Radiating solitary waves in coupled Boussinesq equations

In this paper we are concerned with the analytical description of radiating solitary wave solutions of coupled regularised Boussinesq equations. This type of solution consists of a leading solitary wave with a small-amplitude co-propagating oscillatory tail, and emerges from a pure solitary wave solution of a symmetric reduction of the full system. We construct an asymptotic solution, where the leading order approximation in both components is obtained as a particular solution of the regularised Boussinesq equations in the symmetric case. At the next order, the system uncouples into two linear non-homogeneous ordinary differential equations with variable coefficients, one correcting the localised part of the solution, which we find analytically, and the other describing the co-propagating oscillatory tail. This latter equation is a fourth order ordinary differential equation and is solved approximately by two different methods, each exploiting the assumption that the leading solitary wave has a small amplitude, and thus enabling an explicit estimate for the amplitude of the oscillating tail. These estimates are compared with corresponding numerical simulations

Slow Invariant Manifolds of Slow-Fast Dynamical Systems

Slow-fast dynamical systems, i.e., singularly or non-singularly perturbed dynamical systems possess slow invariant manifolds on which trajectories evolve slowly. Since the last century various methods have been developed for approximating their equations. This paper aims, on the one hand, to propose a classification of the most important of them into two great categories: singular perturbation-based methods and curvature-based methods, and on the other hand, to prove the equivalence between any methods belonging to the same category and between the two categories. Then, a deep analysis and comparison between each of these methods enable to state the efficiency of the Flow Curvature Method which is exemplified with paradigmatic Van der Pol singularly perturbed dynamical system and Lorenz slow-fast dynamical system.Comment: arXiv admin note: text overlap with arXiv:1808.0805

Proceedings for the ICASE Workshop on Heterogeneous Boundary Conditions

Domain Decomposition is a complex problem with many interesting aspects. The choice of decomposition can be made based on many different criteria, and the choice of interface of internal boundary conditions are numerous. The various regions under study may have different dynamical balances, indicating that different physical processes are dominating the flow in these regions. This conference was called in recognition of the need to more clearly define the nature of these complex problems. This proceedings is a collection of the presentations and the discussion groups

Asymptotic soliton-like and asymptotic peakon-like solutions of the modified Camassa-Holm equation with variable coefficients and singular perturbation

The paper deals with the construction of the asymptotic soliton-like and the asymptotic peakon-like solutions to the modified Camassa-Holm equation with variable coefficicents and a singular perturbation. This equation is a generalization of the well known modified Camassa-Holm equation (\ref{CHE_cons_mod}) which is integrable system and in addition to the soliton solutions the equation has the peakon solutions. The novelty of the ideas of this paper lies in the development of a technique for constructing asymptotic peakon-like solutions. In the paper a general scheme of finding asymptotic approximation of any order is presented and accuracy of the asymptotic approximation is found. The obtained results are illustrated by examples both the soliton-like and the peakon-like solutions. For the examples the equations for the phase function as well as the main and the first terms of the soliton-like and peakon-like solutions are found. Moreover, for different values of a small parameter the graphs that demonstrate kind of the solutions are presented. The considered examples demonstrate that for an adequate description of the wave process it is enough obtain the main and the first terms of correspond asymptotic solutions. The results also confirm that the proposed technique can be used for constructing asymptotic wave-like solutions of other equations.Comment: 33 pages and 12 figure

Scattering of bulk strain solitary waves in bi-layers with delamination

We study the scattering of longitudinal bulk strain solitary waves in delaminated bi-layers with different types of bonding. The direct numerical modelling of these problems is challenging and has natural limitations. We develop a semi-analytical approach, based on the use of several matched asymptotic multiple-scale expansions and the Integrability Theory of the Korteweg - de Vries equation by the Inverse Scattering Transform. We show that the semi-analytical approach agrees well with the direct numerical simulations and use it to study the scattering of different types of longitudinal bulk strain solitary waves in a wide range of bi-layers with delamination. In particular, we model the dynamics of a long longitudinal strain solitary wave in a symmetric perfectly bonded bi-layer with delamination. The numerical modelling confirms that delamination causes fission of an incident solitary wave and, thus, can be used to detect the defect. We then extend our approaches to the modelling of the waves in bi-layers with soft ("imperfect") bonding, described by a system of coupled Boussinesq equations and supporting radiating solitary waves. The results may help us to control the integrity of layered structures

