Periodic solutions of coupled Boussinesq equations and Ostrovsky-type models free from zero-mass contradiction

Coupled Boussinesq equations describe long weakly-nonlinear longitudinal strain waves in a bi-layer with a soft bonding between the layers (e.g. a soft adhesive). From the mathematical viewpoint, a particularly difficult case appears when the linear long-wave speeds in the layers are significantly different (high-contrast case). The traditional derivation of the uni-directional models leads to four uncoupled Ostrovsky equations, for the right- and left-propagating waves in each layer. However, the models impose a ``zero-mass constraint'' i.e. the initial conditions should necessarily have zero mean, restricting the applicability of that description. Here, we bypass the contradiction in this high-contrast case by constructing the solution for the deviation from the evolving mean value, using asymptotic multiple-scale expansions involving two pairs of fast characteristic variables and two slow-time variables. By construction, the Ostrovsky equations emerging within the scope of this derivation are solved for initial conditions with zero mean while initial conditions for the original system may have non-zero mean values. Asymptotic validity of the solution is carefully examined numerically. We apply the models to the description of counter-propagating waves generated by solitary wave initial conditions, or co-propagating waves generated by cnoidal wave initial conditions, as well as the resulting wave interactions, and contrast with the behaviour of the waves in bi-layers when the linear long-wave speeds in the layers are close (low-contrast case). One local (classical) and two non-local (generalised) conservation laws of the coupled Boussinesq equations for strains are derived, and these are used to control the accuracy of the numerical simulations.Comment: 25 pages, 11 figures; previously this version appeared as arXiv:2210.14107 which was submitted as a new work by acciden

Dispersive hydrodynamics: Preface

This Special Issue on Dispersive Hydrodynamics is dedicated to the memory and work of G. B. Whitham who was one of the pioneers in this field of physical applied mathematics. Some of the papers appearing here are related to work reported on at the workshop \Dispersive Hydrodynamics: The Mathematics of Dispersive Shock Waves and Applications" held in May 2015 at the Banff International Research Station. This Preface provides a broad overview of the field and summaries of the various contributions to the Special Issue, placing them in a unified context

Solitary waves propagating over variable topography

Solitary water waves are long nonlinear waves that can propagate steadily over long distances. They were first observed by Russell in 1837 in a now famous report [26] on his observations of a solitary wave propagating along a Scottish canal, and on his subsequent experiments. Some forty years later theoretical work by Boussinesq [8] and Rayleigh [25] established an analytical model. Then in 1895 Korteweg and de Vries [21] derived the well-known equation which now bears their names. Significant further developments had to wait until the second half of the twentieth century, when there were two parallel developments. On the one hand it became realised that the Korteweg-de Vries equation was a valid model for solitary waves in a wide variety of physical contexts. On the other hand came the discovery of the soliton by Kruskal and Zabusky [27], with the subsequent rapid development of the modern theory of solitons and integrable systems

Mathematical modelling of nonlinear internal waves in a rotating fluid

Large amplitude internal solitary waves in the coastal ocean are commonly modelled with the Korteweg-de Vries (KdV) equation or a closely related evolution equation. The characteristic feature of these models is the solitary wave solution, and it is well documented that these provide the basic paradigm for the interpretation of oceanic observations. However, often internal waves in the ocean survive for several inertial periods, and in that case, the KdV equation is supplemented with a linear non-local term representing the effects of background rotation, commonly called the Ostrovsky equation. This equation does not support solitary wave solutions, and instead a solitary-like initial condition collapses due to radiation of inertia-gravity waves, with instead the long-time outcome typically being an unsteady nonlinear wave packet. The KdV equation and the Ostrovsky equation are formulated on the assumption that only a single vertical mode is used. In this thesis we consider the situation when two vertical modes are used, due to a near-resonance between their respective linear long wave phase speeds. This phenomenon can be described by a pair of coupled Ostrovsky equations, which is derived asymptotically from the full set of Euler equations and solved numerically using a pseudo-spectral method. The derivation of a system of coupled Ostrovsky equations is an important extension of coupled KdV equations on the one hand, and a single Ostrovsky equation on the other hand. The analytic structure and dynamical behaviour of the system have been elucidated in two main cases. The first case is when there is no background shear flow, while the second case is when the background state contains current shear, and both cases lead to new solution types with rich dynamical behaviour. We demonstrate that solitary-like initial conditions typically collapse into two unsteady nonlinear wave packets, propagating with distinct speeds corresponding to the extremum value in the group velocities. However, a background shear flow allows for several types of dynamical behaviour, supporting both unsteady and steady nonlinear wave packets, propagating with the speeds which can be predicted from the linear dispersion relation. In addition, in some cases secondary wave packets are formed associated with certain resonances which also can be identified from the linear dispersion relation. Finally, as a by-product of this study it was shown that a background shear flow can lead to the anomalous version of the single Ostrovsky equation, which supports a steady wave packet

