8 problems

Dynamical splitting conjecture. In the case $K=1$, the shifted state with $a<0$ leads to a solitary wave that moves towards the vertex point at $x=0$ along the only incoming edge,…

Let $\Omega$ be a domain, let $\boldsymbol{\beta}\in [L^\infty(\Omega)]^d$ be divergence-free, and let $L$ be the transport operator with inflow boundary conditions, … For a discre…

Subsonic-or-sonic stability conjecture. A standing or traveling wave profile for the QHD systems with linear or nonlinear viscosity is spectrally stable if and only if its velocity…

Algebraic stability conjecture. The operator $\Omega(k)$ has negative eigenvalues if and only if $\lambda_{\ell,1}<0$ for every $\ell=1,\ldots,N$.

Consider the modified system … where the coupling term $\beta v$ has been replaced by $\beta\sigma(x)v$, and suppose that the Fourier transform of $\sigma(x)$ has full support. Res…

Let $\psi_0 \in H^{\alpha}_{\rm per}(\mathbb{T})$ be the solution to the boundary-value problem with $c=c_0$ obtained from the main existence theorem, and let $X_0$ and…

Let $(u_{n,\omega}^{\pm},0)$ be the stationary standing-wave branch indexed by $n\in\mathbb{N}$, and let the spectral stability problem be the linearized eigenvalue problem for the…

Let $U^c(\xi,y)$ be a regular internal solitary wave (ISW), and let $D_N$ denote its $N$-th order truncated Evans function. Write…