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PreprintCC BY-NC-SA 4.0arXiv

On Periodic Solutions of the Zakharov-Rubenchik System

  • Yeison Alejandro Gómez Hernández
  • Juan Carlos Cordero Ceballos
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Abstract

We study the one-dimensional Zakharov-Rubenchik system with periodic initial conditions. This system models the interaction of low-frequency waves with high-frequency waves in different physical phenomena. Using semigroup theory, we prove the global well-posedness of the associated linear equation in Hq(𝕋)× Hs(𝕋)× Hs+1(𝕋), q,s ∈ ℝ. Subsequently, we consider the modified Zakharov-Rubenchik system and, by means of a change of variables and the Banach Fixed Point Theorem, we obtain local well-posedness in the space Hs(𝕋)× Hs(𝕋)× Hs+1(𝕋), s>3/2.

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Cite

BibTeXRIS
BibTeX
@misc{hernandez2026periodic,
  title = {{On Periodic Solutions of the Zakharov-Rubenchik System}},
  author = {Yeison Alejandro Gómez Hernández and Juan Carlos Cordero Ceballos},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.14158},
  eprint = {2609.14158},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.14158},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - On Periodic Solutions of the Zakharov-Rubenchik System
AU  - Yeison Alejandro Gómez Hernández
AU  - Juan Carlos Cordero Ceballos
PY  - 2026
DA  - 2026/09/12
PB  - arXiv
DO  - 10.48550/arxiv.2609.14158
UR  - https://arxiv.org/abs/2609.14158
AB  - We study the one-dimensional Zakharov-Rubenchik system with periodic initial conditions. This system models the interaction of low-frequency waves with high-frequency waves in different physical phenomena. Using semigroup theory, we prove the global well-posedness of the associated linear equation in H^q(𝕋)× H^s(𝕋)× H^s+1(𝕋), q,s ∈ ℝ. Subsequently, we consider the modified Zakharov-Rubenchik system and, by means of a change of variables and the Banach Fixed Point Theorem, we obtain local well-posedness in the space H^s(𝕋)× H^s(𝕋)× H^s+1(𝕋), s>3/2.
ER  -