Evolution of solitary waves and undular bores in shallow-water flows over a gradual slope with bottom friction

This paper considers the propagation of shallow-water solitary and nonlinear periodic waves over a gradual slope with bottom friction in the framework of a variable-coefficient Korteweg-de Vries equation. We use the Whitham averaging method, using a recent development of this theory for perturbed integrable equations. This general approach enables us not only to improve known results on the adiabatic evolution of isolated solitary waves and periodic wave trains in the presence of variable topography and bottom friction, modelled by the Chezy law, but also, importantly, to study the effects of these factors on the propagation of undular bores, which are essentially unsteady in the system under consideration. In particular, it is shown that the combined action of variable topography and bottom friction generally imposes certain global restrictions on the undular bore propagation so that the evolution of the leading solitary wave can be substantially different from that of an isolated solitary wave with the same initial amplitude. This non-local effect is due to nonlinear wave interactions within the undular bore and can lead to an additional solitary wave amplitude growth, which cannot be predicted in the framework of the traditional adiabatic approach to the propagation of solitary waves in slowly varying media

Periodic solutions of coupled Boussinesq equations and Ostrovsky-type models free from zero-mass contradiction

Coupled Boussinesq equations are used to describe long weakly-nonlinear longitudinal strain waves in a bi-layer with a soft bonding between the layers (e.g. a soft adhesive). From the mathematical viewpoint, a particularly difficult case appears when the linear long-wave speeds in the layers are significantly different (high-contrast case). The traditional derivation of the uni-directional models leads to four uncoupled Ostrovsky equations, for the right-and left-propagating waves in each layer. However, the models impose a "zero-mass constraint" i.e. the initial conditions should necessarily have zero mean, restricting the applicability of that description. Here, we bypass the contradiction in this high-contrast case by constructing the solution for the deviation from the evolving mean value, using asymptotic multiple-scale expansions involving two pairs of fast characteristic variables and two slow-time variables. By construction, the Ostro-vsky equations emerging within the scope of this derivation are solved for initial conditions with zero mean while initial conditions for the original system may have non-zero mean values. Asymptotic validity of the solution is carefully examined numerically. We apply the models to the description of counter-propagating waves generated by solitary wave initial conditions, or co-propagating waves generated by cnoidal wave initial conditions, as well as the resulting wave interactions, and contrast with the behaviour of the waves in bi-layers when the linear long-wave speeds in the layers are close (low-contrast case). One local (classical) and two non-local (generalised) conservation laws of the coupled Boussinesq equations for strains are derived and used to control the accuracy of the numerical simulations. A weakly-nonlinear solution to the coupled Boussinesq equations on a finite interval with periodic boundary conditions is constructed, resolving the zero-mass contradiction. The solution is shown to be asymptotically valid by comparison to direct numerical simulations of the original coupled Boussinesq equations, with the additional control of derived generalised conservation laws. Examples include counter-propagating radiating solitary waves and Ostrovsky-type wave packets when the period of the solution is large compared to the size of a localised initial condition, while decreasing the period of the solution for the localised perturbations and using non-localised initial conditions leads to more complicated scenarios. We observe that, in many cases, the waves appear to interact in a nearly-elastic manner, similarly to that of solitary waves, with small phase shift and amplitude changes compared to the case with no interaction, while in other cases strong interactions lead to formation of new wave structures

Optical solitons and modulation instability analysis to the quadratic-cubic nonlinear Schrödinger equation

This paper obtains the dark, bright, dark-bright, dark-singular optical and singular soliton solutions to the nonlinear Schrödinger equation with quadratic-cubic nonlinearity (QC-NLSE), which describes the propagation of solitons through optical fibers. The adopted integration scheme is the sine-Gordon expansion method (SGEM). Further more, the modulation instability analysis (MI) of the equation is studied based on the standard linear-stability analysis, and the MI gain spectrum is got. Physical interpretations of the acquired results are demonstrated. It is hoped that the results reported in this paper can enrich the nonlinear dynamical behaviors of the PNSE

Dispersive hydrodynamics: Preface

This Special Issue on Dispersive Hydrodynamics is dedicated to the memory and work of G.B. Whitham who was one of the pioneers in this field of physical applied mathematics. Some of the papers appearing here are related to work reported on at the workshop “Dispersive Hydrodynamics: The Mathematics of Dispersive Shock Waves and Applications” held in May 2015 at the Banff International Research Station. This Preface provides a broad overview of the field and summaries of the various contributions to the Special Issue, placing them in a unified context

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