The "Nonlinear Waves and Coherent Structures Online Colloquium" is a monthly seminar that will have speakers from a diverse range of disciplines to speak on a broad range of topics connected to nonlinear waves and coherent structures. The organizers hope this series will help maintain a sense of community and provide a broad audience with access to research developments on the forefront of this field.
NWCS Colloquium
Karima Khusnutdinova (Mathematical Sciences, Loughborough University)
Tuesday, November 11th, 11:00am (Eastern Time)
Title: Generalised travelling waves of the KdV equation and the effect of rotation
Abstract: 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. We develop simultaneous asymptotic expansions for the evolving mean fields and deviations from them to describe a large class of uni-directional waves via the Ostrovsky equation, which fully decouples from mean-field variations. The latter generate additive inertial oscillations in the shear and in the phase of both the interfacial displacement and shear. Unlike conventional derivations leading to the Ostrovsky equation, our formulation does not impose the zero-mean constraints on the initial conditions of any variable. 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 with local defects, both with and without rotation. We show that rotation leads to the emergence of bursts of internal waves and shear currents, qualitatively similar to the wavepackets generated from solitons and modulated cnoidal waves in earlier studies, but emerging much faster. We also 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 corner condition 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 leads to the emergence of strong bursts. The talk is based on the joint paper with K. Nirunwiroj and D. Tseluiko ("Evolution of internal cnoidal waves with local defects in a two-layer fluid with and without rotation", accepted to JFM).
How to join: An email containing the zoom link and password for the seminar will be sent out by 9:30am (Eastern Time) the day of the talk. To join the email list please register here
Speakers: To see the full list of upcoming speakers and recordings of prior talks, click here
Organizers:
Stathis Charalampidis, SDSU, San Diego
Sathya Chandramouli, UMass Amherst
Panos Kevrekidis, UMass Amherst