8 problems

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…

Gaussian absolute-continuity conjecture. If $4\mu$ is a suitable Gaussian measure on $h^{s+\frac{1}{2}}$, then the corresponding measure $4\rho_s$ is absolutely continuous with res…

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…

Invariance conjecture. The measure $\rho_{k/2,R}$ is invariant under a well-defined flow of the Benjamin–Ono equation.

Benjamin–Ono uniqueness conjecture. The function $U(X,T)$ is uniquely characterized as the solution of the string equation satisfying this boundary behaviour.

Benjamin–Ono string-equation conjecture. The critical behaviour of solutions to the Benjamin–Ono equation is described by a particular solution of this singular integro-differentia…

Invariance conjecture. The measures $\rho_{k/2,R}$ are invariant under the Benjamin–Ono flow, at least for even values of $k$.