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- W3207986918 abstract "Let $q(x,y)$ be an nondegenerate lump solution to KP-I (Kadomtsev-Petviashvili-I) equation $$partial_x^4q-2sqrt{2}partial_x^2q-3sqrt{2}partial_x((partial_xq) ^2)-2partial_y^2q=0. $$ We prove the existence of a traveling wave solution $ u_{e} (x-ct, y)$ to GP (Gross-Pitaevskii) equation $$ ipartial_{t}Psi+DeltaPsi+(1-|Psi|^{2})Psi=0, mbox{in} {mathbb R}^2 $$ 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 $$partial_x^4-(2sqrt{2}-e^2)partial_x^2-2partial_y^2+e^2partial_x^2partial_y^2+e^4partial_y^4.$$" @default.
- W3207986918 created "2021-11-08" @default.
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- W3207986918 date "2021-10-28" @default.
- W3207986918 modified "2023-09-27" @default.
- W3207986918 title "From KP-I lump solution to travelling waves of Gross-Pitaevskii equation" @default.
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- W3207986918 doi "https://doi.org/10.48550/arxiv.2110.15472" @default.
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