Recurrence in the high-order nonlinear Schrödinger equation: A low-dimensional analysis

Area of Science:

• Fluid dynamics
• Nonlinear physics
• Wave propagation

Background:

• The nonlinear Schrödinger equation (NLSE) is a fundamental model for wave phenomena.
• Deep-water wave dynamics exhibit complex behaviors not fully captured by the basic NLSE.
• The Dysthe equation offers a higher-order approximation for these dynamics.

Purpose of the Study:

• To investigate a three-wave truncation of the high-order nonlinear Schrödinger equation (Dysthe equation).
• To validate this model against numerical simulations.
• To analyze the impact of fourth-order terms and classify solution topologies.

Main Methods:

• Numerical simulation for model validation.
• Analysis of fourth-order nonlinear terms.
• Topological classification of wave solutions.

Main Results:

• Successful validation of the three-wave Dysthe equation model.
• Distinguished contributions of individual fourth-order terms.
• Classification of solutions based on their topological properties.
• Precise definition of the temporary spectral upshift during Benjamin-Feir instability.

Conclusions:

• The validated model provides a robust tool for studying deep-water wave dynamics.
• The findings facilitate a deeper understanding of nonlinear wave phenomena.
• This work lays the groundwork for further generalizations of the Dysthe equation model.