A distinguished-limit Gardner–Kawahara model for internal-wave–forced near-critical gravity–capillary ripples and Wilton-type spectra

Abstract

We investigate gravity–capillary surface ripples radiated by a nonlinear internal solitary-wave–type disturbance near the critical Bond number, where third-order dispersion weakens and fifth-order effects become essential. A coupled unidirectional reduction is developed, in which the internal mode follows Gardner dynamics and the free surface obeys a forced near-critical Kawahara model, enabling a phase-speed–matching interpretation of ripple selection and envelope formation. The stiff dispersive system is solved with a Fourier pseudospectral discretization and an ETDRK4 integrator. Simulations reveal a robust two-component surface response: a low-wavenumber background adjustment and a delayed, high-wavenumber ripple packet that grows in a downstream wake window, exhibits intermittent energization, and becomes increasingly organized in phase. Spectral–time and ridge diagnostics show persistent energy concentration near the predicted radiation neighborhoods, while regime maps in Bond number and coupling strength identify transitions from weak surface signatures to pronounced Wilton-type dual-peak behavior.