发表机构
University of Macau; Academy of Mathematics and Systems Science, Chinese Academy of Sciences; University of Chinese Academy of Sciences; The Chinese University of Hong Kong（澳门大学; 中国科学院数学与系统科学研究院; 中国科学院大学; 香港中文大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了平面Gross--Pitaevskii方程在每一亚声速下存在有限能量行波，解决了长期开放问题，并建立了能量估计与有限气泡紧性。
AI 中文摘要
对于每一亚声速$c\in(0,\sqrt2)$，我们证明了平面Gross--Pitaevskii方程存在有限能量的行波。这解决了二维空间中指定速度行波解存在的长期问题，该问题被Mariş（Ann. of Math., 2013）和Bellazzini与Ruiz（Amer. J. Math., 2023）明确列为开放问题。证明本质上依赖于能量估计\begin{equation*} E(\psi)\le C_J\bigl(I_c(\psi)+\ind(\psi)\bigr), \qquad c\in J, \end{equation*}其中$E$是能量，$I_c$是速度$c$下的作用量，$\ind$是实Morse指标，$J$是包含于$(0,\sqrt2)$中的任意紧区间，$C_J$是仅依赖于$J$的正常数。我们还证明了有限气泡紧性，包括能量、作用量、势能和动量的分裂，以及在Morse指标至多为一的非恒定波中作用量的取得。
英文摘要
For every subsonic speed $c\in(0,\sqrt2)$, we prove the existence of a finite-energy traveling wave for the planar Gross--Pitaevskii equation. This resolves the longstanding problem of the existence of prescribed-speed traveling-wave solutions in two dimensions, explicitly stated as open by Mariş (Ann. of Math., 2013) and Bellazzini and Ruiz (Amer. J. Math., 2023). The proof relies essentially on the energy estimate \begin{equation*} E(ψ)\le C_J\bigl(I_c(ψ)+\ind(ψ)\bigr), \qquad c\in J, \end{equation*} where $E$ is the energy, $I_c$ the action at speed $c$, $\ind$ the real Morse index, $J$ is any compact interval contained in $(0,\sqrt2)$, and $C_J$ is a positive constant depending only on $J$. We also prove finite-bubble compactness, including splitting of the energy, action, potential energy, and momentum, and attainment of the action among nonconstant waves of Morse index at most one.
