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 $${\partial }_x^{4}q-2\sqrt{2}{\partial }_x^{2}q-3\sqrt{2}{\partial }_x(({\partial }_{x}q{)}^2)-2{\partial }_y^{2}q=0.$$ We prove the existence of a traveling wave solution $ u_{\e} (x-ct, y)$ to GP (Gross-Pitaevskii) equation $$ i\partial_{t}Ψ+ΔΨ+(1-|Ψ|^{2})Ψ=0,\ \ \ \mbox{in} \ {\mathbb R}^2 $$ 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 $$\partial_x^4-(2\sqrt{2}-\e^2)\partial_x^2-2\partial_y^2+\e^2\partial_x^2\partial_y^2+\e^4\partial_y^4.$$

55 pages; comments welcome

References in corpus (1)

• Travelling waves for the Gross-Pitaevskii equation II