| Abstract: |
| In the present work, we will explore a number of variants of the standard
Klein-Gordon and NLS problems and study both kinks/dark solitons but also
occasionally pulses/bright solitons in them. We will be motivated by the
recent remarkable experimental realization of not only biharmonic, but
also arbitrary fractional index Laplacian operators in nonlinear optics
and we will see how the tails of the structures behave for different
regimes of the fractional exponent $\alpha$. This, in turn, will guide
us to explore not only the soliton-soliton interactions, but also their
stability. We will seek to present a unifying picture of the relevant
interactions as a function of $\alpha$, starting from the biharmonic
setting and interpolating between that and the Laplacian one, as well
as moving to the sub-Laplacian index case. Finally, we will touch upon
dissipative variant of the problem, presently of interest, including the quartic
Lugiato-Lefever equation and related problems that experiments are
currently getting to. |
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