15 problems

Conjecture on the transmission exponent. The exponent in this asymptotic is exactly

Let a soliton-bearing system be perturbed by an ac force, and suppose that the soliton or solitary wave under consideration has no resonances with other modes intrinsic to the exci…

The $1+1$-dimensional nonlinear Dirac equation admits $mathcal{PT}$-symmetric perturbations, including the Alexeeva-Barashenkov-Saxena (ABS) model. A conservation law is a quanti…

Let $u(t)$ solve the one-dimensional fourth-order mixed-dispersion nonlinear Schrödinger equation with critical power $\alpha=8$, parameter $a$, initial datum $u_0$, and ground sta…

Consider a field theory with solitonic solutions, and distinguish the appearance of an inner substructure in a soliton from the existence of a larger number of normal modes around…

Weakly non-BPS two-scalar-field models are two-field theories in $(1+1)$ dimensions that are small departures from BPS models, so that the BPS structure is only weakly broken. In n…

Consider distant multiple bubbles for the energy-critical wave equation in dimension $5$, with infinite-time blow-up and bubble scales tending to zero. Uniqueness conjecture for th…

Soliton-component independence conjecture. If $\mu$ is $T$-invariant and $\theta$-mixing, then the $\zeta_k$ are independent over $k$, and each $\zeta_k$ is $\theta$-mixing.

Soliton-component decomposition conjecture. If $\mu$ is shift-mixing and $T$-invariant, then its soliton components should be independent and shift-mixing.

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…

Let $u^\varepsilon_s(t)$ and $u^\varepsilon_1(t)$ denote the solutions of the fractional and local Schrödinger Cauchy problems, respectively, and let…

Let $S_1$ be the set of initial data in $\dot{H}^1\times L^2$ whose corresponding radial solutions of the focusing, energy-critical wave equation scatter forward in time, let $S_2$…

Consider equivariant wave-map solutions that are close to a soliton in a stronger topology, with additional regularity and decay at infinity. Type 1 solutions are finite-time blow-…

Holmer–Zworski higher-Sobolev-norm conjecture. The error bounds

Consider oscillations about the minimal mass soliton coupled to the continuous spectrum of the nonlinear Schrödinger equation. Damped convergence conjecture. Such oscillations will…