Turbulent Threshold for Continuum Calogero-Moser Models

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

13 pages

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• Integrable hydrodynamics of Calogero-Sutherland model: Bidirectional Benjamin-Ono equation
• An explicit formula for the Benjamin--Ono equation
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Cited by in corpus (1)

• Dispersive decay for the energy-critical nonlinear Schrödinger equation