Mathematical Theory of Water Waves

Water waves, that is waves on the surface of a fluid (or the interface between different fluids) are omnipresent phenomena. However, as Feynman wrote in his lecture, water waves that are easily seen by everyone, and which are usually used as an example of waves in elementary courses, are the worst possible example; they have all the complications that waves can have. These complications make mathematical investigations particularly challenging and the physics particularly rich. Indeed, expertise gained in modelling, mathematical analysis and numerical simulation of water waves can be expected to lead to progress in issues of high societal impact (renewable energies in marine environments, vorticity generation and wave breaking, macro-vortices and coastal erosion, ocean shipping and near-shore navigation, tsunamis and hurricane-generated waves, floating airports, ice-sea interactions, ferrofluids in high-technology applications, ...). The workshop was mostly devoted to rigorous mathematical theory for the exact hydrodynamic equations; numerical simulations, modelling and experimental issues were included insofar as they had an evident synergy effect

Mathematical modelling of nonlinear waves in layered waveguides with delamination

The propagation of nonlinear bulk strain waves in layered elastic waveguides has many applications, particularly its potential use for non-destructive testing, where a small defect in the bonding between the layers of a waveguide can lead to a catastrophic failure of the structure. Experiments have shown that strain solitons can propagate for significantly longer distances than the waves used in current methods, and therefore they are of great interest. This thesis considers two problems. Firstly, we consider the scattering of nonlinear bulk strain waves in two types of waveguides: a perfectly bonded layered waveguide, and a layered waveguide with a soft bond between the layers, when the materials in the layers have similar properties. In each case we assume that there is a region where the bond is absent - a delamination. This behaviour is described by a system of uncoupled or coupled Boussinesq equations, with conditions on the interface between the sections of the bar. This is a complicated system of equations, and we develop a direct numerical method to solve these equations numerically. A weakly nonlinear solution is then constructed for the system of equations, describing the leading order reflected and transmitted strain waves. In the case of a layered elastic bar with a perfect bond we obtain Korteweg-de Vries equations, and in the case of a soft bond between the layers, where the properties of the layers are close, we obtain coupled Ostrovsky equations describing the propagation of the reflected and transmitted waves in each layer of the waveguide. In the delaminated regions of the bar, Korteweg-de Vries equations are derived in every case and therefore we make use of the Inverse Scattering Transform to provide theoretical predictions in this region. The modelling in each case is extended to the case of a finite delamination in the waveguide, and we study the effect of re-entering a bonded region on a strain wave. In each case considered we develop a measure of the delamination length in terms of the change in amplitude of the incident wave, and furthermore the structure of the wave provides further insight about the structure of the waveguide. Numerical simulations are developed using finite-difference techniques and pseudospectral methods, and these are detailed in the appendices. Finally, we consider the initial value problem for the Boussinesq equation with an Ostrovsky term, on a periodic domain. The initial condition for this equation does not necessary have zero mean on the interval. The mean value is subtracted from the function so that a weakly nonlinear solution to the problem can be constructed where all functions in this expansion have zero mean. This is necessary as the derived Ostrovsky equations have zero mean. The expansion is constructed in increasing powers of $\sqrt{\epsilon}$ up to and including \O{\epsilon}, where $\epsilon$ is a small amplitude parameter in the equation. We compare the results for a wide range of values of $\gamma$ (the coefficient of the Ostrovsky term) and varying mean values for the initial condition, to confirm that the expansion is valid. A comparison of the errors shows that the constructed expansion is correct and the errors behave as predicted by the expansion. This was further confirmed for non-unity coefficients in the equation