The Calogero--Moser Derivative Nonlinear Schrödinger Equation

Abstract

We study the Calogero--Moser derivative NLS equation $$i{\partial }_{t}u+{\partial }_{xx}u+(D+|D|)(|u{|}^2)u=0$$ posed on the Hardy-Sobolev space $H_{+}^s(\mathbb{R})$ with suitable $s>0$. By using a Lax pair structure for this $L^2$-critical equation, we prove global well-posedness for $s\ge 1$ and initial data with sub-critical or critical $L^2$-mass $\Vert u_0{\Vert }_{L^2}^2\le 2\pi$. Moreover, we prove uniqueness of ground states and also classify all traveling solitary waves. Finally, we study in detail the class of multi-soliton solutions $u(t)$ and we prove that they exhibit energy cascades in the following strong sense such that $\Vert u(t){\Vert }_{H^s}{\sim }_s|t{|}^{2s}$ as $t\to \pm \infty$ for every $s>0$. \end{abstract}

43 pages. This is a revised version with an extended introduction. Main results are now stated as theorems in the introduction. Various typos corrected. Comments are welcome

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