## Abstract

We consider the three-dimensional Gross–Pitaevskii equation (Equation presented) and construct traveling wave solutions to this equation. These are solutions of the form ψ (t, x) = (x_{1}, x_{2}, x_{3} - Ct) with a velocity C of order ∊|log ∊| for a small parameter ∊ > 0. We build two different types of solutions. For the first type, the functions u have a zero-set (vortex set) close to a union of n helices for n ≥ 2 and near these helices u has degree 1. For the second type, the functions u have a vortex filament of degree -1 near the vertical axis e_{3} and n ≥ 4 vortex filaments of degree C1 near helices whose axis is e_{3}. In both cases the helices are at a distance of order 1/(∊√|log ∊|) from the axis and are solutions to the Klein–Majda–Damodaran system, supposed to describe the evolution of nearly parallel vortex filaments in ideal fluids. Analogous solutions have been constructed recently by the authors for the stationary Gross–Pitaevskii equation, namely the Ginzburg–Landau equation. To prove the existence of these solutions we use the Lyapunov–Schmidt method and a subtle separation between even and odd Fourier modes of the error of a suitable approximation.

Original language | English |
---|---|

Pages (from-to) | 1319-1367 |

Number of pages | 49 |

Journal | Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire |

Volume | 39 |

Issue number | 6 |

Early online date | 24 May 2022 |

DOIs | |

Publication status | Published - 18 Jan 2023 |

### Bibliographical note

Funding Information:Funding. J. Dávila has been supported by a Royal Society Wolfson Fellowship, UK. M. del Pino has been supported by a Royal Society Research Professorship, UK. M. Medina has been partially supported by Project PDI2019-110712GB-100, MICINN, Spain. R. Rodiac has been partially supported by the ANR project BLADE Jr. ANR-18-CE40-0023.

## Keywords

- helices
- Traveling waves equations

## ASJC Scopus subject areas

- Analysis
- Mathematical Physics
- Applied Mathematics