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

Abstract

Let $q(x,y)$ be an nondegenerate lump solution to KP-I (Kadomtsev-Petviashvili-I) equation $$\partial_x^4q-2\sqrt{2}\partial_x^2q-3\sqrt{2}\partial_x((\partial_xq) ^2)-2\partial_y^2q=0. $$ We prove the existence of a traveling wave solution $ u_{\e} (x-ct, y)$ to GP (Gross-Pitaevskii) equation $$ i\partial_{t}\Psi+\Delta\Psi+(1-|\Psi|^{2})\Psi=0,\ \ \ \mbox{in} \ {\mathbb R}^2 $$ in the transonic limit $$ c=\sqrt{2}-\epsilon^2 $$ with $$ u_\epsilon =1 + i \epsilon q(x,y) + {\mathcal O} (\epsilon^2). $$ This proves the existence of finite energy solutions in the so-called Jones-Roberts program in the transonic range $ c \in (\sqrt{2}-\epsilon^2, \sqrt{2})$. The main ingredients in our proof are detailed point-wise estimates of the Green function associated to a family of fourth order hypoelliptic operators $$\partial_x^4-(2\sqrt{2}-\e^2)\partial_x^2-2\partial_y^2+\e^2\partial_x^2\partial_y^2+\e^4\partial_y^4.$$