Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userNew results on integrability of the Kahan-Hirota-Kimura discretizations
128
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
R. Hirota and K. Kimura discovered integrable discretizations of the Euler and the Lagrange tops, given by birational maps. Their method is a specialization to the integrable context of a general discretization scheme introduced by W. Kahan and applicable to any vector field with a quadratic dependence on phase variables. We report several novel observations regarding integrability of the Kahan-Hirota-Kimura discretization. For several of the most complicated cases for which integrability is known (Clebsch system, Kirchhoff system, and Lagrange top), - we give nice compact formulas for some of the more complicated integrals of motion and for the density of the invariant measure, and - we establish the existence of higher order Wronskian Hirota-Kimura bases, generating the full set of integrals of motion. While the first set of results admits nice algebraic proofs, the second one relies on computer algebra.
rate research
Read More
R. Hirota and K. Kimura discovered integrable discretizations of the Euler and the Lagrange tops, given by birational maps. Their method is a specialization to the integrable context of a general discretization scheme introduced by W. Kahan and applicable to any vector field with a quadratic dependence on phase variables. According to a proposal by T. Ratiu, discretizations of the Hirota-Kimura type can be considered for numerous integrable systems of classical mechanics. Due to a remarkable and not well understood mechanism, such discretizations seem to inherit the integrability for all algebraically completely integrable systems. We introduce an experimental method for a rigorous study of integrability of such discretizations. Application of this method to the Hirota-Kimura type discretization of the Clebsch system leads to the discovery of four functionally independent integrals of motion of this discrete time system, which turn out to be much more complicated than the integrals of the continuous time system. Further, we prove that every orbit of the discrete time Clebsch system lies in an intersection of four quadrics in the six-dimensional phase space. Analogous results hold for the Hirota-Kimura type discretizations for all commuting flows of the Clebsch system, as well as for the $so(4)$ Euler top.
New further integrability conditions of the Riccati equation $dy/dx=a(x)+b(x)y+c(x)y^{2}$ are presented. The first case corresponds to fixed functional forms of the coefficients $a(x)$ and $c(x)$ of the Riccati equation, and of the function $F(x)=a(x)+[f(x)-b^{2}(x)]/4c(x)$, where $f(x)$ is an arbitrary function. The second integrability case is obtained for the reduced Riccati equation with $b(x)equiv 0$. If the coefficients $a(x)$ and $c(x)$ satisfy the condition $pm dsqrt{f(x)/c(x)}/dx=a(x)+f(x)$, where $f(x)$ is an arbitrary function, then the general solution of the reduced Riccati equation can be obtained by quadratures. The applications of the integrability condition of the reduced Riccati equation for the integration of the Schrodinger and Navier-Stokes equations are briefly discussed.
We discuss the main points of the quantum group approach in the theory of quantum integrable systems and illustrate them for the case of the quantum group $U_q(mathcal L(mathfrak{sl}_2))$. We give a complete set of the functional relations correcting inexactitudes of the previous considerations. A special attention is given to the connection of the representations used to construct the universal transfer operators and $Q$-operators.
We give a sufficient condition for quantising integrable systems.
Kahan discretization is applicable to any quadratic vector field and produces a birational map which approximates the shift along the phase flow. For a planar quadratic Hamiltonian vector field, this map is known to be integrable and to preserve a pencil of cubic curves. Generically, the nine base points of this pencil include three points at infinity (corresponding to the asymptotic directions of cubic curves) and six finite points lying on a conic. We show that the Kahan discretization map can be represented in six different ways as a composition of two Manin involutions, corresponding to an infinite base point and to a finite base point. As a consequence, the finite base points can be ordered so that the resulting hexagon has three pairs of parallel sides which pass through the three base points at infinity. Moreover, this geometric condition on the base points turns out to be characteristic: if it is satisfied, then the cubic curves of the corresponding pencil are invariant under the Kahan discretization of a planar quadratic Hamiltonian vector field.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
