From KP-I lump solution to travelling waves of Gross-Pitaevskii equation

Let q be a nondegenerate lump type solution to the KP-I (Kadomtsev-Petviashvili-I) equation${\partial }_x^{4}q-2\sqrt{2}{\partial }_x^{2}q-3\sqrt{2}{\partial }_x\left({\left({\partial }_{x}q\right)}^2\right)-2{\partial }_y^{2}q=0.$ We show the existence of travelling wave solutions with the form $u_{\epsilon }(x-ct,y)$, for the GP (Gross-Pitaevskii) equation$i{\partial }_t\Psi +\Delta \Psi +(1-|\Psi {|}^2)\Psi =0\text{in}{R}^2,$ with travelling speed $c=\sqrt{2}-{\epsilon }^2$, and $u_{\epsilon }=1+i\epsilon q+O\left({\epsilon }^2\right)$. This proves the existence of finite energy solutions in the so-called Jones-Roberts program within the transonic regime $c\in (\sqrt{2}-{\epsilon }^2,\sqrt{2})$. The main ingredients in our proof are detailed point-wise estimates for the Green functions associated to a family of fourth order hypoelliptic operators. In view of the classification of lump type solutions of the KP-I equation, our proof also indicates that for fixed small ε, there should exist a sequence of travelling wave solutions to GP equation, with energy tends to infinity.