New solutions for the cubic Schrödinger equation and solitonic solutions

In this work, we will consider a nonlinear cubic Schrödinger equation. We make use of both the truncated Painleve expansion and symbolic computation method to obtain an auto-Backlund transformation and certain soliton-typed explicit solutions. Also, it is show that the homogeneous balance method is extended to search for Backlund transformation for the Schrödinger equation.