Gauge transformations between reduced $q$-KP and reduced $q$-mKP hierarchies

Shilong Huang, Chuanzhong Li

College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, Shandong, China

Abstract: We reduce the $q$-KP hierarchy and the $q$-mKP hierarchy. By using the link between the $q$-KP hierarchy and the $q$-mKP hierarchy, we obtain a relation between the $q$-KdV and the $q$-mKdV hierarchies, as well as a relation between the $q$-Boussinesq and $q$-mBoussinesq equations. The connection between them can also be established by a gauge transformation. We verify that when in the limit $q\to 1$, these relations correspond to the results for classical systems.

Keywords: $q$-deformation, $q$-KP hierarchy, $q$-mKP hierarchy, gauge transformation.

MSC: 35Q51; 35Q15; 37K10

Received: 30.07.2024
Revised: 30.08.2024

DOI: 10.4213/tmf10799

 English version: Theoretical and Mathematical Physics, 2025, 222:1, 131–139

Bibliographic databases:

• 
• 
•