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An integrable ℤ22-graded extension of Camassa-Holm equation and its bi-Hamiltonian structure

  • N. Aizawa
  • Ichi Fujii
  • Ren Ito
  • L. Vasconcellos
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Abstract

By constructing Lax operators in the loop algebra of the ℤ22-graded extension of the Lie superalgebra osp(1|2), we derive a ℤ22-graded extension of the Camassa-Holm equation. The resulting equation is an integrable nonlinear PDE for a system of four ℤ22-graded commutative functions, each associated with a distinct ℤ22-degree. We further show that the ℤ22-Camassa-Holm equation admits a bi-Hamiltonian structure. As a consequence, it possesses infinitely many conserved quantities, including one with non-trivial ℤ22-degree, which are mutually in involution with respect to the ℤ22-graded Poisson brackets.

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Cite

BibTeXRIS
BibTeX
@misc{aizawa2026integrable,
  title = {{An integrable $\mathbbZ_2^2$-graded extension of Camassa-Holm equation and its bi-Hamiltonian structure}},
  author = {N. Aizawa and Ichi Fujii and Ren Ito and L. Vasconcellos},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.19543},
  eprint = {2609.19543},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.19543},
}
RIS
TY  - UNPB
TI  - An integrable ℤ_2^2-graded extension of Camassa-Holm equation and its bi-Hamiltonian structure
AU  - N. Aizawa
AU  - Ichi Fujii
AU  - Ren Ito
AU  - L. Vasconcellos
PY  - 2026
DA  - 2026/09/17
PB  - arXiv
DO  - 10.48550/arxiv.2609.19543
UR  - https://arxiv.org/abs/2609.19543
AB  - By constructing Lax operators in the loop algebra of the ℤ_2^2-graded extension of the Lie superalgebra osp(1|2), we derive a ℤ_2^2-graded extension of the Camassa-Holm equation. The resulting equation is an integrable nonlinear PDE for a system of four ℤ_2^2-graded commutative functions, each associated with a distinct ℤ_2^2-degree. We further show that the ℤ_2^2-Camassa-Holm equation admits a bi-Hamiltonian structure. As a consequence, it possesses infinitely many conserved quantities, including one with non-trivial ℤ_2^2-degree, which are mutually in involution with respect to the ℤ_2^2-graded Poisson brackets.
ER  -