45 problems

Phase-chaos persistence conjecture. Phase chaos persists in the thermodynamic limit in this regime.

Let $p$ denote the quantity characterizing the phase-chaotic states, and let the SN line be the saddle-node stability boundary for modulated amplitude waves (MAWs) of spatial perio…

Consider two open selection problems: selecting harmonic wavenumbers through fronts, and finding selected fronts that leave hexagonal patterns in the wake. Secondary-front explanat…

Consider modulated invasion in which the leading edge, the front interface, or the state in the wake may be oscillatory or temporally constant. Oscillatory invasion conjecture. The…

Let $u_\mathrm{pl}(\xi,\tau)$ be a modulated pulled pattern-forming front satisfying the stated pinched-double-root, periodic-wake, exponential-asymptotic, marginal-Floquet-stabili…

Let $u_\mathrm{ps}(\xi,\tau)$ be a modulated pushed pattern-forming front, namely a temporally modulated pushed front leaving a pattern-forming wake. Selection of modulated pushed…

Let $u_\mathrm{pl}$ be a rigid pulled pattern-forming front, meaning a rigid front with marginal spectral stability that leaves behind a diffusively stable periodic pattern and has…

Let $u_\mathrm{tr}$ be a pushed-to-pulled transition front. A pushed-to-pulled transition front is a front at the boundary between pushed and pulled invasion, characterized here by…

Differential-diffusion conjecture. Differential diffusion is a necessary condition for the stability, and thus the physical realization, of Turing patterns.

The one-dimensional Brusselator model has source-defect instability curves in the $(\sigma,b)$ parameter space. Source-defect snaking conjecture. These curves are the first fold cu…

Typical diffraction-density conjecture. For any $M,r>0$ there exists $\epsilon>0$ such that $\Lambda_{u,\epsilon}$ is $r$-dense in ${\mathcal L}^*\cap B(0,M)$.

Consider the Hastings–Levitov random aggregation model with parameter $\alpha\in\mathbb{R}$, where $\alpha$ controls the regularization of the growth dynamics. The regimes…

The space-fractional Swift–Hohenberg equation is a non-local reaction-diffusion PDE whose critical modes arise near a bifurcation point. An amplitude equation is a reduced PDE gove…

Conjecture on the small-diffusion limit. As $D\rightarrow 0$, the time to decay of a two-peaked solution tends to infinity, so that two-peaked solutions become stable.

Conjecture on local energy minima and transient states. Local energy minima of the approximate system are qualitatively similar to the asymptotic patterns observed in the aggregati…

A propagating pulse or spot in a heterogeneous medium is subject to an external defect, and may possess a hidden instability of drift–Hopf type. Higher-dimensional annihilation con…

The system under consideration describes interacting populations with population number $N$ and interaction coefficients $gamma_{ii}$. The analysis discussed here focuses on two po…

The paper studies tumor boundary stability for traveling waves and radially symmetric tumors under in vitro and in vivo nutrient supply regimes. In the in vivo regime, low-frequenc…

Consider the dynamic Swift–Hohenberg equation and its asymptotic approximation near the dynamic bifurcation. The available error estimates use an approximation whose residual is co…

Let $\mu(x)$ be a source term in the dynamic Swift–Hohenberg equation, and suppose that the reflection symmetry $u\mapsto -u$ is broken. The resulting modulation equations describe…

Zero-set conjecture.

Nonlinear stability conjecture. All $N$-spot ring patterns are nonlinearly stable unless the spot-spot distance is the smallest binding distance.

Localised dihedral bifurcation conjecture. For any $$ taken sufficiently small, true localised dihedral solutions of the system, arising from the limit $N\to\infty$, should exist a…

Periodic traveling-wave bifurcation conjecture. When the homogeneous ground state destabilizes, a spatially periodic traveling wave of this form bifurcates.

Consider the generalized antiferromagnetic local/nonlocal interaction functionals in dimension $d$, with parameter $\tau$ measuring the relative strength of the short-range attract…