Evolution Dynamics and Cauchy Problems with Quaternionic Variables and Lifting to Banach Bilateral 𝔸-Algebras
Abstract
In this paper, we develop a theory of evolution dynamics in a Banach bilateral ℍ-algebra and establish a well-posedness theory for the Cauchy problem of evolution equations with quaternionic variables. First, we study the geometric properties of slice structures of Banach bilateral ℍ-modules. By employing different slice structures, we construct some noncommutative Banach bilateral ℍ-subalgebras which are compatible with the slice regularity of the functions involved. Moreover, through the quaternionic sliced geometric path integral over Banach bilateral 𝔸-module, an integral representation of the solution to the homogeneous Cauchy problem with quaternionic variables is obtained in a quaternionic σ-ball, and for the first time, a local existence and uniqueness theorem is established for σ-analytic solutions. Then the local σ-analytic solution is extended to the entire quaternionic spherical sectorial region, thereby proving a global existence and uniqueness theorem for the Cauchy problem. Based on it, the notion of an evolution system in a quaternionic ℍ-algebra is introduced and its core properties are established. In addition, the global existence and uniqueness theorems are obtained under hyperbolic-like and parabolic-like conditions. Finally, the structure of Banach bilateral ℍ-modules and ℍ-algebras is lifted to Banach bilateral 𝔸-modules and 𝔸-algebras, and a well-posedness theory for evolution equations in this setting is developed. This reveals, from a new perspective, the universality of well-posedness for the Cauchy problem under the generic noncommutative 𝔸-algebraic structures.
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BibTeX
@misc{wang2026evolution,
title = {{Evolution Dynamics and Cauchy Problems with Quaternionic Variables and Lifting to Banach Bilateral $\mathbbA$-Algebras}},
author = {Chao Wang and Guangzhou Qin and Jibin Li},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06721},
eprint = {2611.06721},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06721},
}
RIS
TY - UNPB
TI - Evolution Dynamics and Cauchy Problems with Quaternionic Variables and Lifting to Banach Bilateral 𝔸-Algebras
AU - Chao Wang
AU - Guangzhou Qin
AU - Jibin Li
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06721
UR - https://arxiv.org/abs/2611.06721
AB - In this paper, we develop a theory of evolution dynamics in a Banach bilateral ℍ-algebra and establish a well-posedness theory for the Cauchy problem of evolution equations with quaternionic variables. First, we study the geometric properties of slice structures of Banach bilateral ℍ-modules. By employing different slice structures, we construct some noncommutative Banach bilateral ℍ-subalgebras which are compatible with the slice regularity of the functions involved. Moreover, through the quaternionic sliced geometric path integral over Banach bilateral 𝔸-module, an integral representation of the solution to the homogeneous Cauchy problem with quaternionic variables is obtained in a quaternionic σ-ball, and for the first time, a local existence and uniqueness theorem is established for σ-analytic solutions. Then the local σ-analytic solution is extended to the entire quaternionic spherical sectorial region, thereby proving a global existence and uniqueness theorem for the Cauchy problem. Based on it, the notion of an evolution system in a quaternionic ℍ-algebra is introduced and its core properties are established. In addition, the global existence and uniqueness theorems are obtained under hyperbolic-like and parabolic-like conditions. Finally, the structure of Banach bilateral ℍ-modules and ℍ-algebras is lifted to Banach bilateral 𝔸-modules and 𝔸-algebras, and a well-posedness theory for evolution equations in this setting is developed. This reveals, from a new perspective, the universality of well-posedness for the Cauchy problem under the generic noncommutative 𝔸-algebraic structures.
ER -