Quasi-periodic breathers and rogue waves to the focusing Davey–Stewartson equation

Abstract

The Davey–Stewartson equation has been instrumental in describing various physical phenomena, especially (2+1)-dimensional breathers and rogue waves. In this paper, we present a direct approach to studying the quasi-periodic breathers of the Davey–Stewartson equation. By employing Hirota’s bilinear method and leveraging certain identities of theta functions, the problem is transformed into an over-determined nonlinear algebraic system, which can be formulated as a nonlinear least square problem and solved by classical numerical iterative algorithms. Through asymptotic analysis and numerical experiments, we categorize these solutions into three cases: quasi-periodic breathers, quasi-periodic stationary breathers, and quasi-periodic homoclinic orbits. The latter exhibit behavior reminiscent of quasi-periodic rogue waves, which are often observed in oceanic rogue wave phenomena. This work advances our understanding of the dynamics and properties of (2+1)-dimensional quasi-periodic breathers and rouge waves.