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

Abstract

We study a three-wave truncation of the high-order nonlinear Schrödinger equation for deepwater waves (HONLS, also named Dysthe equation). We validate our approach by comparing it to numerical simulation, distinguish the impact of the different fourth-order terms and classify the solutions according to their topology. This allows us to properly define the temporary spectral upshift occurring in the nonlinear stage of Benjamin-Feir instability and provides a tool for studying further generalizations of this model.

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Cited by in corpus (11)

• Growth rate of modulation instability driven by superregular breathers
• Spectral up- and downshifting of Akhmediev breathers under wind forcing
• Two-dimensional rogue waves on zero background of the Davey-Stewartson II equation
• Nonlinear stage of Benjamin-Feir instability in forced/damped deep water waves
• Stabilization of uni-directional water-wave trains over an uneven bottom
• Stabilization of unsteady nonlinear waves by phase space manipulation
• Single-spectrum prediction of kurtosis of water waves in a non-conservative model
• Mean flow modelling in high-order nonlinear Schrödinger equations
• Modulational instability of nonuniformly damped, broad-banded waves: applications to waves in sea-ice
• Separatrix crossing and symmetry breaking in NLSE-like systems due to forcing and damping
• Numerical simulations of modulated waves in a higher-order Dysthe equation