ON THE SOLVABILITY OF THE SCHRÖDINGER EQUATION WITH POWER AND LOGARITHMIC-POWER TYPE OF NONLINEARITIES.

This paper deals with the nonlinear Schrödinger equation (NSE) with polynomial (power type) and power logarithmic nonlinearities. A third order generalization of NSE - the so called RKLE with applications in telecommunications is considered too. In some special cases the soliton solutions or the blowing up solutions can be expressed by elementary or special functions (elliptic, hyperbolic). We are looking for the wave solutions ansatz having real non-negative amplitude and linear phase. The amplitude satisfies second order nonlinear ODE and it is constructed by using the method of first integral (in mechanic processes with one degree of freedom are described by such type ODEs). So the solutions are found into integral form and one-parametric family of periodic solutions is constructed. Several illustrative examples of periodic, blowing up and soliton type solutions are proposed. [ABSTRACT FROM AUTHOR]