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Stability conditions for one-dimensional optical solitons in cubic-quintic-septimal media

Conditions for stable propagation of one-dimensional bright spatial solitons in media exhibiting optical nonlinearities up to the seventh-order are investigated. The results show well-defined stability regions even when all the nonlinear terms are focusing. Conditions for onset of the supercritical collapse of the optical beam are identified too. A variational approximation is used to predict dependence of the soliton propagation constant on the norm, and respective stability regions are identified using the Vakhitov-Kolokolov criterion. Analytical results obtained by means of the variational approximation are corroborated by numerical simulations of the cubic-quintic-septimal nonlinear Schroedinger equation.