Related papers: Generalized Gaudin Models and Riccatians
We introduce a generalized Gaudin Lie algebra and a complete set of mutually commuting quantum invariants allowing the derivation of several families of exactly solvable Hamiltonians. Different Hamiltonians correspond to different…
We generalize the classical Lie results on a basis of differential invariants for a one-parameter group of local transformations to the case of arbitrary number of independent and dependent variables. It is proved that if universal…
We report a solution of the inverse Lagrangian problem for the first order Riccati differential equation by means of an analogy with the Friedmann equation of a suitable Friedmann-Lema\^itre-Robertson-Walker universe in general relativity.…
We systematically analyze the nonlinear partial differential equation that determines the behaviour of a bounded radiating spherical mass in general relativity. Four categories of solution are possible. These are identified in terms of…
A general method of obtaining linear differential equations having polynomial solutions is proposed. The method is based on an equivalence of the spectral problem for an element of the universal enveloping algebra of some Lie algebra in the…
The Riccati equations reducible to first-order linear equations by an appropriate change the dependent variable are singled out. All these equations are integrable by quadrature. A wide class of linear ordinary differential equations…
Generalized differential forms are used in discussions of metric geometries and Einstein's vacuum field equations. Cartan's structure equations are generalized and applied. In particular flat generalized connections are associated with any…
The Gaudin models based on the face-type elliptic quantum groups and the $XYZ$ Gaudin models are studied. The Gaudin model Hamiltonians are constructed and are diagonalized by using the algebraic Bethe ansatz method. The corresponding…
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…
We study the Gaudin models associated with $\mathfrak{gl}(1|1)$. We give an explicit description of the algebra of Hamiltonians (Gaudin Hamiltonians) acting on tensor products of polynomial evaluation $\mathfrak{gl}(1|1)[t]$-modules. It…
In this paper, we consider spectral approximation of fractional differential equations (FDEs). A main ingredient of our approach is to define a new class of generalized Jacobi functions (GJFs), which is intrinsically related to fractional…
We introduce a class of quantum integrable systems generalizing the Gaudin model. The corresponding algebras of quantum Hamiltonians are obtained as quotients of the center of the enveloping algebra of an affine Kac-Moody algebra at the…
Let $\mathfrak{B}_{m|n}(\underline{\boldsymbol{z}})$ be the Gaudin algebra of the general linear Lie superalgebra $\mathfrak{gl}_{m|n}$ with respect to a sequence $\underline{\boldsymbol{z}} \in \mathbb{C}^\ell$ of pairwise distinct complex…
In this note, we discuss implications of the results obtained in [MTV4]. It was shown there that eigenvectors of the Bethe algebra of the quantum gl_N Gaudin model are in a one-to-one correspondence with Fuchsian differential operators with…
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…
We derive some analytic closed-form solutions for a class of Riccati equation y'(x)-\lambda_0(x)y(x)\pm y^2(x)=\pm s_0(x), where \lambda_0(x), s_0(x) are C^{\infty}-functions. We show that if \delta_n=\lambda_n s_{n-1}-\lambda_{n-1}s_n=0,…
In the article we discuss the notion of the generalized invariant manifold introduced in our previous study. In the literature the method of the differential constraints is well known as a tool for constructing particular solutions for the…
The XXX Gaudin model with generic integrable boundaries specified by the most general non-diagonal K-matrices is studied by the off-diagonal Bethe ansatz method. The eigenvalues of the associated Gaudin operators and the corresponding Bethe…
It is shown that the expressions for the tangential pressure, the anisotropy factor and the radial pressure in the Einstein equations may serve as generating functions for stellar models. The latter can incorporate an equation of state when…
We study the ODE/IM correspondence for ODE associated to $\hat{\mathfrak g}$-valued connections, for a simply-laced Lie algebra $\mathfrak g$. We prove that subdominant solutions to the ODE defined in different fundamental representations…