Analysis of High-Order Bright–Dark Rogue Waves in (2+1)-D Variable-Coefficient Zakharov Equation via Self-Similar and Darboux Transformations

Hangwei Zhang, Jie Zong, Geng Tian and Guangmei Wei ()
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Hangwei Zhang: School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China
Jie Zong: School of Mathematical Scineces, Beihang University, Beijing 100191, China
Geng Tian: School of Mathematical Scineces, Beihang University, Beijing 100191, China
Guangmei Wei: School of Mathematical Scineces, Beihang University, Beijing 100191, China

Mathematics, 2024, vol. 12, issue 9, 1-25

Abstract: This paper conducts an in-depth study on the self-similar transformation, Darboux transformation, and the excitation and propagation characteristics of high-order bright–dark rogue wave solutions in the (2+1)-dimensional variable-coefficient Zakharov equation. The Zakharov equation is instrumental for studying complex nonlinear interactions in these areas, with specific implications for energy transfer processes in plasma and nonlinear wave propagation systems. By analyzing bright–dark rogue wave solutions—phenomena that are critical in understanding high-energy events in optical and fluid environments—this research elucidates the intricate dynamics of energy concentration and dissipation. Using the self-similar transformation method, we map the (2+1)-dimensional equation to a more tractable (1+1)-dimensional nonlinear Schrödinger equation form. Through the Lax pair and Darboux transformation, we successfully construct high-order solutions that reveal how variable coefficients influence rogue wave features, such as shape, amplitude, and dynamics. Numerical simulations demonstrate the evolution of these rogue waves, offering novel perspectives for predicting and mitigating extreme wave events in engineering applications.This paper crucially advances the practical understanding and manipulation of nonlinear wave phenomena in variable environments, providing significant insights for applications in optical fibers, atmospheric physics, and marine engineering.

Keywords: zakharov equation; higher-order bright–dark rogue waves; self-similar transformation; Darboux transformation; dynamics analysis (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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