Cartan matrices and integrable lattice Toda field equations

Habibullin, Ismagil
Zheltukhın, Kostyantyn
Yangubaeva, Marina
Differential-difference integrable exponential-type systems are studied corresponding to the Cartan matrices of semi-simple or affine Lie algebras. For the systems corresponding to the algebras A(2), B(2), C(2), G(2), the complete sets of integrals in both directions are found. For the simple Lie algebras of the classical series A(N), B(N), C(N) and affine algebras of series D(N)((2)), the corresponding systems are supplied with the Lax representation.


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Using the algebraic method of Gardner's deformations for completely integrable systems, we construct recurrence relations for densities of the Hamiltonians for the Boussinesq and the Kaup-Boussinesq equations. By extending the Magri schemes for these equations, we obtain new integrable systems adjoint with respect to the initial ones and describe their Hamiltonian structures and symmetry properties.
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A systematic construction of the Green's matrix for a second-order self-adjoint matrix differential operator from the linearly independent solutions of the corresponding homogeneous differential equation set is carried out. We follow the general approach of extracting the Green's matrix from the Green's matrix of the corresponding first-order system. This construction is required in the cases where the differential equation set cannot be turned to an algebraic equation set via transform techniques.
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We first consider the Hamiltonian formulation of n=3 systems, in general, and show that all dynamical systems in R-3 are locally bi-Hamiltonian. An algorithm is introduced to obtain Poisson structures of a given dynamical system. The construction of the Poisson structures is based on solving an associated first order linear partial differential equations. We find the Poisson structures of a dynamical system recently given by Bender et al. [J. Phys. A: Math. Theor. 40, F793 (2007)]. Secondly, we show that al...
On the integrability of a class of Monge-Ampere equations
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We give the Lax representations for the elliptic, hyperbolic and homogeneous second order Monge-Ampere equations. The connection between these equations and the equations of hydrodynamical type give us a scalar dispersionless Lax representation. A matrix dispersive Lax representation follows from the correspondence between sigma models, a two parameter equation for minimal surfaces and Monge-Ampere equations. Local as well nonlocal conserved densities are obtained.
Citation Formats
I. Habibullin, K. Zheltukhın, and M. Yangubaeva, “Cartan matrices and integrable lattice Toda field equations,” JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, pp. 0–0, 2011, Accessed: 00, 2020. [Online]. Available: