Coercivity for travelling waves in the Gross-Pitaevskii equation in $\mathbb{R}^2$ for small speed

In the previous paper, we constructed a smooth branch of travelling waves for 2 dimensional Gross-Pitaevskii equation. Here, continue study this branch. We show some coercivity results, and deduce from them kernel linearized operator, spectral stability result, as well uniqueness result in energy space. particular, our proves non degeneracy these waves, which is key step classification construction multi-travelling waves.