8 problems

Let $k\in\mathbb{N}$ and let $c_1,\dots,c_k$ be arbitrary constants. For each $i$, let $\{\vec{J}_i\}$ denote the set of vectors satisfying … For $\vec{J}_i$, write…

Let $w=w(x,t)$ satisfy the CKdV equation, and let $g[w]$ and $h[w]$ denote functions of $w$ and its $x$-derivatives. Let $T'$ be an unknown operator. The Lax-pair conjecture. A Lax…

Let $\mathcal{L}$ be the Lax operator of the extended rationally constrained KP hierarchy, with dependent variables $a_1,\dots,a_N$, and let $\log \mathcal{Q}$ and $\tilde{B}_\ell$…

Rational Lax-pair nonexistence conjecture. The system $$ has no nontrivial linear or nonlinear Lax pairs written in terms of rational, rather than algebraic, functions.

Let $A$ be the free associative algebra of the system, let $[A,A]$ be its commutator subspace, and write the coefficients of the noncommutative spectral curve as … The induced Lie…

Let $A$ be the free associative algebra of the system under consideration, let $[A,A]$ be its commutator subspace, and let $L(\lambda)$ and $M$ be the displayed noncommutative matr…

Consider an integrable hierarchy with a universal space Lax operator, and its type-I and type-II Bäcklund transformations. Universality conjecture. For a given hierarchy, both the…

Let $U^{(l)}$ and $V_N^{(l)}$ be linear Lax operators built from the building blocks $\{\lambda,U^{(0)}\}$ and their $x$-derivatives, with scaling dimensions…