12 problems

Hessian determinant conjecture. The determinant satisfies

Generic phase-plateau behavior conjecture. This behavior should be generic across Hamiltonian models with symmetry: localized perturbations should produce phase profiles whose expa…

Let the traveling periodic wave be a periodic-wave solution of either version of the nonlocal derivative nonlinear Schrödinger equation described in the paper. Linear stability con…

The focusing modified Korteweg–de Vries equation admits two families of periodic traveling waves, and spectral instability bands can exhibit figure-$8$ patterns, crossing at the im…

Periodic-wave existence conjecture. There exist complex-valued periodic solutions of the difference equation with

Let $\varphi$ be a periodic wave with parameter $\mu$, and suppose the equation has an antisymmetric nonlinearity. The two normalization assumptions under consideration are…

Consider the periodic extreme waves in the negative-order dispersive equations under discussion, with dispersion parameter $\alpha$. For $\alpha>1$, these waves are Lipschitz and h…

Uniqueness hypothesis conjecture. The hypothesis $$ should be valid for all $cfrac{1}{2}$.

Let $\psi_0$ be the even, single-lobe periodic wave from the even periodic-wave theorem, write … and define $\omega_0=c_0-6a_0^2$. Let $\phi(\cdot,\omega,a)$ be the associated loca…

Let $\alpha \in \left(\frac{1}{2},2\right]$ and $c \in (-1,\infty)$. Let $\psi$ be the odd, single-lobe periodic wave obtained as in the odd periodic-wave theorem, and let…

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