Bright Fractional Single and Multi-Solitons in a Prototypical Nonlinear Schr{\"o}dinger Paradigm: Existence, Stability and Dynamics

Abstract

In the present work we explore features of single and pairs of solitary waves in a fractional variant of the nonlinear Schr{\"o}dinger equation. Motivated by the recent experimental realization of arbitrary fractional exponents, upon quantifying the tail properties of such coherent structures, we detail their destabilization when the fractional exponent $\alpha$ acquires values $\alpha<1$ and showcase how the relevant destabilization is associated with collapse type phenomena. We then turn to in- and out-of-phase pairs of such waveforms and illustrate how they generically exist for arbitrary $\alpha$ when we cross the harmonic limit, i.e., for $\alpha>2$. Importantly, we use the parameter $\alpha$ as a ``bifurcation parameter'' in order to connect the harmonic ($\alpha=2$) and biharmonic ($\alpha=4$) limits. Remarkably, not only do we retrieve the instability of all solitonic pairs in the biharmonic case, but showcase a stabilization feature of particular branches of such multipulses that is {\it unique} to the fractional case and does not arise -- to our knowledge -- for integer multi-pulse settings. We explain systematically this stabilization via spectral analysis and expand upon the implications of our results for the potential observability of fractional multipulse solitary waves.