17 problems

Let a $\tau$-symmetric Hamiltonian perturbation of the Riemann--Hopf hierarchy be written in generalized standard form, with first Hamiltonian … Here $\mathcal{P}_k$ denotes the se…

Let a $\tau$-symmetric Hamiltonian perturbation of the Riemann--Hopf hierarchy be in generalized standard form, and let $a_0$ denote the coefficient appearing in its first Hamilton…

Geometric classification conjecture. The orbits are labelled by certain finite-dimensional geometrical structures on the manifold $M$.

Consider an integrable hierarchy of scalar conservation laws … which is a perturbation of the Riemann--Hopf hierarchy, and transform it to the Arsie--Lorenzoni--Moro normal form ……

Consider an integrable system of conservation laws that is a deformation of the Riemann--Hopf hierarchy, and transform it by the unique Miura-type transformation to its normal form…

Consider a deformation of the Riemann hierarchy in conservation-law form, with first Hamiltonian density … Assume $c_2(u)\ne0$, and suppose the existence conditions for a deformati…

Consider a deformation of the Riemann hierarchy … where the flow with respect to $t_1$ is in normal form: … with $c_\lambda(u)\in\mathbb C[[u]]$. Arsie–Lorenzoni–Moro conjecture. I…

Consider a PDE … with $Q=uu_x+O(\varepsilon)$, and expand … where $b_\lambda(u)\in\mathbb C[[u]]$. Let $b_k(u)$, $k\ge2$, denote the formal power series obtained from the correspon…

Let a tau-symmetric deformation of the Riemann hierarchy be given. A normal Miura transformation is a Miura transformation of the type defined in the source that preserves the rele…

Let ${\cal G}=sl(n+1)$ and let $E^{(1)}=\sum_{j=1}^{n}E_{\alpha_j}+\lambda E_{-(\alpha_1+\cdots+\alpha_n)}$. Suppose the gauge potentials are related through the proposed solutions…

Let the matrix analog of the Miura transformation be a matrix generalization describing the rectangle $W$ algebra, and let its q-deformation be the corresponding q-deformed constru…

Let a cohomological field theory determine a double ramification hierarchy and a Dubrovin–Zhang hierarchy. In genus $0$, the double ramification hierarchy coincides with the Dubrov…

Let an integrable scalar evolutionary equation be related to another scalar equation by a Miura transformation of the form specified in the source. A system is Miura trivial if a M…

Let a tau-symmetric integrable Hamiltonian deformation of the Riemann hierarchy be considered modulo normal Miura-type transformations. Write $w_k=\partial_x^k w$. The canonical de…

Viscous central-invariant conjecture. Two integrable viscous conservation laws admitting the same viscous central invariant $a(u)$ are Miura equivalent.

Quasilinear-part conjecture. The quasilinear part of a viscous conservation law is uniquely determined by $a(u)$.

Parametrisation conjecture. Normal forms of scalar integrable dispersive conservation laws are parametrised by infinitely many arbitrary functions identifiable with the coefficient…