Conservation laws for a non-linear system describing the propagation of "triadic" waves

When two waves, be they sound waves, waves in a liquid or electromagnetic waves, of different wavenumbers travel at the same speed, they may interact nonlinearly and resonance can arise. In this paper we shall be especially concerned with capillary-gravity waves (that is, waves which are subject to the twin restoring forces of gravity and surface tension) on the surface of deep water. However, our theory could just as well be applied to any physical wave system where the evolution of the waves is governed by an asymptotic balance between the effects of nonlinearity and dispersion. One obvious generalization is to interfacial waves between two fluids, but other examples are light pulses propagating along an optical fibre, waves in a hot electron plasma, or ‘beat’ waves. In this report we shall be studying the case of ‘one-three’ (i.e. ‘triadic’) resonance when the interaction occurs between the fundamental mode and its third harmonic (i.e. a wave with a third of the wavelength of the fundamental)

Integrability and linear stability of nonlinear waves

It is well known that the linear stability of solutions of (Formula presented.) partial differential equations which are integrable can be very efficiently investigated by means of spectral methods. We present here a direct construction of the eigenmodes of the linearized equation which makes use only of the associated Lax pair with no reference to spectral data and boundary conditions. This local construction is given in the general (Formula presented.) matrix scheme so as to be applicable to a large class of integrable equations, including the multicomponent nonlinear Schrödinger system and the multiwave resonant interaction system. The analytical and numerical computations involved in this general approach are detailed as an example for (Formula presented.) for the particular system of two coupled nonlinear Schrödinger equations in the defocusing, focusing and mixed regimes. The instabilities of the continuous wave solutions are fully discussed in the entire parameter space of their amplitudes and wave numbers. By defining and computing the spectrum in the complex plane of the spectral variable, the eigenfrequencies are explicitly expressed. According to their topological properties, the complete classification of these spectra in the parameter space is presented and graphically displayed. The continuous wave solutions are linearly unstable for a generic choice of the coupling constants

Nonlinear edge waves in mechanical topological insulators

We show theoretically that the classical 1D nonlinear Schrödinger (NLS) and coupled nonlinear Schrödinger (CNLS) equations govern the envelope(s) of localised and unidirectional nonlinear travelling edge waves in a 2D mechanical topological insulator (MTI). The MTI consists of a collection of pendula with weak Duffing nonlinearity connected by linear springs that forms a mechanical analogue of the quantum spin Hall effect (QSHE). It is found, through asymptotic analysis and dimension reduction, that the NLS and CNLS respectively describe the unimodal and bimodal properties of the nonlinear system. The governing bimodal CNLS is found to be non-integrable by nature and as such we discover new solutions by exploring the spatial dynamics of the reduced travelling wave ODE with general parameters. Such solutions include travelling fronts and, by numerically continuing these fronts, one can find vector soliton (VS) in non integrable CNLS equations. The equilibria can also undergo both pitchfork and Turing bifurcation in the reversible spatial dynamical system and we discuss relevant conditions for the existence and consequences of such critical values. We briefly discuss the necessity of the developed front condition in forming such structures and present an analytical framework for front-grey soliton collisions by utilising conserved quantities of the non-integrable CNLS. The existence/stability of front and VS solutions can be inferred by spatial hyperbolicity and linear stability of the background fields, with the criteria presented here. VS solutions are considered in the form of bright-bright, bright-dark, and dark-dark solitons and their collision dynamics are explored qualitatively in the non-integrable regime. The Turing analysis presents the existence of periodic and localised patterned states in the CNLS, and we compare these solutions to those found in the analysis of the Swift-Hohenberg equation. Theoretical predictions from the 1D (C)NLS are confirmed by numerical simulations of the original 2D MTI for various types of travelling waves and rogue waves. As a result of topological protection the edge solitons persist over long time intervals and through irregular boundaries. Due to the robustness of topologically protected edge solitons (TPES) it is suggested that their existence may have significant implications on the design of acoustic devices. Spacetime simulations show a clear possibility of utilising MTIs in acoustical cloaking with TPES a vital player in such processes