32 problems

Let $u$ solve the one-dimensional fourth-order mixed-dispersion nonlinear Schrödinger equation, with power $alpha$ and ground-state solutions of the associated stationary equation.…

Let a solution of a nonlinear dispersive equation be considered in the asymptotic regime of time evolution. A modulated soliton is a soliton whose parameters, such as scale and pha…

Let $u$ be a solution of the focusing energy-critical wave equation in five spatial dimensions, with lifespan endpoint that may be finite or infinite. A soliton is a rescaled and t…

For a nonlinear dispersive equation, let $u(t)$ denote a solution at time $t$. A soliton is a localized coherent wave, and a radiation term is the dispersive remainder of the solut…

The energy-critical wave equation is an evolutionary dispersive equation whose maximally extended solutions are expected to exhibit soliton resolution. A dynamically scaled soliton…

Consider an energy-critical wave system … with $f$ homogeneous of degree $\frac{N+2}{N-2}$, and suppose that $f$ derives from a potential: there is a homogeneous function $F$ of de…

Let $\Phi$ be a global finite-energy solution of an energy-critical wave system, with maximal forward existence time $T_+=\infty$. A decomposition into rescaled stationary solution…

Higher-dimensional cubic-system conjecture. The analogs of the continuous soliton resolution and rigidity theorems should remain valid for a general system of this form with $f$ ho…

Let a nonlinear dispersive equation admit soliton solutions, meaning localized coherent structures that persist under the evolution. A global solution is understood to be generic i…

Consider the one-dimensional nonlinear Schrödinger equation … with standing-wave and traveling-wave solutions generated by a ground state $Q$. Soliton resolution conjecture. Genera…

Consider the focusing energy-critical wave equation and its data space. Let the collection of data for which soliton resolution holds denote the set of resolution data. Duyckaerts–…

A solution is considered in the setting of a nonlinear dispersive equation, with coherent structures meaning spatially localized solutions such as solitons, breathers, kinks, black…

Consider a global solution to a nonlinear dispersive equation whose norm is bounded in some function space. A soliton resolution is a decomposition into a sum of solitons, radiatio…

Soliton resolution conjecture. Stationary and travelling waves should be the only obstruction to scattering; consequently, solutions with mass below $m_c$ should scatter in both ti…

For a nonlinear dispersive equation, consider generic initial data and the corresponding solution at sufficiently large time. Soliton resolution conjecture. The solution separates…

Consider the nonlinear Schrödinger equation … where $u:{\mathbb R}^{1+3}\to \mathbb{C}$, $g\in C^\infty({\mathbb R},{\mathbb R})$, $g(0)=0$, and assume a family of positive radial…

For a solution $u$ of the generalized KdV equation, let $Q_c$ denote a rescaled soliton profile, let $a_j$ be shifts, and let $r(x,t)$ denote radiation. The soliton speeds are…

Let $J$ be the number of solitons in a solution of the energy-critical wave equation … on a space of dimension $N\geq 3$, and let … . This conjecture concerns the existence of radi…

Let $u$ be a radial solution of the focusing energy-critical wave equations on $\mathbb{R}^6$, with energy norm bounded in time. In this setting, the solitons are stationary states…

Consider the focusing energy-critical wave equations on $\mathbb{R}^6$ with radial initial data in the energy space $\mathcal{H}$, and suppose the solution is global in time. A sol…

Soliton resolution conjecture. Asymptotically, the $(n,t)$-space splits into regions separated by straight lines through the origin. Some regions are periodic, with period equal to…

The soliton resolution conjecture concerns solutions of a broad class of nonlinear dispersive equations. In the setting of (quasi-)equivariant wave maps, let…

Consider the focusing energy-critical wave equation … in odd space dimension $N\geq 3$, with radial initial data in the energy space…

Consider the Cauchy problem for a wave map … with finite-energy initial data $(u_0,u_1)$. Let $\mathcal{K}$ be a compact subset of $\mathbb{H}^2$, and let…

A soliton is a coherent, non-dispersive solution of a dispersive equation. A global bounded solution is a solution that exists for all $t$ and remains bounded in the relevant norm.…