11 problems

Consider the stationary localized planar Swift–Hohenberg equation … for $u=u(x)$ with $x\in\mathbb{R}^2$, $\lim_{|x|\to\infty}u(x)=0$, $\mu>0$, and $(\nu_1,\nu_2)\in\mathbb{R}^2$.…

Coalescing-saddles replacement conjecture. In certain cases, the coalescing-saddles method can be systematically circumvented or replaced by a multiple-scales approach using an int…

Let $mathbb R^2$ be generated by repeated even extension of solutions from fundamental domains, and consider an isosceles right triangle and the corresponding square. Triangle-squa…

Let $al N_-(\Omega_R)$ denote the ridge-Nehari solutions in the scaled domain $\Omega_R$, and let $eta^*$ be the threshold used in the preceding discussion. Irreducibility-threshol…

Let $\beta>0$, let $\Omega_R$ be the scaled domain under consideration, and let $E$ be the energy functional on $\mathcal W(\Omega_R)$. Constant-minimizer conjecture. The global mi…

The quintic Swift–Hohenberg (SH) equation is the stationary planar equation obtained by including a quintic nonlinearity in the Swift–Hohenberg model. A localized pattern is a spat…

Consider the modulation approximation $\Psi$ for the dynamic Swift–Hohenberg equation. In the symmetric case $\mu(x)\equiv 0$, the error estimate remains exponentially small up to…

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…

Order-one noise contribution conjecture. The noise has an $\mathcal{O}(1)$ contribution to both $v_0$ and $v_2$.

Hexagon patch wavenumber-selection conjecture. Every stationary and invading stable hexagon 2D patch solution selects a unique hexagon cellular pattern. Its wavenumbers are determi…