There exists substantial applications motivating the study of nonlinear longitudinal wave propagation in layered (or laminated) elastic waveguides, in particular within areas related to non-destructive testing, where there is a demand to understand, reinforce, and improve deformation properties of such structures. It has been shown [76] that long longitudinal waves in such structures can be accurately modelled by coupled regularised Boussinesq (cRB) equations, provided the bonding between layers is sufficiently soft. The work in this thesis firstly examines the initial-value problem (IVP) for the system of cRB equations in [76] on the infinite line, for localised or sufficiently rapidly decaying initial conditions. Using asymptotic multiple-scales expansions, a nonsecular weakly nonlinear solution of the IVP is constructed, up to the accuracy of the problem formulation. The asymptotic theory is supported with numerical simulations of the cRB equations. The weakly nonlinear solution for the equivalent IVP for a single regularised Boussinesq equation is then constructed; constituting an extension of the classical d'Alembert's formula for the leading order wave equation. The initial conditions are also extended to allow one to separately specify an O(1) and O(ε) part. Large classes of solutions are derived and several particular examples are explicitly analysed with numerical simulations. The weakly nonlinear solution is then improved by considering the IVP for a single regularised Boussinesq-type equation, in order to further develop the higher order terms in the solution. More specifically, it enables one to now correctly specify the higher order term's time dependence. Numerical simulations of the IVP are compared with several examples to justify the improvement of the solution. Finally an asymptotic procedure is developed to describe the class of radiating solitary wave solutions which exist as solutions to cRB equations under particular regimes of the parameters. The validity of the analytical solution is examined with numerical simulations of the cRB equations. Numerical simulations throughout this work are derived and implemented via developments of several finite difference schemes and pseudo-spectral methods, explained in detail in the appendices.

This thesis addresses solitary waves and their significance for auroral particle acceleration, coronal heating and incoherent scatter radar spectra. Solitary waves are formed due to a balance of nonlinear and dispersive effects. There are several nonlinearities present in ideal magnetohydrodynamics (MHD) and dispersion can be introduced by including theHall termin the generalised Ohm’s law. The resulting system of equations comprise the classical ideal MHD waves, whistlers, drift waves and solitarywave solutions. The latter reside in distinct regions of the phase space spanned by the speed and the angle (to the magnetic field) of the propagating wave. Within each region, qualitatively similar solitary structures are found. In the limit of neglected electron intertia, the solitary wave solutions are confined to two regions of slow and fast waves, respectively. The slow (fast) structures are associated with density compressions (rarefactions) and positive (negative) electric potentials. Such negative potentials are shown to accelerate electrons in the auroral region (solar corona) to tens (hundreds) of keV. The positive electric potentials could accelerate solar wind ions to velocities of 300–800 km/s. The structure widths perpendicular to themagnetic field are in the Earth’s magnetosphere (solar corona) of the order of 1–100 km (m). This thesis also addresses a type of incoherent scatter radar spectra, where the ion line exhibits a spectrally uniform power enhancement with the up- and downshifted shoulder and the spectral region in between enhanced simultaneously and equally. The power enhancements are one order of magnitude above the thermal level and are often localised to an altitude range of less than 20 km at or close to the ionospheric F region peak. The observations are well-described by a model of ion-acoustic solitary waves propagating transversely across the radar beam. Two cases of localised ion line enhancements are shown to occur in conjunction with auroral arcs drifting through the radar beam. The arc passages are associated with large gradients in ion temperature, which are shown to generate sufficiently high velocity shears to give rise to growing Kelvin-Helmholtz (K-H) instabilities. The observed ion line enhancements are interpreted in the light of the low-frequency turbulence associated with these instabilities. / Denna avhandling handlar om solitära vågor och deras roll i norrskensacceleration och koronaupphettning, samt deras signatur i spektra uppmätta med inkoherent spridningsradar. Solitära vågor bildas genom en balans mellan ickelinjära och dispersiva effekter. Ickelinjäriteter finns det gott om i ideal magnetohydrodynamik (MHD) och dispersion kan införas genom att inkludera Halltermen i den generaliserade Ohms lag. Det resulterande ekvationssystemet omfattar de klassiska vågorna inom ideal MHD, visslare, driftvågor och solitära vågor. De sistnämnda återfinns i väldefinierade områden i fasrummet som spänns upp av farten och vinkeln (mot magnetfältet) för den propagerande vågen. Inom varje sådant område återfinns kvalitativt lika solitära våglösningar. Om man försummar elektronernas tröghet begränsas de solitära våglösningarna till två områden med långsamma respektive snabba vågor. De långsamma (snabba) strukturerna är associerade med täthets-kompressioner (förtunningar) och positiva (negativa) elektriska potentialer. De negativa potentialerna visas kunna accelerera elektroner i norrskensområdet (solens korona) till tiotals (hundratals) keV medan de positiva potentialerna accelererar solvindsjoner till hastigheter på 300–800 km/s. Strukturbredderna vinkelrät mot magnetfältet är i jordens magnetosfär (solens korona) av storleksordningen 1–100 km (m). Denna avhandling tar även upp en typ av inkoherent spridningsradarspektra, där jonlinjen uppvisar en spektralt uniform förstärkning. Detta innebär att den upp- och nedskiftade skuldran och spektralbandet däremellan förstärks simultant och i lika hög grad. Effektförstärkningen är en storleksordning över den termiska nivån och är ofta lokaliserad till ett höjd-intervall av mindre än 20 km nära jonosfärens F-skiktstopp. Observationerna beskrivs väl av en modell med solitära vågor som propagerar transversellt genom radarstrålen. Två fall av lokaliserade jonlinjeförstärkningar visas sammanfalla med att norrskensbågar driver genom radarstrålen. I samband med bågarnas passage uppmäts stora gradienter i jontemperatur, vilket visas skapa tillräckligt kraftiga hastighetsskjuvningar för att Kelvin-Helmholtz-instabiliteter ska tillåtas växa. De observerade jonlinjeförstärkningarna tolkas i skenet av den lågfrekventa turbulensen som är kopplad till dessa instabiliteter.