Abstract
We assign some kind of invariant manifolds to a given integrable PDE (its discrete or semidiscrete variant). First, we linearize the equation around its arbitrary solution u. Then we construct a differential (respectively, difference) equation compatible with the linearized equation for any choice of u. This equation defines a surface called a generalized invariant manifold. In a sense, the manifold generalizes the symmetry, which is also a solution to the linearized equation. In this paper, we concentrate on continuous and discrete models of hyperbolic type. It is known that such kinds of equations have two hierarchies of symmetries, corresponding to the characteristic directions. We have shown that a properly chosen generalized invariant manifold allows one to construct recursion operators that generate these symmetries. It is surprising that both recursion operators are related to different parametrizations of the same invariant manifold. Therefore, knowing one of the recursion operators for the hyperbolic type integrable equation (having no pseudo-constants) we can immediately find the second one.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “Method for solving the sine-Gordon equation,” Phys. Rev. Lett., 30, No. 25, 1262–1264 (1973).
A. I. Bobenko and Yu. B. Suris, “Integrable systems on quad-graphs,” Int. Math. Res. Notices., No. 11, 573–611 (2002).
I. T. Habibullin and A. R. Khakimova, “On a method for constructing the Lax pairs for integrable models via a quadratic ansatz,” J. Phys. A: Math. Theor., 50, No. 30, 305206 (2017).
I. T. Habibullin and A. R. Khakimova, “Invariant manifolds and Lax pairs for integrable nonlinear chains,” Teor. Mat. Fiz., 191, No. 3, 369–388 (2017).
I. T. Habibullin, A. R. Khakimova, and M. N. Poptsova, “On a method for constructing the Lax pairs for nonlinear integrable equations,” J. Phys. A: Math. Theor., 49, No. 3, 1–35, 035202 (2016).
R. Hirota and S. Tsujimoto, “Conserved quantities of a class of nonlinear difference-difference equations,” J. Phys. Soc. Jpn., 64, No. 9, 3125–3127 (1995).
N. Kh. Ibragimov and A. B. Shabat, “Evolution equations admitting a nontrivial Lie–Backlund group,” Funkts. Anal. Appl., 14, No. 1, 25–36 (1980).
P. D. Lax, “Integrals of nonlinear equations of evolution and solitary waves,” Commun. Pure Appl. Math., 21, No. 5, 467–490 (1968).
A. G. Meshkov and V. V. Sokolov, “Hyperbolic equations with third-order symmetries,” Teor. Mat. Fiz., 166, No. 1, 51–67 (2011).
F. Nijhoff and H. Capel, “The discrete Korteweg–de Vries equation,” Acta Appl. Math., 39, No. 1– 3, 133–158 (1995).
F. W. Nijhoff and A. J. Walker, “The discrete and continuous Painlev´e VI hierarchy and the Garnier system,” Glasgow Math. J., 43A, 109–123 (2001).
A. K. Svinin, “On some integrable lattice related by the Miura-type transformation to the Itoh–Narita–Bogoyavlenskii lattice,” J. Phys. A: Math. Theor., 44, No. 46, 465210 (2011).
S. I. Svinolupov and V. V. Sokolov, “Evolution equations with nontrivial conservative laws,” Funkts. Anal. Appl., 16, No. 4, 86–87 (1982).
A. Tongas, D. Tsoubelis, V. Papageorgiou, “Symmetries and group invariant reductions of integrable partial difference equations,”, 222–230 (2005).
H. D. Wahlquist and F. B. Estabrook, “Prolongation structures of nonlinear evolution equations,” J. Math. Phys., 16, No. 1, 1–7 (1975).
P. Xenitidis, “Integrability and symmetries of difference equations: The Adler–Bobenko–Suris case,” e-print ArXiV.0902.3954
Yamilov R. I., “On classification of discrete equqtions”, Integrable systems, Ufa, (1982), 95–114.
V. E. Zakharov and A. B. Shabat, “Integration of nonlinear equations of mathematical physics by the method of inverse scattering, I,” Funkts. Anal. Appl., 8, No. 3, 43–53 (1974).
V. E. Zakharov and A. B. Shabat, “Integration of nonlinear equations of mathematical physics by the method of inverse scattering, II,” Funkts. Anal. Appl., 13, No. 3, 13–22 (1979).
A. V. Zhiber and V. V. Sokolov, “Exactly integrable hyperbolic equations of Liouville type,” Usp. Mat. Nauk, 56, No. 1 (337), 63–106 (2001).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 162, Complex Analysis. Mathematical Physics, 2019.
Rights and permissions
About this article
Cite this article
Habibullin, I.T., Khakimova, A.R. Invariant Manifolds of Hyperbolic Integrable Equations and their Applications. J Math Sci 257, 410–423 (2021). https://doi.org/10.1007/s10958-021-05491-3
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-021-05491-3