Fractional Solitons: A Homotopic Continuation from the Biharmonic to the Harmonic $\phi^4$ Model

TL;DR

This paper investigates the transition of soliton structures between harmonic and biharmonic PDEs using fractional derivatives, revealing bifurcation phenomena and stability characteristics across models.

Contribution

It introduces a homotopic continuation method using Riesz fractional derivatives to interpolate between harmonic and biharmonic models, analyzing soliton bifurcations and stability.

Findings

Soliton tails transition from exponential to power-law to oscillatory exponential with fractional order.

A cascade of saddle-center bifurcations occurs near the biharmonic limit.

The existence of stationary soliton pairs varies significantly between the models.

Abstract

In the present work we explore the path from a harmonic to a biharmonic PDE of Klein-Gordon type from a continuation/bifurcation perspective. More specifically, we make use of the Riesz fractional derivative as a tool that allows us to interpolate between these two limits. We illustrate, in particular, how the coherent kink structures existing in these models transition from the exponential tail of the harmonic operator case, via the power-law tails of intermediate fractional orders, to the oscillatory exponential tails of the biharmonic model. Importantly, we do not limit our considerations to the single kink case, but extend to the kink-antikink pair, finding an intriguing cascade of saddle-center bifurcations happening exponentially close to the biharmonic limit. Our analysis clearly explains the transition between the infinitely many stationary soliton pairs of the biharmonic case and the absence of even a single such pair in the harmonic limit. The stability of the different configurations obtained and the associated dynamics and phase portraits are also analyzed.