Related papers: Linear integral equations and two-dimensional Toda…
Discrete analogs of the finite and affine Toda field equations are found corresponding to the Lie algebras of series $C_N$ and $\tilde{C_N}$. Their Lax pairs are represented.
We define an integrable hamiltonian system of Toda type associated with the real Lie algebra $\so{p}{q}$. As usual there exists a periodic and a non-periodic version. We construct, using the root space, two Lax pair representations and the…
For each one of the Lie algebras $\mathfrak{gl}_{n}$ and $\widetilde {\mathfrak{gl}}_{n}$, we constructed a family of integrable generalizations of the Toda chains characterized by two integers $m_{+}$ and $m_{-}$. The Lax matrices and the…
The paper is devoted to real Hamiltonian forms of 2-dimensional Toda field theories related to exceptional simple Lie algebras, and to the spectral theory of the associated Lax operators. Real Hamiltonian forms are a special type of…
The new examples are found of the constraints which link the 1+2-dimensional and multifield integrable equations and lattices. The vector and matrix generalizations of the Nonlinear Schr\"odinger equation and the Ablowitz-Ladik lattice are…
This note constructs completely integrable convex Hamiltonians on the cotangent bundle of certain k-dimensional torus bundles over an l-dimensional torus. A central role is played by the Lax representation of a Bogoyavlenskij-Toda lattice.…
A direct linearisation scheme, based on an elliptic Cauchy kernel, is set up for the lattice CKP equation. This leads to an elliptic parametrisation of the lattice CKP equation, together with its Lax triplet, which allows us to perform…
These lecture notes are devoted to the integrability of discrete systems and their relation to the theory of Yang-Baxter (YB) maps. Lax pairs play a significant role in the integrability of discrete systems. We introduce the notion of Lax…
A new class of infinite-dimensional Lie algebras given a name of Lax operator algebras, and the related unifying approach to finite-dimensional integrable systems with spectral parameter on a Riemann surface, such as Calogero--Moser and…
We explicitly construct the series expansion for a certain class of solutions to the 2D Toda hierarchy in the zero dispersion limit, which we call symmetric solutions. We express the Taylor coefficients through some universal combinatorial…
A systematic analysis of a continuous version of a binomial lattice, containing a real parameter $\gamma$ and covering the Toda field equation as $\gamma\to\infty$, is carried out in the framework of group theory. The symmetry algebra of…
We consider a generalization of the full symmetric Toda hierarchy where the matrix $\tilde {L}$ of the Lax pair is given by $\tilde {L}=LS$, with a full symmetric matrix $L$ and a nondegenerate diagonal matrix $S$. The key feature of the…
The direct linearization structure is presented of a "mild" but significant generalization of the lattice BSQ system. Some of the equations in this system were recently discovered in [J. Hietarinta, J. Phys {\bf A}: Math. Theor. {\bf 44}…
Our goal is to develop a more general scheme for constructing integrable lattice regularisations of integrable quantum field theories. Considering the affine Toda theories as examples, we show how to construct such lattice regularisations…
Integrable boundary conditions in 1+1 and 2+1 dimensions are discussed from the higher symmetries point of view. Boundary conditions consistent with the discrete Landau-Lifshitz model and infinite 2D Toda lattice are represented.
A tau function of the 2D Toda hierarchy can be obtained from a generating function of the two-partition cubic Hodge integrals. The associated Lax operators turn out to satisfy an algebraic relation. This algebraic relation can be used to…
This paper is the continuation of the work "On an inverse problem for finite-difference operators of second order". We consider the Cauchy problem for the Toda lattice in the case when the corresponding L-operator is a Jacobi matrix with…
The general solution of the two-dimensional integrable generalization of the f-Toda chain with fixed ends is explicitly presented in terms of matrix elements of various fundamental representations of the SL(n|n-1) supergroup. The dominant…
We have derived a non-abelian analog for the two-dimensional discrete Toda lattice which possesses solutions in terms of quasideterminants and admits Lax pairs of different forms. Its connection with non-abelian analogs for several…
The theory of Poisson-Lie groups and Lie bialgebras plays a major role in the study of one dimensional integrable systems; many families of integrable systems can be recovered from a Lax pair which is constructed from a Lie bialgebra…