215 problems

Dubrovin's universality conjecture. Near a generic gradient-catastrophe point, the dispersive solution should be modeled by this particular solution $\mathcal{D}(X;T)$.

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…

Ablowitz–Ramani–Segur conjecture. There exists a connection between nonlinear integrable PDEs and nonlinear ODEs with the Painlevé property such that any ODE emerging from the redu…

Gamayun–Iorgov–Lisovyy conjecture. The tau functions should admit explicit formal series solutions of the form

Let $y(\lambda)$ be the tritronquée solution of Painlevé I, and let $\alpha\in\mathbb{C}$ be a pole of this solution. Dubrovin's conjecture. If $\alpha$ is a pole of the tritronqué…

Let $\tau_{\mathrm{I}}$ be the tau function of the first Painlevé equation, and let an irregular conformal block be a Virasoro conformal block constructed from an irregular vector.…

Hardy Z-function joint moment conjecture. As $T\to\infty$,

Painlevé I hierarchy asymptotics conjecture. A special solution to the second equation in the Painlevé I hierarchy provides an asymptotic description of the break-up of Camassa–Hol…

Let $q(x;\alpha,\kappa)$ be a Clarkson–McLeod solution of the fourth Painlevé equation, with $\alpha-\frac{1}{2}\notin\mathbb{Z}$, and let $\kappa^*>0$ denote the critical value se…

For the quartic oscillator, let the algebraic spectrum be determined by a finite-dimensional matrix depending on a parameter, and let its discriminant be the resulting polynomial i…

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…

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.

Let $y1, ldots,yn$ be solutions of a fixed Painlevé equation whose complex parameters are generic. Write $operatorname{td}(L/K)$ for the transcendence degree of a field extension…

Algebraic-solution relation. For $s=\pm1$,

Vortex-filament tritronquée conjecture. The generic behavior near the gradient-catastrophe point is

Tricolour graph construction. For any tricolour graph there exists a function

The Garnier system has analogues of the Painlevé VI transformations $w_1,w_3,w_4,l_1,l_2,l_3$, defined by Tsuda, while $w_2$ denotes the remaining generator of the affine extension…

For each generalized Cartan matrix of affine type $A=(a_{ij})_{i,j\in I}$, let $\tau_i$ be the tau-functions in the Weyl-group representation described above. Strong regularity con…

The spectrum singularity average $A_N(t;)$ is related to the isomonodromy theory of the sixth Painlevé system. Jimbo's extension conjecture. The expansions of Jimbo given in Theore…

Let $V(\underline{t})$ be the elliptic hypergeometric function, and let $\mathcal H$ be the unconstrained Hamiltonian associated with the corresponding elliptic hypergeometric mode…

Algebraic solutions of the Painlevé VI equation are considered up to equivalence under Okamoto transformations and simple deformation. Non-abelian Schwarz list conjecture. There ar…

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…

Let $y_0\in\mathbb R$ and $y_l\leq y_0$ define a first solution of the first Painlevé equation, and let the second solution be defined by the data $-y_l$ and $-y_0$. Let $\Delta$ b…