Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userQuantization of higher hamiltonian analogues of the Painleve I and Painleve II equations with two degrees of freedom
460
0
0.0
(
0
)
Added by
Bulat Suleimanov Irekovich
Publication date
2012
fields
Physics
and research's language is
English
Authors
Bulat Suleimanov
Ask ChatGPT about the research
No Arabic abstract
We construct a solution of an analog of the Schr{o}dinger equation for the Hamiltonian $ H_I (z, t, q_1, q_2, p_1, p_2) $ corresponding to the second equation $P_1^2$ in the Painleve I hierarchy. This solution is produced by an explicit change of variables from a solution of the linear equations whose compatibility condition is the ordinary differential equation $P_1^2$ with respect to $z$. This solution also satisfies an analog of the Schr{o}dinger equation corresponding to the Hamiltonian $ H_{II} (z, t, q_1, q_2, p_1, p_2) $ of Hamiltonian system with respect to $t$ which is compatible with $P_1^2$. A similar situation occurs for the $P_2^2$ equation in the Painleve II hierarchy.
rate research
Read More
In this paper a comprehensive review is given on the current status of achievements in the geometric aspects of the Painleve equations, with a particular emphasis on the discrete Painleve equations. The theory is controlled by the geometry of certain rational surfaces called the spaces of initial values, which are characterized by eight point configuration on $mathbb{P}^1timesmathbb{P}^1$ and classified according to the degeration of points. We give a systematic description of the equations and their various properties, such as affine Weyl group symmetries, hypergeomtric solutions and Lax pairs under this framework, by using the language of Picard lattice and root systems. We also provide with a collection of basic data; equations, point configurations/root data, Weyl group representations, Lax pairs, and hypergeometric solutions of all possible cases.
In this paper we study the equation $$ w^{(4)} = 5 w (w^2 - w) + 5 w (w)^2 - w^5 + (lambda z + alpha)w + gamma, $$ which is one of the higher-order Painleve equations (i.e., equations in the polynomial class having the Painleve property). Like the classical Painleve equations, this equation admits a Hamiltonian formulation, Backlund transformations and families of rational and special functions. We prove that this equation considered as a Hamiltonian system with parameters $gamma/lambda = 3 k$, $gamma/lambda = 3 k - 1$, $k in mathbb{Z}$, is not integrable in Liouville sense by means of rational first integrals. To do that we use the Ziglin-Morales-Ruiz-Ramis approach. Then we study the integrability of the second and third members of the $mathrm{P}_{mathrm{II}}$-hierarchy. Again as in the previous case it turns out that the normal variational equations are particular cases of the generalized confluent hypergeometric equations whose differential Galois groups are non-commutative and hence, they are obstructions to integrability.
We study Fredholm determinants related to a family of kernels which describe the edge eigenvalue behavior in unitary random matrix models with critical edge points. The kernels are natural higher order analogues of the Airy kernel and are built out of functions associated with the Painleve I hierarchy. The Fredholm determinants related to those kernels are higher order generalizations of the Tracy-Widom distribution. We give an explicit expression for the determinants in terms of a distinguished smooth solution to the Painleve II hierarchy. In addition we compute large gap asymptotics for the Fredholm determinants.
Discrete Painleve equations are nonlinear, nonautonomous difference equations of second-order. They have coefficients that are explicit functions of the independent variable $n$ and there are three different types of equations according to whether the coefficient functions are linear, exponential or elliptic functions of $n$. In this paper, we focus on the elliptic type and give a review of the construction of such equations on the $E_8$ lattice. The first such construction was given by Sakai cite{SakaiH2001:MR1882403}. We focus on recent developments giving rise to more examples of elliptic discrete Painleve equations.
We construct solutions of analogues of the nonstationary Schrodinger equation corresponding to the polynomial isomonodromic Hamiltonian Garnier system with two degrees of freedom. This solutions are obtained from solutions of systems of linear ordinary differential equations whose compatibility condition is the Garnier system. This solutions upto explicit transform also satisfy the Belavin --- Polyakov --- Zamolodchikov equations with four time variables and two space variables.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
