Abstract

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 $𝜖>0$ there exists initial data $u_0\in H_{+}^{\infty }$ of mass $2\pi +𝜖$ such that the corresponding maximal lifespan solution $u:(T_{-},T_{+})\times ℝ\to ℂ$ satisfies $\underset{t\to T_{\pm }}{\lim <!--FUNCTION\; APPLICATION-->}\parallel u(t){\parallel }_{H^s}=\infty$ 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.

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