Solitary waves and their critical behavior in a nonlinear nonlocal medium with power-law response

We discuss a nonlocal generalization of the nonlinear Schrödinger equation and study propagation of solitary waves in a nonlinear nonlocal medium at its critical state, the response of which obeys the power law with the exponent $k.\mathrm{}$ Using the time-dependent variational principle, we derive a set of dynamical equations and develop the fixed-point analysis. A critical behavior is found in the $k$ dependence of the width of the wave packet. We also present a proof of the stability of the system and discuss an oscillatory phenomenon in the self-focusing process.