53 problems

Consider the integrodifference equation (IDE1) with logistic growth, and let the analytically proven spatially inhomogeneous 2-cycle be the one described in the paper. Spectral sta…

Nondegeneracy conjecture. The kernel of the real linearized operator is trivial:

Let $L_+$ be the linearized operator in the Hartree problem, and let $gKer[L_+]$ denote its generalized kernel. At the mass-critical exponent … the linearized operator has an extra…

The heteroclinic waves arise as component waves in the multi-bump limit of a family of homoclinic orbits. Their associated linearized operators have point spectra, considered in th…

Consider the linearized operator … about a stationary profile, and let $L$ act on a suitable space such as $L^2$. Assume the nonpositive point-spectrum condition … Let the tangent…

Let $v$ denote the relevant traveling-wave profile, and consider the eigenvalues associated with speed $1$ along the corresponding branch up to the critical speed. Stability conjec…

Recurrent bifurcation conjecture. The four bifurcations are repeated recurrently in the same sequence infinitely many times before the wave profile becomes peaked.

Let $\lambda_\varepsilon”(0)$ denote the numerically computed second derivative of the critical spectral curve, and let $-k\varepsilon^{2/3}$ be its leading-order analytical approx…

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…

Consider the integrodifference equation (IDE1) with Ricker growth, and let $n_+$ and $n_-$ denote the two states to be connected by a spatially inhomogeneous 2-cycle. Ricker two-cy…

Let $H$ denote the Hamiltonian and $c$ the speed of a family of traveling periodic Stokes waves, parameterized by the steepness $s$, which approaches a limiting peaked wave as $s$…

Let $\lambda_{1}(3,\omega)$ denote the principal eigenvalue at endpoint parameter $\mu=3$ for the first Fourier mode, and let $s$ be the spatial variable. For $\omega\in[-18,-3]$,…

Let $u\in H^{1,a}(\mathbb{R}^d)\backslash\{0\}$ be a ground state for the degenerate nonlinear Schrödinger equation, meaning that … where … and … is the associated Nehari manifold.…

Critical-frequency stability conjecture. For each $p>6$, there exists a critical value $\omega^*(p)$ such that the waves $\phi_{\omega,p}$ are unstable for $\omega>\omega^*(p)$ and…

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 $\ell=1,\ldots,N$.

Right-lobe mode conjecture. The most unstable mode on the right lobe is either the $\mu=0$ mode or the $\mu=1/2$ mode, and these modes interchange an infinite number of times as…

Benjamin–Feir branch conjecture. An infinite number of Benjamin–Feir instability branches exist as the steepest wave is approached, and all of them experience a universal sequence…

Instability conjecture. Other lattices should still give a convolution structure, but with greater computational complexity, leading to a complex, non-real matrix $A_\tau(\xi)$ and…

Discriminant asymptotic conjecture. In general,

Let $E(\lambda,\eta,\xi)$ be the Evans function associated with the hydraulic-wave spectral problem, and let $\varepsilon(\lambda,\eta,\xi)$ be the characteristic determinant of th…

Let $(F,H_L,H_R)$ define a discontinuous planar hydraulic shock in the whole space, with $2<F<F_{\text{2d}}$. A dispersion relation is $(\mu_L,\mu_R)$-weighted stable when its esse…

The model admits shock-fronted travelling wave solutions, including monotone, nonmonotone, and canard-type families. Geometric destabilization conjecture. Nonmonotonicity may be a…

The effective potential of a black hole may admit two competing purely imaginary quasinormal modes, with one mode spectrally stable and dominant and the remaining modes appearing a…

Weak-profile spectral stability conjecture. If $1<\gamma<3$ and