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Register a new userSolutions of the Painleve VI Equation from Reduction of Integrable Hierarchy in a Grassmannian Approach
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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.
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Iorgov, Lisovyy, and Teschner established a connection between isomonodromic deformation of linear differential equations and Liouville conformal field theory at $c=1$. In this paper we present a $q$ analog of their construction. We show that the general solution of the $q$-Painleve VI equation is a ratio of four tau functions, each of which is given by a combinatorial series arising in the AGT correspondence. We also propose conjectural bilinear equations for the tau functions.
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 study the Cauchy problem for the Korteweg-de Vries (KdV) hierarchy in the small dispersion limit where $eto 0$. For negative analytic initial data with a single negative hump, we prove that for small times, the solution is approximated by the solution to the hyperbolic transport equation which corresponds to $e=0$. Near the time of gradient catastrophe for the transport equation, we show that the solution to the KdV hierarchy is approximated by a particular Painleve transcendent. This supports Dubrovins universality conjecture concerning the critical behavior of Hamiltonian perturbations of hyperbolic equations. We use the Riemann-Hilbert approach to prove our results.
We introduce the concept of multisymplectic formalism, familiar in covariant field theory, for the study of integrable defects in 1+1 classical field theory. The main idea is the coexistence of two Poisson brackets, one for each spacetime coordinate. The Poisson bracket corresponding to the time coordinate is the usual one describing the time evolution of the system. Taking the nonlinear Schrodinger (NLS) equation as an example, we introduce the new bracket associated to the space coordinate. We show that, in the absence of any defect, the two brackets yield completely equivalent Hamiltonian descriptions of the model. However, in the presence of a defect described by a frozen Backlund transformation, the advantage of using the new bracket becomes evident. It allows us to reinterpret the defect conditions as canonical transformations. As a consequence, we are also able to implement the method of the classical r matrix and to prove Liouville integrability of the system with such a defect. The use of the new Poisson bracket completely bypasses all the known problems associated with the presence of a defect in the discussion of Liouville integrability. A by-product of the approach is the reinterpretation of the defect Lagrangian used in the Lagrangian description of integrable defects as the generating function of the canonical transformation representing the defect conditions.
In this paper, we study the Dirac equation for an electron constrained to move on a catenoid surface. We decoupled the two components of the spinor and obtained two Klein-Gordon-like equations. Analytical solutions were obtained using supersymmetric quantum mechanics for two cases, namely, the constant Fermi velocity and the position-dependent Fermi velocity cases.
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