21 problems

Klein–Roudenko–Stoilov's conjecture. Blow-up happens in finite time, and blow-up solutions have some resemblance to being self-similar: the blow-up core forms a rightward-moving se…

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…

Consider the generalized Benjamin–Ono equation with nonlinearity exponent $k\geq 4$, and let $u$ be a solution. For $\lambda>0$, define the scaling … Then $u_\lambda$ is again a so…

Scattering conjecture for critical gKdV. There exist functions

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…

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…

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…

Let a nonlinear dispersive equation have dispersion of order $l$ and a nonlinearity of order $k\geq 3$ with a loss of $m$ derivatives. Let $s_0$ be such that, for $s>s_0$, the…

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…

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…

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

Global existence and blow-up threshold conjecture. If

Let $m>3$, let $u_0\in X\cap H^{1/2}(\mathbb R)$ be sufficiently localized, where $X$ is the corresponding local well-posedness space for the given $m$, and let $Q$ be the ground s…

Let $u_0\in H^{1/2}(\mathbb R)$ be sufficiently localized, let $Q$ be the ground state solution of the equation defining $Q$ with $m=3$, and let $u(x,t)$ be the mBO time evolution…

Large-velocity solitary-wave conjecture. For every $c>1$, there exist smooth and rapidly decaying solitary-wave solutions with velocity $c$ such that: (i) the profiles $\zeta_c$ an…

The long-time behavior of solutions to nonlinear dispersive equations is considered in Euclidean spaces, with $t\to +\infty$. A solution is said to resolve into a train of solitons…

A nonlinear dispersive equation has a dispersionless solution whose shocks are surrounded by regions of modulated oscillations. Complex-singularity conjecture. These regions can be…

Let the focusing Davey–Stewartson II equation be posed with generic localized initial data, and let $t$ denote time. Generic blow-up or radiation conjecture. Generic localized init…

A nonlinear dispersive equation is considered, with global-in-time solutions and coherent structures called solitons. Soliton resolution conjecture. Generic global-in-time solution…

Suppose that $u \in C^0([0,T);H^{1/2}(\mathbb{R}^3))$ solves the boson star equation with $m\geqslant 0$ and blows up at a finite time $0<T<+\infty$. Let $u^*\in L^2(\mathbb{R}^3)$…

Conjecture on Maxwell–Dirac convergence. A similar consideration should allow one to prove convergence to nonlinear eigenstates in systems such as coupled Maxwell–Dirac equations,…