Note on nonlinear Schrödinger equation, KdV equation and 2D topological Yang–Mills–Higgs theory

Abstract

In this paper, we discuss the relation between the [Formula: see text]D nonlinear Schrödinger equation and the KdV equation. By applying the boson/vortex duality, we can map the classical nonlinear Schrödinger equation into the classical KdV equation in the small coupling limit, which corresponds to the UV regime of the theory. At quantum level, the two theories satisfy the Bethe ansatz equations of the spin-[Formula: see text] [Formula: see text] chain and the [Formula: see text] chain in the continuum limit, respectively. Combining these relations with the dualities discussed previously in the literature, we propose a duality web in the UV regime among the nonlinear Schrödinger equation, the KdV equation and the 2D [Formula: see text] topological Yang–Mills–Higgs theory.