An analogous Wu-Yang monopole in superfluid

Abstract

By identifying the Schr\"{o}dinger equation with the hydrodynamic equations in superfluid ${^3}$He, the effective potential is introduced in the Schr\"{o}dinger equation to solve the quantum pressure in steady state. The pure gauge velocity solutions of hydrodynamic equations provide an analogous Wu-Yang monopole potential. The two cases of velocity in superfluid are equivalent to the two regions of Wu-Yang monopole potential. Due to the compressibility of superfluid, the physical models are limited, such as hard core and harmonic oscillator. It is important that the constraint condition of $\nabla\rho\cdot \textbf{v}=0$ plays a key role, which determines that the integral quantum numbers are selected in monopole harmonics and shows the special analogous Wu-Yang monopole. The results provide a new possibility for the simulation of Wu-Yang monopole.