  Ultra-light axions have sparked attention because their tiny mass $m\sim
10^{-22}$ eV, which leads to a Kiloparsec-scale de Broglie wavelength
comparable to the size of dwarf galaxy, could alleviate the so-called
small-scale crisis of massive cold dark matter (CDM) candidates.However, recent
analyses of the Lyman-$\alpha$ forest power spectrum set a tight lower bound on
their mass of $m\gtrsim 10^{-21}$ eV which makes them much less relevant from
an astrophysical point of view. An important caveat to these numerical studies
is that they do not take into account attractive self-interactions among
ultra-light axions, which can counteract the quantum "pressure" induced by the
strong delocalization of the particles. In this work, we show that even a tiny
attractive interaction among ultra-light axions can have a significant impact
on the stability of cosmic structures at low redshift. After a brief review of
known results about solitons in the absence of gravity, we discuss the
stability of filamentary and pancake-like solutions when quantum pressure,
attractive interactions and gravity are present. The analysis based on one
degree of freedom, namely the breathing mode, reveals that pancakes are stable,
while filaments are unstable if the mass per unit length is larger than a
critical value. However, we show that pancakes are unstable against transverse
perturbations. We expect this to be true for halos and filaments as well.
Instabilities driven by the breathing mode will not be seen in the low column
density Lyman-$\alpha$ forest unless the axion decay constant is extremely
small, $f\lesssim 10^{13}$ GeV. Notwithstanding, axion solitonic cores could
leave a detectable signature in the Lyman-$\alpha$ forest if the normalization
of the unknown axion core - filament mass relation is $\sim 100$ larger than it
is for spherical halos.
