Fourth Order Dispersion in Nonlinear Media

Author

Georgios Tsolias, University of Massachusetts AmherstFollow

Author ORCID Identifier

https://orcid.org/0000-0002-0616-4749

AccessType

Open Access Dissertation

Document Type

dissertation

Degree Name

Doctor of Philosophy (PhD)

Degree Program

Mathematics

Year Degree Awarded

2023

Month Degree Awarded

February

First Advisor

Panayotis Kevrekidis

Second Advisor

Qian-Yong Chen

Third Advisor

Wei Zhu

Fourth Advisor

Dimitrios Maroudas

Subject Categories

Non-linear Dynamics | Numerical Analysis and Computation | Optics | Partial Differential Equations

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

DOI

https://doi.org/10.7275/33233379

Recommended Citation

Tsolias, Georgios, "Fourth Order Dispersion in Nonlinear Media" (2023). Doctoral Dissertations. 2785.
https://doi.org/10.7275/33233379 https://scholarworks.umass.edu/dissertations_2/2785

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