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Solution of the Finite Toda Lattice with Self-Consistent Source

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Abstract

In the present paper it is showed that the finite Toda lattice with a self-consistent source is also an important theoretical model as it is a completely integrable system. Namely, it will be determined the time evolution of the spectral data for the Jacobi matrix associated with the solution of a finite Toda chain with a self-consistent source. Then, using the solution of the inverse spectral problem with respect to the time-dependent spectral data, we recover the time-dependent Jacobi matrix and the desired solution of the finite Toda chain with a self-consistent source. For the case \(N=2\), explicit formulas for the solution of the problem under consideration are obtained.

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REFERENCES

  1. M. Toda, ‘‘Waves in nonlinear lattice,’’ Progr. Theor. Phys. Suppl. 45, 174–200 (1970).

    Article  Google Scholar 

  2. M. Henon, ‘‘Integrals of the Toda lattice,’’ Phys. Rev. B 9, 1921–1923 (1974).

    Article  MathSciNet  MATH  Google Scholar 

  3. H. Flaschka, ‘‘The Toda lattice, I,’’ Phys. Rev. B 9, 1924–1925 (1974).

    Article  MathSciNet  MATH  Google Scholar 

  4. H. Flaschka, ‘‘The Toda lattice, II,’’ Progr. Theor. Phys. 51, 703–716 (1974).

    Article  MATH  Google Scholar 

  5. S. V. Manakov, ‘‘Complete integrability and stochastization in discrete dynamic media,’’ Sov. Phys. JETP 40, 269 (1974).

    Google Scholar 

  6. M. Kac and P. van Moerbeke, ‘‘On an explicitly soluble system of nonlinear differential equations related to certain Toda lattices,’’ Adv. Math. 16, 160–169 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  7. V. Muto, A. C. Scott, and P. L. Christiansen, ‘‘Thermally generated solitons in a Toda lattice model of DNA,’’ Phys. Lett. A 136, 33–36 (1989).

    Article  Google Scholar 

  8. M. Toda, Theory of Nonlinear Lattices (Springer, New York, 1981).

    Book  MATH  Google Scholar 

  9. Yu. M. Berezanskii, ‘‘The integration of semi-infinite Toda chain by means of inverse spectral problem,’’ Rep. Math. Phys. 24, 21–47 (1986).

    Article  MathSciNet  Google Scholar 

  10. G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices, Vol. 72 of Math. Surveys Monographs (Am. Math. Soc., Providence, RI, 2000).

  11. A. I. Aptekarev and A. Branquinbo, ‘‘Padé approximants and complex high order Toda lattices,’’ J. Comput. Appl. Math. 155, 231–237 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  12. Yu. M. Berezanskii and A. A. Mokhonko, ‘‘Integration of some differential-difference nonlinear equations using the spectral theory of normal block Jacobi matrices,’’ Funct. Anal. Appl. 42, 1–18 (2003).

    Article  Google Scholar 

  13. W. Bulla, F. Gesztesy, H. Holden, and G. Teschl, ‘‘Algebro-geometric quasi-periodic finite-gap solutions of the Toda and Kac–van Moerbeke hierarchies,’’ Mem. Am. Math. Soc., 135–641 (1998).

  14. Ag. Kh. Khanmamedov, ‘‘The rapidly decreasing solution of the initial-boundary value problem for the Toda lattice,’’ Ukr. Math. J. 57, 1144–1152 (2005).

    MATH  Google Scholar 

  15. V. G. Samoilenko and A. K. Prikarpatskii, ‘‘Periodic problem for a Toda chain,’’ Ukr. Math. J. 34, 380–385 (1982).

    Article  Google Scholar 

  16. J. Garnier and F. Kh. Abdullaev, ‘‘Soliton dynamics in a random Toda chain,’’ Phys. Rev. E 67, 026609-1 (2003).

    Article  Google Scholar 

  17. C. David, G-J. Niels, A. R. Bishop, A. T. Findikoglu, and D. Reago, ‘‘A perturbed Toda lattice model for low loss nonlinear transmission lines,’’ Phys. D (Amsterdam, Neth.) 123, 291–300 (1998).

  18. J. Moser, ‘‘Three integrable Hamiltonian systems connected with isospectral deformations,’’ Adv. Math. 16, 197–220 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  19. M. I. Gekhtman, ‘‘Non-Abelian nonlinear lattice equations on finite interval,’’ J. Phys. A: Math. Gen. 26, 6303–6317 (1993)

    Article  MATH  Google Scholar 

  20. Yu. M. Berezansky, M. I. Gekhtman, and M. E. Shmoish, ‘‘Integration of some chains of nonlinear difference equations by the method of the inverse spectral problem,’’ Ukr. Mat. Zh. 38, 74–78 (1986).

