Related papers: The triple-zero Painlev\'e I transcendent
An analysis of possible extension of the Painlev\'e test, to encompass the one-dimensional Vlasov equation, is performed. The extending requires a nontrivial generalization of the test. The proposed singularity analysis provides…
A birational transformation is one which leaves invariant an ordinary differential equation, only changing its parameters. We first recall the consistent truncation which has allowed us to obtain the first degree birational transformation…
In this paper we propose a geometric approach to study Painlev\'e equations appearing as constrained systems of three first-order ordinary differential equations. We illustrate this approach on a system of three first-order differential…
We review some results of calculations, having the property of maximal transcendentality.
The fourth-order analog to the first Painlev\'{e} equation is studied. All power expansions for solutions of this equation near points $z=0$ and $z=\infty$ are found. The exponential additions to the expansion of solution near $z=\infty$…
Consider the solution $y(t)$ for the ordinary differential equation $y' = f(t, y)$ with $t$ complex. Second-order nonlinear differential equations often exhibit patterns in their poles, branch points, and essential singularities, explored…
We study the asymptotic behaviour of solutions of the fourth Pain\-lev\'e 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…
The existence and multiplicity of positive periodic solutions for first non-autonomous singular systems are established with superlinearity or sublinearity assumptions at infinity for an appropriately chosen parameter. The proof of our…
A nonlinear coupled system descriptive of multi-ion electrodiffusion is investigated and all parameters for which the system admits a single-valued general solution are isolated. This is achieved \textit{via} a method initiated by Painleve'…
The integrability of a four-dimensional sixth-order bilinear equation associated with the exceptional affine Lie algebra $D_4^{(1)}$ is studied by means of the singularity analysis. This equation is shown to pass the Painlev\'{e} test in…
It is well-known that the first and second Painlev\'e equations admit solutions characterised by divergent asymptotic expansions near infinity in specified sectors of the complex plane. Such solutions are pole-free in these sectors and…
The integrability (solvability via an associated single-valued linear problem) of a differential equation is closely related to the singularity structure of its solutions. In particular, there is strong evidence that all integrable…
Starting from the second Painlev\'{e} equation, we obtain Painlev\'{e} type equations of higher order by using the singular point analysis.
We present the bilinear forms of the (continuous) Painlev\'e equations obtained from the continuous limit of the analogous expresssions for the discrete ones. The advantage of this method is that it leads to very symmetrical results. A new…
All of the six Painlev\'e equations except the first have families of rational solutions, which are frequently important in applications. The third Painlev\'e equation in generic form depends on two parameters $m$ and $n$, and it has…
We utilise a recent approach via the so-called re-scaling method to derive a unified and comprehensive theory of the solutions to Painleve's differential equations (I), (II) and (IV), with emphasis on the most elaborate equation (IV).
We consider a connection problem of the first Painlev\'{e} equation ($\mathrm{P_I}$), trying to connect the local behavior (Laurent series) near poles and the asymptotic behavior as the variable $t$ tends to negative infinity for real…
We examine the relationship between the homogeneous second Painlev\'e equation and equation ${\bf XX}$ from the master list recorded by Ince.
In this paper, we revisit the asymptotic formulas of real Painlev\'e I transcendents as the independent variable tends to negative infinity, which were initially derived by Kapaev with the complex WKB method. Using the Riemann-Hilbert…
In this note, we will do analysis of accessible singular points for a polynomial Hamiltonian system obtained by taking a double covering of the Painlev\'e I equation. We will show that this system passes the Painlev\'e $\alpha$-test for all…