Realization of several related (2+1)-dimensional integrable systems via an anti-self-dual Yang--Mills reduction
Abstract
We derive several integrable equations in (2+1) dimensions from the anti-self-dual Yang--Mills (ASDYM) equations with gauge group G=GL(2,ℂ) by imposing a single one-dimensional constraint on the ASDYM matrices. The resulting systems include the (2+1)-dimensional nonlinear Schrödinger (NLS) equation, a (2+1)-dimensional generalized derivative NLS family encompassing the Gerdjikov--Ivanov, Chen--Lee--Liu, and Kaup--Newell equations, and the (2+1)-dimensional Heisenberg ferromagnetic (HF) equation. The (1+1)-dimensional Wadati--Konno--Ichikawa equation is also recovered through a reciprocal transformation of the HF equation. The nonlinear variables and the associated gauge transformations are expressed explicitly in terms of the entries of the ASDYM matrices. The same one-dimensional constraint is implemented in the Cauchy matrix framework, yielding explicit solutions of the reduced integrable equations. In this formulation, the ASD Yang--Mills Lagrangian density is expressed in terms of a τ-function; for suitable exponential data, this τ-function becomes a Grammian-type τ-function of the Kadomtsev--Petviashvili equaton. The studies are extended to GL(M,ℂ) at the matrix level.
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BibTeX
@misc{li2026realization,
title = {{Realization of several related $(2+1)$-dimensional integrable systems via an anti-self-dual Yang--Mills reduction}},
author = {Shangshuai Li and Ken-ichi Maruno and Da-jun Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22753},
eprint = {2610.22753},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22753},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Realization of several related (2+1)-dimensional integrable systems via an anti-self-dual Yang--Mills reduction
AU - Shangshuai Li
AU - Ken-ichi Maruno
AU - Da-jun Zhang
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.22753
UR - https://arxiv.org/abs/2610.22753
AB - We derive several integrable equations in (2+1) dimensions from the anti-self-dual Yang--Mills (ASDYM) equations with gauge group G=GL(2,ℂ) by imposing a single one-dimensional constraint on the ASDYM matrices. The resulting systems include the (2+1)-dimensional nonlinear Schrödinger (NLS) equation, a (2+1)-dimensional generalized derivative NLS family encompassing the Gerdjikov--Ivanov, Chen--Lee--Liu, and Kaup--Newell equations, and the (2+1)-dimensional Heisenberg ferromagnetic (HF) equation. The (1+1)-dimensional Wadati--Konno--Ichikawa equation is also recovered through a reciprocal transformation of the HF equation. The nonlinear variables and the associated gauge transformations are expressed explicitly in terms of the entries of the ASDYM matrices. The same one-dimensional constraint is implemented in the Cauchy matrix framework, yielding explicit solutions of the reduced integrable equations. In this formulation, the ASD Yang--Mills Lagrangian density is expressed in terms of a τ-function; for suitable exponential data, this τ-function becomes a Grammian-type τ-function of the Kadomtsev--Petviashvili equaton. The studies are extended to GL(M,ℂ) at the matrix level.
ER -