Related papers: Riccati-type equations, generalised WZNW equations…
We consider a Lie algebra generalizing the Virasoro algebra to the case of two space variables. We study its coadjoint representation and calculate the corresponding Euler equations. In particular, we obtain a bi-Hamiltonian system that…
Square grid circle patterns with prescribed intersection angles, mimicking holomorphic maps z^a and log(z) are studied. It is shown that the corresponding circle patterns are imbedded and described by special separatrix solutions of…
For any classical Lie algebra $g$, we construct a family of integrable generalizations of Toda mechanics labeled a pair of ordered integers $(m,n)$. The universal form of the Lax pair, equations of motion, Hamiltonian as well as Poisson…
The purpose of this article is to develop an algebraic approach to the problem of integrable classification of differential-difference equations with one continuous and two discrete variables. As a classification criterion, we put forward…
The discrete-time two-dimensional Toda lattice of $A_\infty$-type is studied within the direct linearisation framework, which allows us to deal with several nonlinear equations in this class simultaneously and to construct more general…
Lie's linearizability criteria for scalar second-order ordinary differential equations had been extended to systems of second-order ordinary differential equations by using geometric methods. These methods not only yield the linearizing…
We consider the question of diagonal Riccati stability for a pair of real matrices A, B. A necessary and sufficient condition for diagonal Riccati stability is derived and applications of this to two distinct cases are presented. We also…
The structure of Hamiltonian reductions of the Wess-Zumino-Novikov-Witten (WZNW) theory by first class Kac-Moody constraints is analyzed in detail. Lie algebraic conditions are given for ensuring the presence of exact integrability,…
The existence of a Lagrangian description for the second-order Riccati equation is analyzed and the results are applied to the study of two different nonlinear systems both related with the generalized Riccati equation. The Lagrangians are…
We prove that with a $(2+1)$-dimensional Toda type system are associated algebraic skeletons which are (compatible assemblings) of particle-like Lie algebras of dyons and triadons type. We obtain trix-coaxial and dyx-coaxial Lie algebra…
The Relationship between the Neumann system and the Jacobi system in arbitrary dimensions is elucidated from the point of view of constrained Hamiltonian systems. Dirac brackets for canonical variables of both systems are derived from the…
We consider the Ricatti equation in the context of population dynamics, quantum scattering and a more general context. We examine some exactly solvable cases of real life interest.
Stochastic algebraic Riccati equations, also known as rational algebraic Riccati equations, arising in linear-quadratic optimal control for stochastic linear time-invariant systems, were considered to be not easy to solve. The-state-of-art…
The characterization of systems of differential equations admitting a superposition function allowing us to write the general solution in terms of any fundamental set of particular solutions is discussed. These systems are shown to be…
We present a multi-parameter non-constant-invariant class of Abel ordinary differential equations with the following remarkable features. This one class is shown to unify, that is, contain as particular cases, all the integrable classes…
A general Riccati equation is integrated in quadratures in case one of its coefficients is an arbitrary function and two others are expressed through it.
The questions of global topological, smooth and holomorphic classifications of the differential systems, defined by covering foliations, are considered. The received results are applied to nonautonomous linear differential systems and…
We prove a conjecture about the minimal nonnegative solutions of algebraic Riccati equations associated with reducible singular M-matrices. The result enhances our understanding of the behaviour of doubling algorithms for finding the…
A novel recipe for exactly solving in finite terms a class of special differential Riccati equations is reported. Our procedure is entirely based on a successful resolution strategy quite recently applied to quantum dynamical time-dependent…
We present a framework for enlarging the construction of $\mathbb{Z}_2^2$-graded classical Toda theory from the class of $\mathbb{Z}_2^2$-graded Lie algebras to the class of $\mathbb{Z}_2^2$-graded Lie superalgebras. This scheme is applied…