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We introduce a general model which augments the one-dimensional nonlinearSchr\"{o}dinger (NLS) equation by nonlinear-diffraction terms competing withthe linear diffraction. The new terms contain two irreducible parameters andadmit a Hamiltonian representation in a form natural for optical media. Theequation serves as a model for spatial solitons near the supercollimation pointin nonlinear photonic crystals. In the framework of this model, a detailedanalysis of the fundamental solitary waves is reported, including thevariational approximation (VA), exact analytical results, and systematicnumerical computations. The Vakhitov-Kolokolov (VK) criterion is used toprecisely predict the stability border for the solitons, which is found in anexact analytical form, along with the largest total power (norm) that the wavesmay possess. Past a critical point, collapse effects are observed, caused bysuitable perturbations. Interactions between two identical parallel solitarybeams are explored by dint of direct numerical simulations. It is found thatin-phase solitons merge into robust or collapsing pulsons, depending on thestrength of the nonlinear diffraction.

Spatial solitons under competing linear and nonlinear diffractions