Nodal Curves and Riccati Solutions of Painlevé Equations
نویسنده
چکیده
In this paper, we study Riccati solutions of Painlevé equations from a view point of geometry of Okamoto-Painlevé pairs (S,Y ). After establishing the correspondence between (rational) nodal curves on S − Y and Riccati solutions, we give the complete classification of the configurations of nodal curves on S − Y for each Okamoto–Painlevé pair (S, Y ). As an application of the classification, we prove the non-existence of Riccati solutions of Painlevé equations of types PI , P D̃8 III and P D̃7 III . We will also give a partial answer to the conjecture in [STT] and [T] that the dimension of the local cohomology H Yred(S,ΘS(− log Yred)) is one.
منابع مشابه
Hexagonal circle patterns with constant intersection angles and discrete Painlevé and Riccati equations
Hexagonal circle patterns with constant intersection angles mimicking holomorphic maps z and log(z) are studied. It is shown that the corresponding circle patterns are immersed and described by special separatrix solutions of discrete Painlevé and Riccati equations. The general solution of the Riccati equation is expressed in terms of the hypergeometric function. Global properties of these solu...
متن کاملUltradiscrete Painlevé VI with Parity Variables
We introduce a ultradiscretization with parity variables of the q-difference Painlevé VI system of equations. We show that ultradiscrete limit of Riccati-type solutions of q-Painlevé VI satisfies the ultradiscrete Painlevé VI system of equations with the parity variables, which is valid by using the parity variables. We study some solutions of the ultradiscrete Riccati-type equation and those o...
متن کاملSolutions structure of integrable families of Riccati equations and their applications to the perturbed nonlinear fractional Schrodinger equation
Some preliminaries about the integrable families of Riccati equations and solutions structure of these equations in several cases are presented in this paper, then by using of definitions for fractional derivative we apply the new extended of tanh method to the perturbed nonlinear fractional Schrodinger equation with the kerr law nonlinearity. Finally by using of this method and solutions of Ri...
متن کاملThe B"{a}cklund transformation method of Riccati equation to coupled Higgs field and Hamiltonian amplitude equations
In this paper, we establish new exact solutions for some complex nonlinear wave equations. The B"{a}cklund transformation method of Riccati equation is used to construct exact solutions of the Hamiltonian amplitude equation and the coupled Higgs field equation. This method presents a wide applicability to handling nonlinear wave equations. These equations play a very important role in mathemati...
متن کاملExplicit exact solutions for variable coefficient Broer-Kaup equations
Based on symbolic manipulation program Maple and using Riccati equation mapping method several explicit exact solutions including kink, soliton-like, periodic and rational solutions are obtained for (2+1)-dimensional variable coefficient Broer-Kaup system in quite a straightforward manner. The known solutions of Riccati equation are used to construct new solutions for variable coefficient Broer...
متن کامل