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arXiv · 2609.08906

Planar Gross--Pitaevskii traveling waves at every subsonic speed

Changfeng Gui·Shanfa Lai·Guolin Qin·Juncheng Wei
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Abstract

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.

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Changfeng Gui, Shanfa Lai, Guolin Qin, Juncheng Wei. 2026-09-08. Planar Gross--Pitaevskii traveling waves at every subsonic speed. https://arxiv.org/abs/2609.08906

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