The intertwined derivative Schrödinger system of Calogero-Moser-Sutherland type (Q6561408)

The paper studies the intertwined derivative Schrödinger system of Calogero-Moser-Sutherland (CMS) type, specifically for a cubic \(M \times N\) matrix-valued equation. The system is described by the equations:\N\[\Ni \frac{\partial}{\partial t} U + \frac{\partial^2}{\partial x^2} U = - \frac{1}{2} U (D + |D|)(V^* U) - \frac{1}{2} V (D + |D|)(U^* U),\N\]\N\[\Ni \frac{\partial}{\partial t} V + \frac{\partial^2}{\partial x^2} V = - \frac{1}{2} V (D + |D|)(U^* V) - \frac{1}{2} U (D + |D|)(V^* V),\N\]\Nwhere \( D = -i \frac{\partial}{\partial x} \), and \( U, V \in \mathbb{C}^{M \times N} \). The operator \( |D| \) is the Fourier multiplier defined by \( \xi \mapsto |\xi| \). These equations exhibit invariance under transformations such as scaling, translation, multiplication by unitary matrices, Galilean transformations, and pseudo-conformal transformations. These symmetries are vital in analyzing the physical properties of the system.\N\NThe paper also considers a reduction of the system when \( V = \mu U \) (with \( \mu \in \mathbb{C} \)), leading to a simpler form of the equation, which can represent both focusing and defocusing CMS derivative cubic Schrödinger equations. The equations are analyzed in various contexts: when \( \text{Re}(\mu) > 0 \), the system describes focusing dynamics; when \( \text{Re}(\mu) < 0 \), it corresponds to defocusing dynamics. Additionally, when \( \text{Re}(\mu) = 0 \), the system reduces to the linear Schrödinger equation.\N\NA critical part of the study is the connection to the Calogero-Moser-Sutherland (CMS) models, both on the 1-dimensional torus \( T \) and the real line \( \mathbb{R} \). For instance, the scalar version of the focusing equation is interpreted as the thermodynamic limit of the CMS model on a 1-dimensional torus, and the defocusing equation can be related to the classical Calogero-Moser model in the continuum limit.\N\NThe phase spaces of these equations are studied using filtered Sobolev spaces \( H^+_s(M; \mathbb{C}) \), and global well-posedness results are derived for the focusing and defocusing equations. The paper leverages the Lax pair structure, inherited from the classical scalar cases, to establish well-posedness in these spaces. In particular, the defocusing equation is globally well-posed in \( H^+_s(M; \mathbb{C}) \), \( s > \frac{3}{2} \) for arbitrary initial data, and the focusing equation is globally well-posed for small initial data in \( L^+_2(M; \mathbb{C}) \).\N\NAn essential result of the paper is the derivation of an explicit formula for the solution \( (U(t), V(t)) \) in terms of the initial conditions \( (U(0), V(0)) \) and the time variable \( t \). This formula is obtained by extending the Lax pair structure from the scalar equations to the matrix-valued intertwined system. The proof of local well-posedness follows from applying classical iterative schemes to control the nonlinearity in Sobolev spaces.