Quasi-periodic breathers and data-driven quasi-periodic breathers of the (2+1)-dimensional Hirota equation in fluid mechanics

School of Mathematics, North University of China - Taiyuan, Shanxi 030051, PRC

Received: 28 December 2025
Accepted: 7 April 2026

Abstract

This paper investigates the quasi-periodic breathers of the (2+1)-dimensional Hirota equation, which can describe the wave phenomena in nonlinear optics and fluid mechanics. By combining the theta function and Hirota bilinear method, the quasi-periodic breather solutions are constructed. The solvability problem of the quasi-periodic breathers is transformed into a least squares problem, which can be solved via a new algorithm called the global Levenberg-Marquardt (LM) method. Different types of quasi-periodic breathers are obtained, including quasi-periodic Kuznetsov-Ma breather, quasi-periodic Akhmediev breather as well as regular quasi-periodic breather. By introducing an analysis method related to the characteristic line, the dynamical characteristics of the quasi-periodic breathers are derived. In addition, the data-driven quasi-periodic breathers are studied by using the physics-informed neural network-adaptive residual distribution optimization (PINN-ARDO) method.