9 problems

Let $X_{s,b}$ denote the Bourgain space associated with the KdV dispersion relation, consisting of distributions whose space-time Fourier transform is weighted by…

In an integrable system such as KdV, scattering is elastic and factorised: soliton spectral parameters are preserved and multi-soliton scattering decomposes into two-soliton events…

Let $k\in(0,1)$ be the elliptic modulus and let $c_d=c_d(\lambda)$ denote the velocity of a dark breather parametrized by $\lambda\in(\lambda_2(k),\lambda_3(k))$. Dark breather vel…

Let $k$ be the elliptic modulus and let $c_b=c_b(\lambda)$ denote the velocity of a bright breather parametrized by $\lambda\in(-\infty,\lambda_1(k))$. Bright breather velocity con…

KPZ with flat initial condition is described by a KdV tau function, with $u(x,t;\nu)=2\partial_x^2\log\tau(x,t;\nu)$ solving the KdV equation. In the short-time regime, the KdV fie…

Let $U(t)$ denote the Koopman operator associated with the KdV flow on the underlying probability space, acting on observables by composition with the flow, and let the projection…

Korotyaev–Kuksin conjecture. The function $H^{*}$ extends analytically to a function on $\ell^{0,2}_{+}(\mathbb{N})$ and is strictly concave near $0$.

For the KdV equation, let $\Psi$ denote a Birkhoff map and let $\Psi_0$ be its linearization at the equilibrium point $q=0$. A map is 1-smoothing when its nonlinear part has the sm…

Benjamin's stability conjecture. Cnoidal waves are stable with regard to small perturbations. This conjecture concerns the stability of periodic solitary waves under small perturba…