10 problems

Let $N$ be the number of nodes in a ring lattice, and let $m$ denote the interaction-length parameter for sparse coupling, with nearest-neighbour and next-nearest-neighbour couplin…

For parameters $d$ and $bc$, consider the two-dimensional Nagumo lattice dynamical system … A multi-localized solution is a localized solution with multiple distinct regions of loc…

A lattice dynamical system is assumed to possess localized asymmetric multi-pulse solutions organized into isolas, meaning closed bifurcation curves in parameter space. Consider pe…

Radial-decay exponent conjecture. The radial-decay hypothesis is true for some $p^*\in(1,5]$. Numerical investigations support this range, but establishing it still requires a comp…

Fifth-moment finiteness conjecture. In the infinite-lattice limit, $M_5^r(\alpha)$ is finite for the simulated values $\alpha=0.1,0.5,1.0$. The claim is based only on finite-lattic…

Radial deviation moment conjecture. There exists $p\geq1$ such that $M_p^r(\alpha)<\infty$ for sufficiently small $\alpha\geq0$. Numerical computations on finite lattices provide h…

Let $\{(\bar r_{i,j}(\alpha),\bar\theta_{i,j}(\alpha))\}_{(i,j)\in\mathbb{Z}^2}$ be a rotating-wave solution of the lattice Lambda–Omega system, and let $a$ be the root of the radi…

Let $\{(\bar r_{i,j}(\alpha),\bar\theta_{i,j}(\alpha))\}_{(i,j)\in\mathbb{Z}^2}$ be the rotating-wave solution, with phase values considered on concentric rings about the centre fo…

Let $m$ be a positive integer. An $m$-armed rotating wave solution is a solution of the lattice dynamical system of the form … where $(\bar{r}_{i,j},\bar{\theta}_{i,j})$ is a stead…

Let $U^{(\epsilon)}_j(t)$ be a family of slow/oscillatory solutions, let ${\underline u}^{k}(\Theta)$ denote a one-periodic periodic traveling-wave profile with wavenumber $k$, and…