Bäcklund Transformation and <i>N</i>-Soliton Solutions for the Cylindrical Nonlinear Schrödinger Equation from the Diverging Quasi-Plane Envelope Waves

This paper investigates a cylindrical nonlinear Schrödinger (cNLS) equation, which describes the cylindrically diverging quasi-plane envelope waves in a nonlinear medium. With the Hirota method and symbolic computation, bilinear form and N-soliton solutions in the form of an Nth-order polynomial in N exponentials are obtained for the cNLS equation. By means of the properties of double Wronskian, the N-soliton solutions in terms of the double Wronskian is testified through the direct substitution into the bilinear form. Based on the bilinear form and exchange formulae, the bilinear Bäcklund transformation is also given. Those solutions are graphically depicted to understand the soliton dynamics of the cylindrically diverging quasi-plane envelope waves. Soliton properties are discussed and physical quantities are also analyzed. Dispersion parameter has the effect that it may extend (or shorten) the periodic time of soliton interaction and change the direction of soliton propagation. Amplitudes of solitons are related to the cubic nonlinearity parameter.