The usual(1+1)-dimensional Schwartz Boussinesq equation is extended to the(1+1)-dimensional space-time sym-metric form and the general(n+1)-dimensional space-time symmetric form.These extensions are Painlev integrable in the sense that they possess the Painlev property.The single soliton solutions and the periodic travelling wave solutions for arbitrary dimensional space-time symmetric form are obtained by the Painlev-Bcklund transformation.
This paper analyses bright and dark spatial self-similar waves propagation and collision in graded-index nonlinear waveguide amplifiers with self-focusing and self-defocusing Kerr nonlinearities.It finds an appropriate transformation for the first time such that the nonlinear Schro¨dinger equation (NLSE) with varying coefficients transform into standard NLSE.It obtains one-solitonlike,two-solitonlike and multi-solitonlike self-similar wave solutions by using the transformation.Furthermore,it analyses the features of the self-similar waves and their collisions.
