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PreprintarXiv licencearXiv

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∈(0,√2), 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 E(ψ)≤ CJ(Ic(ψ)+ind(ψ)), c∈ J, where E is the energy, Ic the action at speed c, ind the real Morse index, J is any compact interval contained in (0,√2), and CJ 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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Cite

BibTeXRIS
BibTeX
@misc{gui2026planar,
  title = {{Planar Gross--Pitaevskii traveling waves at every subsonic speed}},
  author = {Changfeng Gui and Shanfa Lai and Guolin Qin and Juncheng Wei},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.08906},
  eprint = {2609.08906},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.08906},
}
RIS
TY  - UNPB
TI  - Planar Gross--Pitaevskii traveling waves at every subsonic speed
AU  - Changfeng Gui
AU  - Shanfa Lai
AU  - Guolin Qin
AU  - Juncheng Wei
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.08906
UR  - https://arxiv.org/abs/2609.08906
AB  - For every subsonic speed c∈(0,√2), 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 E(ψ)≤ C_J(I_c(ψ)+ind(ψ)), c∈ J, 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,√2), 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.
ER  -