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Abstract
In recent years, there has been an explosion of interest in media bearing quartic dispersion. After the experimental realization of so-called pure-quartic solitons, a significant number of studies followed both for bright and for dark solitonic struc- tures exploring the properties of not only quartic, but also setic, octic, decic etc. dispersion, but also examining the competition between, e.g., quadratic and quartic dispersion among others. In the first chapter of this Thesis, we consider the interaction of solitary waves in a model involving the well-known φ4 Klein-Gordon theory, bearing both Laplacian and biharmonic terms with different prefactors. As a result of the competition of the respective linear operators, we obtain three distinct cases as we vary the model parameters. In the first the biharmonic effect dominates, yielding an oscillatory inter-wave interaction; in the third the harmonic effect prevails yielding exponen- tial interactions, while we find an intriguing linearly modulated exponential effect in the critical second case, separating the above two regimes. For each case, we calculate the force between the kink and antikink when initially separated with suf- ficient distance. Being able to write the acceleration as a function of the separation distance, and its corresponding ordinary differential equation, we test the corre- sponding predictions, finding very good agreement, where appropriate, with the corresponding partial differential equation results. Where the two findings differ, we explain the source of disparities. Finally, we offer a first glimpse of the interplay of harmonic and biharmonic effects on the results of kink-antikink collisions and the corresponding single- and multi-bounce windows. In the next two Chapters, we explore the competition of quadratic and quar- tic dispersion in producing kink-like solitary waves in a model of the nonlinear Schroedinger type bearing cubic nonlinearity. We present 6 families of multikink so- lutions and explore their bifurcations as a prototypical parameter is varied, namely the strength of the quadratic dispersion. We reveal a rich bifurcation structure for the system, connecting two-kink states with ones involving 4-, as well as 6-kinks. The stability of all of these states is explored. For each family, we discuss a “lower branch” adhering to the energy landscape of the 2-kink states (also discussed in the previous Chapter). We also, however, study in detail the “upper branches” bearing higher numbers of kinks. In addition to computing the stationary states and analyzing their stability at the PDE level, we develop an effective particle the- ory that is shown to be surprisingly efficient in capturing the kink equilibria and normal (as well as unstable) modes. Finally, the results of the bifurcation analysis are corroborated with direct numerical simulations involving the excitation of the states in a targeted way in order to explore their instability-induced dynamics. While the previous two studies were focused on the one-dimensional problem, in the fourth and final chapter, we explore a two-dimensional realm. More specif- ically, we provide a characterization of the ground states of a higher-dimensional quadratic-quartic model of the nonlinear Schr ̈odinger class with a combination of a focusing biharmonic operator with either an isotropic or an anisotropic defocusing Laplacian operator (at the linear level) and power-law nonlinearity. Examining principally the prototypical example of dimension d = 2, we find that instability arises beyond a certain threshold coefficient of the Laplacian between the cubic and quintic cases, while all solutions are stable for powers below the cubic. Above the quintic, and up to a critical nonlinearity exponent p, there exists a progressively narrowing range of stable frequencies. Finally, above the critical p all solutions are unstable. The picture is rather similar in the anisotropic case, with the dif- ference that even before the cubic case, the numerical computations suggest an interval of unstable frequencies. Our analysis generalizes the relevant observations for arbitrary combinations of Laplacian prefactor b and nonlinearity power p. We conclude the thesis with a summary of its main findings, as well as with an outlook towards interesting future problems
Type
Dissertation (Open Access)
Date
2023-02
Publisher
Degree
License
Attribution-NonCommercial 4.0 International
License
http://creativecommons.org/licenses/by-nc/4.0/