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PreprintCC BY 4.0arXiv

Non-uniqueness for the Complex Ginzburg-Landau Equation

  • Joel Dahne
  • Vladimír Šverák
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Abstract

We show that singularities of the three-dimensional cubic complex Ginzburg-Landau equation developing from smooth initial data can induce non-uniqueness of solutions after the time of blowup. The solutions we study start as backward self-similar solutions, and the source of the non-uniqueness is an unstable eigenvalue of the linearization around the forward self-similar profile generated by the singularity. We establish the existence of this eigenvalue, and hence the non-uniqueness, by a computer-assisted proof for specific values of the parameters in the equation. The work is in part motivated by similarities to conjectured behavior for the Navier-Stokes equation, with which the complex Ginzburg-Landau equation shares several important properties, such as an energy inequality and the scaling symmetry.

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Cite

BibTeXRIS
BibTeX
@misc{dahne2026nonuniqueness,
  title = {{Non-uniqueness for the Complex Ginzburg-Landau Equation}},
  author = {Joel Dahne and Vladimír Šverák},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.08985},
  eprint = {2609.08985},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.08985},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Non-uniqueness for the Complex Ginzburg-Landau Equation
AU  - Joel Dahne
AU  - Vladimír Šverák
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.08985
UR  - https://arxiv.org/abs/2609.08985
AB  - We show that singularities of the three-dimensional cubic complex Ginzburg-Landau equation developing from smooth initial data can induce non-uniqueness of solutions after the time of blowup. The solutions we study start as backward self-similar solutions, and the source of the non-uniqueness is an unstable eigenvalue of the linearization around the forward self-similar profile generated by the singularity. We establish the existence of this eigenvalue, and hence the non-uniqueness, by a computer-assisted proof for specific values of the parameters in the equation. The work is in part motivated by similarities to conjectured behavior for the Navier-Stokes equation, with which the complex Ginzburg-Landau equation shares several important properties, such as an energy inequality and the scaling symmetry.
ER  -