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

TL;DR

This paper investigates the existence, stability, and dynamics of fractional solitons in a nonlinear Schr{"o}dinger equation, revealing unique stabilization phenomena and connecting harmonic and biharmonic limits through the fractional parameter.

Contribution

It introduces the first detailed analysis of fractional solitons, highlighting their destabilization, bifurcation behavior, and unique stabilization features not seen in integer-order cases.

Findings

Destabilization of solitary waves for fractional exponent $oldsymbol{ extit{ extalpha}<1}$.

Existence of in- and out-of-phase soliton pairs for $oldsymbol{ extalpha>2}$.

Unique stabilization of certain multipulse branches in the fractional case.

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.