77 problems

In infinite depth, consider nontrivial periodic water-wave solutions and a branch of such solutions bifurcating from a half-space. Stokes's conjecture. There is a branch of nontriv…

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…

Whitham's conjecture. The equation should be capable of predicting both breaking and peaking of waves. Here, breaking means the evolutionary formation of bounded solutions with inf…

The tWhitham equation is a model for traveling water waves, and a traveling-wave solution has a wave height, with maximal height meaning the greatest height attained among such sol…

The Whitham equation is … for $(x,t)\in\mathbb{R}\times\mathbb{R}^+$, where $\zeta(x,t)\in\mathbb{R}$ is the surface elevation and $\sqrt{\mathcal{K}_{\mu}}(D)$ is the square root…

We consider periodic traveling waves at the free surface of a two-dimensional, infinitely deep, incompressible inviscid fluid of constant vorticity, under gravity and without surfa…

Consider the $(r,\mathcal S)$-plane, where $r$ is the Bernoulli constant and $\mathcal S$ is the flow force constant. The cuspidal region is the region bounded by the two curves re…

Benjamin–Lighthill conjecture. The Bernoulli constant satisfies

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

The Ross Ice Shelf contains rifts whose growth has been associated with ocean-wave forcing. Flexural-gravity-wave conjecture. Flexural-gravity waves play an important role in the e…

Consider the large-amplitude solution shown in panel (B), for which the stagnation point in the upper layer above the crest is very close to the interface. Corner formation conject…

Let $\theta=0$ in the infinite-wavelength case, and consider perturbative solutions of the steady Euler equations with piecewise constant vorticity. Bore conjecture. Perturbative s…

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

Let $s$ denote the wave steepness, let $h$ be the conformal depth, and let $\bar h(s)$ be the corresponding physical depth, related by … The computations concern the fixed conforma…

Let $c$ denote the velocity and $\mathcal{H}$ the Hamiltonian along the tertiary branch of two-crested class $\mathrm{II}$ waves, which bifurcates from the primary branch at…

Let $s$ denote wave steepness and consider class $\mathrm{II}$ waves on the secondary branch, with one steeper crest that forms a corner in the limiting shape and another crest of…

Let $c$ be the velocity and $\mathcal{H}$ the Hamiltonian along the secondary branch of class $\mathrm{II}$ waves, and let…

Consider the primary branch of Stokes waves and the class $\mathrm{II}$ two-crested branches that originate at double-period bifurcation points, such as the secondary, tertiary and…

Let $s$ denote wave steepness and $c$ the velocity along the tertiary branch of class $\mathrm{II}$ two-crested waves, which bifurcates from the primary branch at $s_2=0.140487$. T…

Let $s$ denote wave steepness and $c$ the velocity along the secondary branch of $2\pi$-periodic class $\mathrm{II}$ Stokes waves, whose two crests have different amplitudes and wh…

Let $d>0$ and let $\beta_d$ be the kernel of the periodic Hilbert transform on the strip of depth $d$, defined for $s\in\mathbb{R}\setminus 2\pi\mathbb{Z}$ by the kernel formulas i…

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…