30 problems

Let $d\geq 3$ and consider the mass-critical nonlinear Schrödinger equation with nonlinearity $F(z)=+|z|^{4/d}z$. For initial data $u_0\in L^2_x(\mathbf{R}^d)$ and initial time…

Let a one-dimensional dispersive flow on the real line have a cubic defocusing nonlinearity, and let its initial data be sufficiently small. Non-localized data defocusing global we…

Consider the bilinear restriction estimate for free wave solutions, with data $f$ and $g$ localized at frequency approximately $1$ and with separated Fourier supports. The correspo…

Kadomtsev–Petviashvili stability conjecture. $Q(x-t)$ is stable under the KP-II flow and unstable under the KP-I flow.

Let a weakly nonlinear dispersive flow evolve from generic initial data. Soliton resolution conjecture. As $t\to+\infty$, the solution should split asymptotically into a finite col…

Let $u$ solve the one-dimensional biharmonic nonlinear Schrödinger equation with lower-dispersion parameter $a$, and consider the critical and supercritical regimes. A blow-up prof…

Consider a cubic quasilinear dispersive flow with small initial data in spatial dimension at least two. Higher-dimensional non-localized data global well-posedness conjecture. The…

Consider a one-dimensional dispersive flow on the real line with a cubic conservative nonlinearity and initial data of size $epsilon \ll 1$. Non-localized data long-time well-posed…

Small-data gauge-scattering conjecture. There exist unique $\varphi^{\pm},\psi^{\pm}\in H^1$ such that

Let $u$ be a solution of the generalized Korteweg–de Vries initial-value problem referenced in the source, with nonlinearity exponent $k\in[1,2]$. Rammaha's conjecture. Such soluti…

The Intermediate Long Wave equation has small initial data of size $epsilon$ and dispersion relation … which is KdV-like at low frequencies and Benjamin–Ono-like at high frequencie…

Consider the post-adiabatic regime of the one-dimensional damped nonlinear Schrödinger equation, where the dynamics are described by the linear Schrödinger equation … Let $M_c$ den…

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…

The paper considers a complex spectral method for analysing linear dispersive partial differential equations posed on a finite spatial interval, including their initial and boundar…

Consider the inhomogeneous nonlinear Schrödinger equation and a general solution $u$ that is neither assumed to have finite variance nor to be radially symmetric. Suppose that … fo…

Eventual energy-retraction conjecture. All the energy will eventually retract into any given light cone; equivalently, for every real number $\eta$, the above quantity should conve…

Consider the defocusing semilinear wave equation on $\mathbb{R}^{1+2}$ with power $p$, for sufficiently smooth and localized initial data. Sharp-decay conjecture. If $p>3$, the sol…

Consider a system of nonlinear wave equations with sufficiently small initial data localized around $N$ points, as in the stated global-existence theorem. Let $d_\Pi$ denote the ra…

Let $u=u(t,x)$ be a real-valued function on $\textnormal{I}\!\textnormal{R}$ satisfying … where $\mathcal{F}_x\{L_pu\}(t,\xi)=|\xi|^p\mathcal{F}_xu(t,\xi)$ and $0\leq p\leq 2$. Set…

Consider the Maxwell–Klein–Gordon equations on three-dimensional Euclidean space with finite-energy initial data, including data with non-vanishing charge and arbitrarily large siz…

Let $\psi$ solve the three-dimensional Gross–Pitaevskii equation, and write $u=\psi-1$. For traveling waves of the form $\psi(t,x)=v(x-ct)$, define … where $E$ is the conserved Gro…

For the defocusing supercritical wave equation on $\mathbb R^3$, let $u$ be a solution on a time interval $I$ with energy $E[u]$. Defocusing dispersion conjecture. All solutions sh…

Let $V$ be a smooth, radially symmetric, unimodal potential well that decays rapidly at spatial infinity, and suppose that $-\aDelta+V$ has finitely many bound states. Consider the…

Consider initial data that are the $x$-derivative of a rapidly decreasing smooth function in $L_{2}(\mathbb{R}^{2})$ with a single maximum. Write the two-dimensional dispersionless…

Breaking-dynamics conjecture. All, or at least most, even initial data characterized by zero initial velocity and a single vacuum point in the density should follow a similar break…