47 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…

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

Consider the nonlinear Schrödinger equation in the setting of the paper, and let $Q$ denote the ground state with critical mass $\|Q\|_{L^2}^2$. Let the symmetries of the equation…

Let $\tau$ be a polynomial defining a KP lump solution of the form … with $k\in\mathbb{N}$ and $\deg\tau=k(k+1)$. A symmetric solution is one invariant under the reflections…

Nondegeneracy conjecture. For every $k\in\mathbb{N}$, the only smooth, evanescent solutions of this linearised equation are linear combinations of $\partial_xu_k^\star$ and…

Let $u_k^\star(x,y)=-2\partial_x^2\log \tau_k^\star(x,y)$ be the explicit symmetric lump solutions of the normalised KP equation … where $k\in\mathbb{N}$ and $\tau_k^\star$ is a po…

Stability conjecture. Let $1\leq d\leq 6$. For $\alpha\leq 2/d$, the solution $\varphi_\omega$ minimizes $\mathcal{E}$ at mass $M=M(\omega)$ for all $\omega\in(0,\omega^*)$ and is…

The Euler–Poisson system admits small-amplitude solitary waves whose weakly transverse long-wave regime is described by the KP-II equation. Transverse-stability conjecture. Such so…

Embedded-spectrum conjecture. No embedded spectrum appears for $\mathcal{H}$ if $Q$ is a ground state; equivalently, $\mathcal{H}$ has no eigenvalues embedded in its essential spec…

Asymptotic stability conjecture. The line solitary waves of the Amick-Schonbek system are asymptotically stable.

Let $\Omega$ be a regular polygon with at least seven edges and a center, in the setting of the damped nonlinear Klein–Gordon equation and the multi-soliton construction described…

Radiation–multisolitary-wave asymptotic decomposition conjecture. For nonlinear dispersive PDEs such as gKdV, NLS, and NLW, a generic solution evolves asymptotically as the sum of…

A solitary water wave is considered in a two-dimensional channel of finite depth with rotational, unidirectional flow. Its Bernoulli constant, mass flux, and flow force are the ass…

The transverse stability problem concerns perturbations that depend weakly on the transverse spatial variables. In the small-amplitude, weakly transverse long-wave regime, the gove…

Asymptotic stability conjecture. Asymptotic stability of solitary waves holds for this semilinear model.

Let $v$ be a solitary-wave solution of the Benjamin equation, with spatial variable $x\in\mathbb R$. The known asymptotic result gives algebraic decay of the profile at spatial inf…

Consider the Benjamin solitary-wave equation … Here $v$ is a solitary-wave profile, $c$ is its velocity, and $\alpha,\beta$ are the Benjamin-equation parameters. Solitary-wave exis…

Asymptotic stability conjecture. A perturbation of the line solitary wave leads, for sufficiently large times, to the unperturbed line solitary wave plus radiation.

Defocusing conjecture. There are no stable SGN solitary waves localized in two dimensions, and the equation has a defocusing effect.

Main conjecture I. The line solitons $Q_c$ for $p=2$ and $c>c^*$ are unstable against the formation of lumps.

Let $m\in\mathbb{N}$ and $\tilde{c}>0$. For sufficiently small $\varepsilon>0$, consider the FPU system and the speed $c=1-\varepsilon^2\tilde{c}$. A generalized moving pulse solut…

Let $F(u)=1-\cos(u)$, so that the variable-coefficient sine-Gordon equation is a perturbation of … and let the coefficients satisfy the conditions denoted by. The kink is the stati…

Consider the constant-coefficient $phi^4$ equation … with nonlinearity $F'(u)=u^3-u$, and its kink solution. Asymptotic stability means that solutions starting near the kink in the…

Let $I_c$ be the variational functional associated with the stationary nonlinear Maxwell–Klein–Gordon equation, and let $v_infty$ be the global minimizer of $I_\infty$ described ab…

Let $u^*$ and $\mathring{\varrho}$ denote the background velocity and density profiles in each layer, and let $\Omega_+$ and $\Omega_-$ be the upper and lower fluid domains. Let…