    Article  MATH  Google Scholar 

  21. A. Huseynov and Sh. Guseinov, ‘‘Solution of the finite complex Toda lattice by the method of inverse spectral problem,’’ Appl. Math. Comput. 219, 5550–5563 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  22. H. Bereketoglu and A. Huseynov, ‘‘Boundary value problems for nonlinear second-order difference equations with impulse,’’ Appl. Math. Mech. 30, 1045–1054 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  23. V. K. Melnikov, ‘‘A direct method for deriving a multisoliton solution for the problem of interaction of waves on the x,y plane,’’ Commun. Math. Phys. 112, 639–652 (1987).

    Article  MathSciNet  Google Scholar 

  24. V. K. Melnikov, ‘‘Integration method of the Korteweg–de Vries equation with a self-consistent source,’’ Phys. Lett. A 133, 493–496 (1988).

    Article  MathSciNet  Google Scholar 

  25. V. K. Melnikov, ‘‘Integration of the nonlinear Schroedinger equation with a self-consistent source,’’ Commun. Math. Phys. 137, 359–381 (1991).

    Article  MathSciNet  Google Scholar 

  26. V. K. Melnikov, ‘‘Integration of the Korteweg–de Vries equation with a source,’’ Inverse Probl. 6, 233–246 (1990).

    Article  MathSciNet  Google Scholar 

  27. J. Leon and A. Latifi, ‘‘Solution of an initial-boundary value problem for coupled nonlinear waves,’’ J. Phys. A: Math. Gen. 23, 1385–1403 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  28. C. Claude, A. Latifi, and J. Leon, ‘‘Nonlinear resonant scattering and plasma instability: An integrable model,’’ J. Math. Phys. 32, 3321–3330 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  29. V. S. Shchesnovich and E. V. Doktorov, ‘‘Modified Manakov system with self-consistent source,’’ Phys. Lett. A 213, 23–31 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  30. B. A. Babajanov, A. K. Babadjanova, and A. S. Azamatov, ‘‘Integration of the differential-difference sine-Gordon equation with a self-consistent source,’’ Theor. Math. Phys. 210, 327–336 (2022).

    Article  MathSciNet  MATH  Google Scholar 

  31. X. Liu and Y. Zeng, ‘‘On the Toda lattice equation with self-consistent sources,’’ J. Phys. A: Math. Gen. 38, 8951–8965 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  32. A. Cabada and G. U. Urazboev, ‘‘Integration of the Toda lattice with an integral-type source,’’ Inverse Probl. 26, 085004 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  33. G. U. Urazboev, ‘‘Toda lattice with a special self-consistent source,’’ Theor. Math. Phys. 154, 305–315 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  34. G. U. Urazboev, ‘‘Integrating the Toda lattice with self-consistent source via inverse scattering method,’’ Math. Phys., Anal. Geom. 15, 401–412 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  35. B. A. Babajanov, M. Feckan, and G. U. Urazbayev, ‘‘On the periodic Toda Lattice with self-consistent source,’’ Commun. Sci. Numer. Simul. 22, 1223–1234 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  36. B. A. Babajanov and A. B. Khasanov, ‘‘Periodic Toda chain with an integral source,’’ Theor. Math. Phys. 184, 1114–1128 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  37. B. A. Babajanov, M. Fečkan, and G. U. Urazbayev, ‘‘On the periodic Toda Lattice hierarchy with an integral source,’’ Commun. Sci. Numer. Simul. 52, 110–123 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  38. B. A. Babajanov and M. M. Ruzmetov, ‘‘On the construction and integration of a Hierarchy for the periodic Toda lattice with a self-consistent source,’’ Vestn. Irkut. Univ., Ser. Mat. 38, 3–18 (2021).

    MATH  Google Scholar 

  39. G. Sh. Guseinov, ‘‘Inverse spectral problems for tridiagonal \(N\) by \(N\) complex Hamiltonians,’’ SIGMA 5 (018), 18–46 (2009).

    MATH  Google Scholar 

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Authors and Affiliations

  1. Urgench State University, 220100, Urgench, Uzbekistan

    B. A. Babajanov & M. M. Ruzmetov

  2. Romanovskii Institute of Mathematics, Academy of Sciences of Uzbekistan, 100174, Tashkent, Uzbekistan

    B. A. Babajanov

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  1. B. A. Babajanov
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  2. M. M. Ruzmetov
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Correspondence to B. A. Babajanov or M. M. Ruzmetov.

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(Submitted by T. K. Yuldashev)

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Babajanov, B.A., Ruzmetov, M.M. Solution of the Finite Toda Lattice with Self-Consistent Source. Lobachevskii J Math 44, 2587–2600 (2023). https://doi.org/10.1134/S1995080223070089

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  • DOI: https://doi.org/10.1134/S1995080223070089

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