MIR@W - W^3

Long nonlinear ring waves and their relatives

In this talk, I will give a brief overview of recent developments in our studies of long nonlinear ring waves, together with related advances on perturbed line solitons and on hybrid waves that consist, to leading order, of an arc of a ring wave connected to two tangent plane waves. For waves propagating in a stratified fluid over a parallel shear flow, we establish a linear modal decomposition (separation of variables) in the far-field set of the Euler equations with boundary conditions relevant to oceanographic applications. This decomposition is more intricate than the classical one for plane waves. The resulting modal equations form a new spectral problem and require the construction of a singular solution of a nonlinear first-order ODE that governs directional adjustments of the wave speed for the description of ring waves. Depending on whether the coefficients of the amplitude equation are evaluated using the general or the singular solution of this ODE, two distinct evolution equations arise. The validity ranges of these models are assessed through direct numerical simulations of the two-dimensional Boussinesq–Peregrine system for surface waves. Analytical results for perturbed line solitons are obtained by mapping the problem to the KdV equation while retaining the polar-angle dependence in the initial conditions. We also confirm the instability of inward-propagating waves predicted by Ostrovsky and Shrira (1978) and provide insight into the nonlinear development of this instability. Finally, a case study shows that the extended cKdV model offers a significantly more accurate description of surface ring waves and substantially extends the range of validity of weakly nonlinear modelling to waves of moderate amplitude.

GENERIC formulation and small-angle limit for kinetic wave equations

In this talk, we discuss the three-wave and four-wave kinetic equations. We formulate them into the GENERIC framework and formally derive a small-angle limit for the four-wave equation. This limit is akin to the well-known grazing limit from the kinetic Boltzmann equation to the kinetic Landau equation. We also show the GENERIC structure of the limiting system. This talk is based on a joint work with Zihui He (Bielefeld University).

Extreme Event Quantification for Rogue Waves in Deep Sea

A central problem in uncertainty quantification is how to characterize the impact that our incomplete knowledge about models has on the predictions we make from them. This question naturally lends itself to a probabilistic formulation, by making the unknown model parameters random with given statistics. Here this approach is used in concert with tools from large deviation theory (LDT) and optimal control to estimate the probability that some observables in a dynamical system go above a large threshold after some time, given the prior statistical information about the system's parameters and its initial conditions. We use this approach to quantify the likelihood of extreme surface elevation events for deep sea waves, so-called rogue waves, described by the nonlinear Schrödinger (NLS) equation, and compare the results to experimental measurements. We show that our approach offers a unified description of rogue wave events in the one-dimensional NLS, covering a vast range of paramters. In particular, this includes both the predominantly linear regime as well as Peregrine-type solitons in the highly nonlinear regime as limiting cases, and is able to predict the experimental data regardless of the strength of the nonlinearity.

Edge solitons in 1D and 2D nonlinear topological insulators

I will first discuss edge solitons in a 2D nonlinear mechanical topological insulator via reduction to a 1D coupled nonlinear Schrödinger equation. A bulk lattice consists of pendulums with cubic nonlinearity connected by linear springs realizing quantum spin Hall effect. On the edge separating two bulk lattices, linear springs are constructed to enable vector edge solitons and domain walls. I will then describe edge solitons in a 1D nonlinear photonic topological insulator via spatial dynamics. The bulk lattice consists of coupled waveguides realizing the Su-Schrieffer-Heeger model with cubic nonlinearity. I will outline some potentially interesting mathematical questions in these and related systems.

Advancing the argument on elastic vector solitons found by Deng (2017) in a dynamical Grima–Evans squares model developed as the basis for a flexible mechanical metamaterial

In 2017 Deng derived a continuum approximation from a discrete rotating squares system, which resulted in an ODE - effectively the time-independent Klein-Gordon equation. In going through the lengthy calculation involving many harmonic terms, series approximations were used to simplify, but at the same time keeping the essential physics. Deng compared a computer simulation of the discrete system with his continuum approximation and was happy with the results, which to be fair were pretty good. When we approached the problem we realised he had been too quick to accept his findings. The good results hid the fact that he had been inconsistent in his approximation scheme. In being somewhat more careful we found the system to be much more interesting with extra terms. However miraculously it was in fact the same kind of system but gave 1/9 better agreement with simulation. We present our arguments with a quasi-Feynman-diagram creation to aid computation and intuition for others that will follow our efforts.

Nonlinear waves in nonreciprocal systems

In this work, we show how nonlinearity fundamentally reshapes wave dynamics in non-Hermitian lattices with non-reciprocal couplings. First, we demonstrate that the non-Hermitian skin effect persists beyond the linear regime: nonlinear stationary modes remain localized at the favored edge, becoming more confined under focusing nonlinearity or more extended under defocusing conditions. We then show that nonlinearity can balance amplification, dispersion, and dissipation in active nonreciprocal systems, enabling the emergence of stable unidirectional solitonic pulses even when all bulk modes are exponentially localized. Together, these results highlight nonlinearity as a key ingredient for controlling localization and robust pulse propagation in strongly non-Hermitian media.

A weakly nonlinear model for sound propagation in brass instruments

We present a summary of work on a new weakly nonlinear model of sound in curved 2D or 3D waveguides without mean flow. Modelling begins by considering a general duct geometry, allowing for curvature and width variation, and also in 3D for torsion. Equations are projected onto a Frenet-Serret frame with arclength s, then expanded at each s in terms of the modes of an equiradial straight duct. This is the multi-modal method. We introduce a matricial admittance that encodes the radiative properties of the duct; this allows for the prescription of a radiation condition at the duct outlet. Previous modelling has taken this to be the characteristic admittance: while mathematically tractable, this is highly physically idealised and corresponds to a duct opening into a room of equal radius. We instead derive a radiation condition consisting of a greatly enlarged duct with the original duct at the centre, which tells us more about the passage of sound from the outlet. This allows for more direct physical comparisons with the model, e.g. the end correction for organ pipes or the harmonic series of a trumpet.

Graded metamaterials: rainbow effects for sensing and energy harvesting

Abstract TBC