Integral preserving discretization of 2D Toda lattices. (English) Zbl 1523.37073
This paper describes a method to develop a semi-discrete analog of two-dimensional Toda lattices, based on Cartan matrices of simple Lie algebras, as proposed in [I. Habibullin et al., J. Phys. A, Math. Theor. 44, No. 46, Article ID 465202, 20 p. (2011; Zbl 1230.37085)]. This kind of discretization is based on the idea of Darboux integrability.
In the continuous case, generalized Toda lattices are known to be Darboux integrable, and so admit complete families of characteristic integrals in both directions. Here the author proves that semi-discrete analogs of Toda lattices that are associated with Cartan matrices of all simple Lie algebras are Darboux integrable. The author also shows that if a function is a characteristic integral in the continuous case, then the same function is also a characteristic integral in the semi-discrete case.
After a review of the concepts of Darboux integrability, characteristic algebras, and the relationships between them, the author describes Habibullin’s work and justifies Habibullin’s integral-preserving discretization method. The last part of the paper proves the existence of a complete family of independent \(x\)-integrals for semi-discrete exponential systems that correspond to the Cartan matrices of all simple Lie algebras. Since some of the basic terminology and notation are not provided in the paper, a potential reader would be advised to review at least the cited paper of Habibullin first.
In the continuous case, generalized Toda lattices are known to be Darboux integrable, and so admit complete families of characteristic integrals in both directions. Here the author proves that semi-discrete analogs of Toda lattices that are associated with Cartan matrices of all simple Lie algebras are Darboux integrable. The author also shows that if a function is a characteristic integral in the continuous case, then the same function is also a characteristic integral in the semi-discrete case.
After a review of the concepts of Darboux integrability, characteristic algebras, and the relationships between them, the author describes Habibullin’s work and justifies Habibullin’s integral-preserving discretization method. The last part of the paper proves the existence of a complete family of independent \(x\)-integrals for semi-discrete exponential systems that correspond to the Cartan matrices of all simple Lie algebras. Since some of the basic terminology and notation are not provided in the paper, a potential reader would be advised to review at least the cited paper of Habibullin first.
Reviewer: William J. Satzer Jr. (St. Paul)
MSC:
| 37J70 | Completely integrable discrete dynamical systems |
| 37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |
| 37J37 | Relations of finite-dimensional Hamiltonian and Lagrangian systems with Lie algebras and other algebraic structures |
| 37M15 | Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems |
| 39A36 | Integrable difference and lattice equations; integrability tests |
Keywords:
two-dimensional Toda lattice; Darboux integrability; discretization; characteristic algebraCitations:
Zbl 1230.37085References:
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