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Register a new userBilinear Structure and Exact Solutions of the Discrete Painleve I Equation
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Bilinear structure for the discrete Painleve I equation is investigated. The solution on semi-infinite lattice is given in terms of the Casorati determinant of discrete Airy function. Based on this fact, the discrete Painleve I equation is naturally extended to a discrete coupled system. Corresponding matrix model is also mentioned.
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In this paper, we consider the discrete power function associated with the sixth Painleve equation. This function is a special solution of the so-called cross-ratio equation with a similarity constraint. We show in this paper that this system is embedded in a cubic lattice with $widetilde{W}(3A_1^{(1)})$ symmetry. By constructing the action of $widetilde{W}(3A_1^{(1)})$ as a subgroup of $widetilde{W}(D_4^{(1)})$, i.e., the symmetry group of P$_{rm VI}$, we show how to relate $widetilde{W}(D_4^{(1)})$ to the symmetry group of the lattice. Moreover, by using translations in $widetilde{W}(3A_1^{(1)})$, we explain the odd-even structure appearing in previously known explicit formulas in terms of the $tau$ function.
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Kenji Kajiwara (Department of Applied Physics
, Faculty of Engineering
, n University of Tokyo
1993
A q-discrete version of the two-dimensional Toda molecule equation is proposed through the direct method. Its solution, Backlund transformation and Lax pair are discussed. The reduction to the q-discrete cylindrical Toda molecule equation is also discussed.
We present an explicit method to perform similarity reduction of a Riemann-Hilbert factorization problem for a homogeneous GL (N, C) loop group and use our results to find solutions to the Painleve VI equation for N=3. The tau function of the reduced hierarchy is shown to satisfy the sigma-form of the Painleve VI equation. A class of tau functions of the reduced integrable hierarchy is constructed by means of a Grassmannian formulation. These solutions provide rational solutions of the Painleve VI equation.
This paper is an addendum to earlier papers cite{R1,R2} in which it was shown that the unstable separatrix solutions for Painleve I and II are determined by $PT$-symmetric Hamiltonians. In this paper unstable separatrix solutions of the fourth Painleve transcendent are studied numerically and analytically. For a fixed initial value, say $y(0)=1$, a discrete set of initial slopes $y(0)=b_n$ give rise to separatrix solutions. Similarly, for a fixed initial slope, say $y(0)=0$, a discrete set of initial values $y(0)=c_n$ give rise to separatrix solutions. For Painleve IV the large-$n$ asymptotic behavior of $b_n$ is $b_nsim B_{rm IV}n^{3/4}$ and that of $c_n$ is $c_nsim C_{rm IV} n^{1/2}$. The constants $B_{rm IV}$ and $C_{rm IV}$ are determined both numerically and analytically. The analytical values of these constants are found by reducing the nonlinear Painleve IV equation to the linear eigenvalue equation for the sextic $PT$-symmetric Hamiltonian $H=frac{1}{2} p^2+frac{1}{8} x^6$.
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.
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