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Ultra-light axions have sparked attention because their tiny mass $m\sim10^{-22}$ eV, which leads to a Kiloparsec-scale de Broglie wavelengthcomparable to the size of dwarf galaxy, could alleviate the so-calledsmall-scale crisis of massive cold dark matter (CDM) candidates.However, recentanalyses of the Lyman-$\alpha$ forest power spectrum set a tight lower bound ontheir mass of $m\gtrsim 10^{-21}$ eV which makes them much less relevant froman astrophysical point of view. An important caveat to these numerical studiesis that they do not take into account attractive self-interactions amongultra-light axions, which can counteract the quantum "pressure" induced by thestrong delocalization of the particles. In this work, we show that even a tinyattractive interaction among ultra-light axions can have a significant impacton the stability of cosmic structures at low redshift. After a brief review ofknown results about solitons in the absence of gravity, we discuss thestability of filamentary and pancake-like solutions when quantum pressure,attractive interactions and gravity are present. The analysis based on onedegree of freedom, namely the breathing mode, reveals that pancakes are stable,while filaments are unstable if the mass per unit length is larger than acritical value. However, we show that pancakes are unstable against transverseperturbations. 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 columndensity Lyman-$\alpha$ forest unless the axion decay constant is extremelysmall, $f\lesssim 10^{13}$ GeV. Notwithstanding, axion solitonic cores couldleave a detectable signature in the Lyman-$\alpha$ forest if the normalizationof the unknown axion core - filament mass relation is $\sim 100$ larger than itis for spherical halos.

The impact of ultra-light axion self-interactions on the large scale structure of the Universe