13 problems

Let $b1$ denote the dimensionless depth parameter for a small-amplitude Stokes wave, and let the high-frequency instability closest to the origin be the spectral instability associ…

Let $\beta_1^{(\mathtt{p})}:(0,+\infty)\to\mathbb{R}$ be the analytically depth-dependent coefficient associated with the isola of order $\mathtt{p}$, where $\mathtt{p}$ is an inte…

Transverse-instability conjecture. All small-amplitude finite-depth Stokes waves are unstable with respect to two-dimensional, or transverse, perturbations; in particular, the inde…

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…

Let $s^{\mu=0}_k$ denote the steepness of the $k$th coperiodic turning point of the wave speed, and let $s^{\mu=1/2}_k$ denote the steepness of the $k$th double-period bifurcation…

Let $\hat S_1$ be the linearization operator of the Babenko equation, and let a zero eigenvalue mean an eigenvalue equal to zero. Let the extreme wave denote the limiting Stokes wa…

Let a Stokes wave be a periodic traveling wave, and let the limiting Stokes wave denote the limiting-wave regime. A secondary bifurcation is a bifurcation from the primary branch o…

A Stokes wave is a periodic traveling wave of the water-wave equations, and a secondary bifurcation point is a bifurcation from the primary branch of Stokes waves. The limiting Sto…

Let a small-amplitude Stokes wave be a periodic traveling-wave solution of the water-wave equations, and call it spectrally unstable if its linearized operator has spectrum with po…

In the limit toward the steepest Stokes wave, the two main crest singularities must be recombined into Stokes' single crest singularity. The matching region between outer and inner…

In the complex potential description of a Stokes wave, the steepest possible wave has a crest singularity with complex velocity scaling like $f^{1/3}$, while away from the steepest…

Consider the solutions $z_1,z_2,\dots$ of … with $\operatorname{Re}(z)>-1$. These solutions are simple. Amick–Fraenkel conjecture. The set … is linearly independent over the ration…