20 problems

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…

A diatomic FPUT lattice has a mass ratio, denoted by $\mu$, and a one-sided nanopteron is a travelling wave with an oscillatory tail on only one side of its central leading wave. G…

Non-trapping obstacle conjecture. The local energy of solutions should decay for any non-trapping obstacle, and consequently such solutions should scatter.

Let $K_\alpha$ be the kernel of the resolvent operator $(D^\alpha+2)^{-1}$ on $\mathbb R$, with $\alpha\in(2,4)$. Le–Pelinovsky conjecture. The kernel $K_\alpha$ is sign-changing f…

Let the $J$-bF equation be the higher-order $b$-family equation denoted by. A $b$-dependent peakon is a real peaked travelling-wave solution whose parameters depend on $b$. The $b$…

Let the $J$-bF equation be the higher-order $b$-family equation denoted by, with wave speed $c$ and constants $a_i$. A $b$-independent peakon is a peaked travelling-wave solution w…

Equivariant wave maps from the $2+1$-dimensional wormhole to the $2$-sphere admit explicit harmonic map solutions which, in suitable coordinates, are sine-Gordon kinks and anti-kin…

Existence conjecture. Space- and time-periodic solutions to the Euler equations should exist.

The paper considers periodic localized traveling-wave solutions of the two-dimensional suspension bridge equation, including solutions with one-peak and two-peak patterns and a sol…

Revolution conjecture. For each positive integer $n$, there exists an initial condition such that a trajectory coming into $R_2$ from $R_1$ performs $n$ clockwise revolutions aroun…

Let $C_5$ be the coefficient obtained in the asymptotic analysis of the tailing oscillations, and let $\alpha$ denote the parameter governing the nonlinear woodpile chain. For…

Integrability–reflectionless transmission conjecture. If the nonlinear wave equation on a network is integrable, then the transmission of solitons through the branching points beco…

Long-range interactions (LRI) can generate mean-field potentials through the coherent addition of microscopic interactions across the sample. Generic coherent addition conjecture.…

Let the discrete nonlinear Schrödinger equation with saturation be … where $h$ is the lattice spacing and $\theta$ is the saturation parameter. Consider the transparent points for…

Third-peakompacton birth conjecture. The marked shortening of the shorter peakompacton after an overtaking collision is due to the birth of a third, trailing peakompacton at the mo…

Uniqueness conjecture. The solution is unique in the class of $L^2\times L^2$ functions.

The Fermi–Pasta–Ulam chain has $N$ identical masses connected by nonlinear springs, with the elastic force expanded as … The original $\alpha$-FPU system is the case $\alpha\ne0$ a…

Universality conjecture. The critical behaviour at a dispersive shock is universal in this sense.

Quasi-equilibrium conjecture. Quasisolitons and quasibreathers exist only in quasi-equilibrium with a weakly nonlinear wave-noise environment.

Let $0<\kappa\leq\pi/2$, $q\in\mathbb{R}$, and $\gamma>0$. Theorem 1 gives a three-parameter family of wavetrains with profiles $V\in\mathcal{C}$, where $\mathcal{C}$ is the admiss…