Korsarun Nirunwiroj | Searcharxiv

arXiv2026

Boussinesq-Klein-Gordon and Ostrovsky equations: evolution of cnoidal waves with local defects

The Boussinesq-Klein-Gordon (BKG) equation has emerged in the studies of nonlinear bulk strain waves in layered solid waveguides. The developed bi-directional weakly-nonlinear solution leads to two copies of the Ostrovsky equation, for the right- and left-propagating waves. Importantly, the derivation avoids the so-called `zero-mean contradiction' between the type of initial conditions in the parent equation and in the reduced model. In this paper, we apply the solution to describe the evolution of cnoidal waves with local periodicity defects and generic localised perturbations, and compare the results with the direct numerical simulations for the full BKG equation. The cnoidal waves with the periodicity defects discussed in our work constitute generalised travelling waves of the Korteweg-de Vries equation, while the Ostrovsky equation leads to a strong burst (and may lead to a rogue wave), qualitatively similar to the wavepacket emerging from a soliton initial condition, but appearing much faster. We compare the weakly-nonlinear solution with the direct numerical simulations within the bi-directional setting of the BKG equation and show that the discussed uni-directional waves and evolution scenarios remain stable in the presence of counter-propagating perturbations.

nlin.PS↗

arXiv2025

Evolution of internal cnoidal waves with local defects in a two-layer fluid with rotation

Internal waves in a two-layer fluid with rotation are considered within the framework of Helfrich's f-plane extension of the Miyata-Maltseva-Choi-Camassa (MMCC) model. Within the scope of this model, we develop an asymptotic procedure which allows us to obtain a description of a large class of uni-directional waves leading to the Ostrovsky equation and allowing for the presence of shear inertial oscillations and barotropic transport. Importantly, unlike the conventional derivations leading to the Ostrovsky equation, the constructed solutions do not impose the zero-mean constraint on the initial conditions for any variable in the problem formulation. Using the constructed solutions, we model the evolution of quasi-periodic initial conditions close to the cnoidal wave solutions of the Korteweg-de Vries (KdV) equation but having a local amplitude and/or periodicity defect, and show that such initial conditions can lead to the emergence of bursts of large internal waves and shear currents. As a by-product of our study, we show that cnoidal waves with expansion defects discussed in this work are generalised travelling waves of the KdV equation: they satisfy all conservation laws of the KdV equation (appropriately understood), as well as the Weirstrass-Erdmann conditions for broken extremals of the associated variational problem and a natural weak formulation. Being smoothed in numerical simulations, they behave, in the absence of rotation, as long-lived states with no visible evolution, while rotation changes this behaviour and leads to the emergence of strong bursts.

physics.flu-dyn↗