The intertwined derivative Schrödinger system of Calogero–Moser–Sutherland type

This paper is dedicated to extending the focusing/defocusing Calogero–Moser–Sutherland cubic derivative Schrödinger equations (CMSdNLS)

$$\begin{aligned} \small i\partial _t u + \partial _x^2 u = \pm u \left( \textrm{D} + |\textrm{D}| \right) \left( |u|^2 \right) , \quad \textrm{D}= -i\partial _x, \quad x \in \mathbb {R} \quad \textrm{or} \quad x \in \mathbb {T}:= \mathbb {R}/2 \pi \mathbb {Z}, \end{aligned}$$

which were initially introduced in Matsuno (Phys Lett A 278(1–2):53–58, 2000; Inverse Probl 18:1101–1125, 2002; J Phys Soc Jpn 71(6):1415–1418, 2002; Inverse Prob 20(2):437–445, 2004), Abanov et al. (J Phys A 42(13): 135201, 2009), Gérard and Lenzmann (The Calogero–Moser derivative nonlinear Schrödinger equation, Communications on Pure and Applied Mathematics. arXiv:2208.04105) and Badreddine (On the global well-posedness of the Calogero–Sutherland derivative nonlinear Schrödinger equation, Pure and Applied Analysis. arXiv:2303.01087; Traveling waves and finite gap potentials for the Calogero–Sutherland derivative nonlinear Schrödinger equation, Annales de l’Institut Henri Poincaré, Analyse Non Linéaire. arXiv:2307.01592), to a system of two matrix-valued variables, leading to the following intertwined system,

$$\begin{aligned} {\left\{ \begin{array}{ll} i\partial _t U + \partial _x^2 U = - \tfrac{1}{2} U \left( \textrm{D} + |\textrm{D}| \right) \left( V^* U\right) - \tfrac{1}{2} V \left( \textrm{D} + |\textrm{D}| \right) \left( U^* U\right) ,\\ i\partial _t V + \partial _x^2 V = - \tfrac{1}{2} V \left( \textrm{D} + |\textrm{D}| \right) \left( U^* V\right) - \tfrac{1}{2} U \left( \textrm{D} + |\textrm{D}| \right) \left( V^* V\right) .\\ \end{array}\right. } \end{aligned}$$

This system enjoys a Lax pair structure, enabling the establishment of an explicit formula for general solutions on both the 1-dimensional torus and the real line. Consequently, this system can be regarded as an integrable perturbation and extension of both the linear Schrödinger equation and the CMSdNLS equations.