  In this paper we discuss the relation between the (1+1)D nonlinear
Schr\"odinger equation and the KdV equation. By applying the boson/vortex
duality, we can map the classical nonlinear Schr\"odinger 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-$\frac{1}{2}$ XXX chain and the XXZ 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 Schr\"odinger equation, the KdV equation and the 2D
$\mathcal{N}=(2,2)^*$ topological Yang-Mills-Higgs theory.
