The nonlinear Schrödinger equation with variable coefficients: concentrated solution and its break

The nonlinear Schrödinger equation with variable coefficients is applied to the description of the wave processes in inhomogeneous media. The Cauchy problem is considered with an initial value from the Schwartz class. Conditions of the conservation of a concentrated solution over the whole time are developed basing on the research of the local solvability of this problem. A possibility is discussed for a concentrated solution to break down within a finite period of time.