Turbulent Threshold for Continuum Calogero-Moser Models

We determine the sharp mass threshold for Sobolev norm growth for the focusing continuum Calogero–Moser model. It is known that below the mass of $2\pi$, solutions to this completely integrable model enjoy uniform-in-time $H^s$ bounds for all $s\ge 0$. In contrast, we show that for arbitrarily small $\epsilon >0$ there exists initial data $u_0\in H_{+}^{\mathrm{\infty }}$ of mass $2\pi +\epsilon$ such that the corresponding maximal lifespan solution $u:(T_{-},T_{+})\times \mathbb{R}\to \mathbb{C}$ satisfies ${\lim }_{t\to T_{\pm }}\mathrm{\parallel }u(t){\mathrm{\parallel }}_{H^s}=\mathrm{\infty }$ for all $s>0$. As part of our proof, we demonstrate an orbital stability statement for the soliton and a dispersive decay bound for solutions with suitable initial data.