Nondegeneracy of the traveling lump solution to the $2+1$ Toda lattice

We consider the $2+1$ Toda system $$\frac{1}{4}\mathrm{\Delta }{q}_n=e^{q_{n-1}-q_n}-e^{q_n-q_{n+1}}\text{ in }{\mathbb{R}}^2,n\in \mathbb{Z}\mathrm{.}$$ It has a traveling wave type solution $\left\{Q_n\right\}$ satisfying $Q_{n+1}(x,y)=Q_n(x+\frac{1}{2\sqrt{2}},y)$, and is explicitly given by $$Q_n(x,y)=\ln \frac{\frac{1}{4}+{(n-1+2\sqrt{2}x)}^2+4y^2}{\frac{1}{4}+{(n+2\sqrt{2}x)}^2+4y^2}\mathrm{.}$$ In this paper we prove that {$Q_n$} is nondegenerate.