97 problems

Consider the full constrained 3D system studied in the paper, rather than either of its restrictions to an invariant hypersurface. The Painlevé property means that its solutions ha…

Let $X=(P_1,\dots,P_n)$ be an autonomous $C^1$-vector field on $\mathbb{R}^n$ with a unique singularity at the origin. For each $x\in\mathbb{R}^n$, let $DX(x)$ denote its Jacobian…

A two-point boundary value problem (BVP) for an ordinary differential operator is called dissipative when its associated operator is dissipative, and Birkhoff-regular when its boun…

Let an ODE system be given by polynomial equations of the form described above, and let a monomial quadratization mean a quadratization obtained by introducing monomial functions o…

Consider the second member of the Painlevé I hierarchy, the fourth-order equation … where $t\in\mathbb{R}$, and seek solutions $u(x,t)$ that are real and pole-free for real $x$. Du…

Grassmann convexity conjecture. The number of real zeros of this Wronskian in $I$ is at most

ARS conjecture. Any ordinary differential equation which arises as a reduction of an integrable partial differential equation possesses the Painlevé property, possibly after a tran…

The fourth-order equation … was given by Cosgrove and denoted by F-VI. Cosgrove's conjecture. This equation defines a new Painlevé transcendental, meaning that it is not reducible…

New-transcendent conjecture. The general solutions of these equations cannot be expressed in terms of the six Painlevé transcendents; hence the equations define a new transcendent.

Consider the second-order boundary-value problem … with … where $K$ and $A$ are positive constants and $L(x)=a_0^2+(a_1^2-a_0^2)x$, with $a_0,a_1>0$. Exact multiplicity conjecture.…

Let $Y(t;\kappa)$ denote the solution of … Denote the eigenvalues of $Y(t;\kappa)$ by $\lambda_n(t;\kappa)$, ordered so that each $\lambda_n$ is continuous as a function of $t$. Le…

Uniqueness conjecture for the periodic solution. The value $p=\sqrt{3/8}$ is the only value corresponding to a periodic solution satisfying these conditions.

Pugh's conjecture. This equation should have at most $k$ solutions.

Let $m\geqslant 2$, let $x\in\mathbb{K}$, and let $y=(y^1,\dots,y^m)\in\mathbb{K}^m$. Consider a linear system of $m$ second-order differential equations … where the coefficient fu…

The discussion concerns a family of linear ordinary differential equations exhibiting a singular perturbation, meaning that for certain parameter values the leading coefficient van…

Let $x_1<x_2<0$, and let $X(x)$ and $X_{min}(x)$ denote the corresponding extremal location functions for real solutions of the first Painlevé equation. Comparison conjecture. … Th…

Let $y_{max}(x_0;x)$ and $y_{min}(x_0;x)$ be the two real solutions with pole at $x_0\leq0$, with $x$ in the interval of existence of $y_{min}(x_0;x)$. Ordering conjecture. … The i…

Uniqueness conjecture. For any $x_0$, the solution with a pole at $x_0$ and minimum at $\Xi_{min}(x_0)$ is unique. The claim is presented as a numerical and heuristic extension of…

Let $x_0<0$ be a pole location, and let $X_{min}(x_0)$ denote the extremal location of a minimum among the relevant real solutions of the first Painlevé equation. Uniqueness conjec…

Let $d>2$, and let ${\cal S}_1$ be the set of solutions $\varphi$ of … for $\eta>0$ that have exactly one intersection with the singular solution $\varphi_S(\eta)=2d/\eta^2$. Relax…

Let ${\rm P}_{\rm II}^{(n)}$ denote the $n$th equation in the second Painlevé hierarchy, with parameter $\alpha_n$. Irreducibility means that, for generic values of $\alpha_n$, no…

The second Painlevé hierarchy is an infinite sequence of nonlinear ordinary differential equations containing the second Painlevé equation … where $V=V(x)$ and $\alpha$ is constant…

For real parameters $T$ and real $X$, consider the fourth-order ordinary differential equation … where primes denote derivatives with respect to $X$. Main Conjecture, Part 2. The O…

Rational integrating-factor conjecture. For the algorithm proposed in the earlier work, the integrating factor $R$ should necessarily be rational whenever the corresponding second-…

Rationality conjecture. If the equation has an elementary solution, then it is always possible to choose a first integral $I$ such that $RS$ is a rational function of $(x,y,y')$.