DOI | Zbl

[13] Garnier J., Abdullaev F.Kh., “Soliton dynamics in a random Toda chain”, Phys. Rev. E, 67 (2003), 026609, 1 pp. | DOI

[14] Grinevich P. G., Taimanov I. A., “Spectral conservation laws for periodic nonlinear equations of the Melnikov type”, Amer. Math. Soc. Transl. Ser. 2, 224 (2008), 125–138 | Zbl

[15] Hochstadt H., “On the theory of Hill's matrices and related inverse spectral problems”, Linear Algebra Appl., 11 (1975), 41–52 | DOI | Zbl

[16] Hurwitz A., Courant R., Vorlesungen uber allgemeine Funktionentheorie und elliptische Funktionen, Springer, Berlin, 1964 | Zbl

[17] Kac M., van Moerbeke P., “A complete solution of the periodic Toda problem”, Proc. Nat. Acad. Sci. USA, 72 (1975), 1627–1629 | DOI | Zbl

[18] Krichever I. M., “Algebraic curves and non-linear difference equations”, Uspekhi Mat. Nauk, 33:4 (1978), 215–216 | DOI | Zbl

[19] Leon J., Latifi A., “Solution of an initial-boundary value problem for coupled nonlinear waves”, J. Phys.A. Math. Gen., 23 (1990), 1385–403 | DOI

[20] Lin R., Du Y., “Generalized Darboux Transformation for the Discrete Kadomtsev-Petviashvili Equation with Self-Consistent Sources”, Theor. Math. Phys., 196 (2018), 1320–133 | DOI

[21] Liu X., Zeng Y., “On the Toda lattice equation with self-consistent sources”, J. Phys. A: Math. Gen., 38 (2005), 8951–8965 | DOI | Zbl

[22] Lou S. Y., Tang X. Y., Method of Nonlinear Mathematical Physics, Science Press, Beijing, 2006

[23] Mel`nikov V. K., “Integration of the Korteweg-de Vries equation with a source”, Inverse Problems, 6 (1990), 233–246 | DOI | Zbl

[24] Mel'nikov V. K., “A direct method for deriving a multisoliton solution for the problem of interaction of waves on the x,y plane”, Commun. Math. Phys., 112 (1987), 639–52 | DOI

[25] Mel'nikov V. K., “Integration of the nonlinear Schroedinger equation with a self-consistent source”, Commun. Math. Phys., 137 (1991), 359–381 | DOI

[26] Muto V., Scott A. C., Christiansen P. L., “Thermally generated solitons in a Toda lattice model of DNA”, Physics Letters A, 136 (1989), 33–36 | DOI

[27] Reyimberganov A. A., Rakhimov I. D., “The Soliton Solutions for the Nonlinear Schrodinger Equation with Self-consistent Sources”, The Bulletin of Irkutsk State University. Series Mathematics, 35 (2021), 84–94 | DOI

[28] Shchesnovich V. S., Doktorov E. V., “Modified Manakov system with self-consistent source”, Phys. Lett. A, 213 (1996), 23–31 | DOI | Zbl

[29] Toda M., “Waves in nonlinear lattice. Suppl”, Progress Theor. Physics, 45 (1970), 74–200 | DOI

[30] Urazboev G. U., “Toda lattice with a special self-consistent source”, Theoret. and Math. Phys., 154 (2008), 305–315 | DOI | Zbl

[31] Urazboev G., “Integrating the Toda Lattice with Self-Consistent Source via Inverse Scattering Method”, Math. Phys. Anal. Geom., 15 (2012), 401–412 | DOI | Zbl

[32] Yakhshimuratov A. B., Babajanov B. A., “Integration of equations of Kaup system kind with self-consistent source in class of periodic functions”, Ufa Mathematical Journal, 12 (2020), 103–113 | DOI | Zbl