Supersolidity-related phenomena in holographic superfluid: cnoidal wave

The candidate of supersolid-like state, cnoidal wave, is investigated in finite temperature holographic superfluid model for the first time. We find by giving different source $a_x$ for the particle current density $\langle j^{x}\rangle$, different kinds of superflow states can be obtained numerically. And based on $a_x$, three limiting cases of cnoidal waves can be found easily as well as the general cnoidal wave solution. The chemical potential $\mu$ and particle current density $\langle j^x \rangle$, as functions of source $a_x$, are calculated for these superflow states and are compared with homogeneous superflow. We find as we increasing $a_x$, cnoidal waves will enter into the uniform limit, which reflects in the confluence of the behavior of $\mu(a_x)$ and $\langle j^x(a_x) \rangle$ between cnoidal wave and homogeneous superflow. Based on the quasi-normal modes, we show the cnodial waves are always dynamically unstable due to the finite temperature, while the energetic stability is influenced by the particle current density. All these evidences manifest that the cnoidal wave can be seen as an \textcolor{black}{unstable excited states} with supersolidity in finite temperature superfluid system.