Quadratic fractional solitons

We introduce a system combining the quadratic self-attractive or composite quadratic-cubic nonlinearity, acting in combination with fractional diffraction, which is characterized by its Lévy index ?. The model applies to gas of quantum particles moving flights, term representing Lee-Huang-Yang correction mean-field interactions. A family fundamental solitons constructed numerical form, while dependence norm on chemical potential characteristic obtained an exact analytical form. quasi-Townes solitons, appearing limit case ?=1/2, investigated means variational approximation. nonlinear lattice, represented spatially periodical modulation term, briefly addressed too. consideration interplay competing (attractive) and cubic (repulsive) terms lattice reveals families single-, double-, triple-peak gap (GSs) two finite bandgaps. nonlinearity gives rise alternating regions stability instability GS, intervals shrinking increase number peaks GS.