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Register a new userAsymptotic behaviour of the fourth Painleve transcendents in the space of initial values
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We study the asymptotic behaviour of solutions of the fourth Pain-leve equation as the independent variable goes to infinity in its space of (complex) initial values, which is a generalisation of phase space described by Okamoto. We show that the limit set of each solution is compact and connected and, moreover, that any non-special solution has an infinite number of poles and infinite number of zeroes.
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We study the solutions of the second Painleve equation in the space of initial conditions first constructed by Okamoto, in the limit as the independent variable, x, goes to infinity. Simultaneously, we study solutions of the related equation known as the thirty-fourth Painleve equation. By considering degenerate cases of the autonomous flow, we recover the known special solutions, which are either rational functions or expressible in terms of Airy functions. We show that the solutions that do not vanish at infinity possess an infinite number of poles. An essential element of our construction is the proof that the union of exceptional lines is a repellor for the dynamics in Okamotos space. Moreover, we show that the limit set of the solutions exists and is compact and connected.
We consider the generalized Painleve--Ince equation, begin{equation*} ddot{x}+alpha xdot{x}+beta x^{3}=0 end{equation*} and we perform a detailed study in terms of symmetry analysis and of the singularity analysis. When the free parameters are related as $beta =alpha ^{2}/9~$the given differential equation is maximally symmetric and well-known that it pass the Painlev{e} test. For arbitrary parameters we find that there exists only two Lie point symmetries which can be used to reduce the differential equation into an algebraic equation. However, the generalized Painlev{e}--Ince equation fails at the Painlev{e} test, except if we apply the singularity analysis for the new second-order differential equation which follows from the change of variable $x=1/y.$ We conclude that the Painlev{e}--Ince equation is integrable is terms of Lie symmetries and of the Painlev{e} test.
We construct the initial-value space of a $q$-discrete first Painleve equation explicitly and describe the behaviours of its solutions $w(n)$ in this space as $ntoinfty$, with particular attention paid to neighbourhoods of exceptional lines and irreducible components of the anti-canonical divisor. These results show that trajectories starting in domains bounded away from the origin in initial value space are repelled away from such singular lines. However, the dynamical behaviours in neighbourhoods containing the origin are complicated by the merger of two simple base points at the origin in the limit. We show that these lead to a saddle-point-type behaviour in a punctured neighbourhood of the origin.
The classical Painleve equations are so well known that it may come as a surprise to learn that the asymptotic description of its solutions remains incomplete. The problem lies mainly with the description of families of solutions in the complex domain. Where asymptotic descriptions are known, they are stated in the literature as valid for large connected domains, which include movable poles of families of solutions. However, asymptotic analysis necessarily assumes that the solutions are bounded and so these domains must be punctured at locations corresponding to movable poles, leading to asymptotic results that may not be uniformly valid. To overcome these issues, we recently carried out asymptotic analysis in Okamotos geometric space of initial values for the first and second Painleve equations. In this paper, we review this method and indicate how it may be extended to the discrete Painleve equations.
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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