Polar compactons and solitons in a two dimensional optical waveguide: Theory and simulations

Abstract

The propagation of optical cylindrical compactons and solitons in a two dimensional nonlocal nonlinear waveguide is deeply investigated by carrying a particular emphasis on its nonlinear response function. Considering the weakly nonlocal medium, the two dimensional nonlocal nonlinear Schrödinger equation describing the propagation of signal light is derived. This equation contains the basic nonlinear Schrödinger equation with additional nonlinear terms proportional to the second order space derivative of coordinates in the transverse directions. By seeking the solution in polar form, this equation is separated into real and imaginary parts, followed by the analysis of the equilibrium points and their stability, predicting the type of solutions. These solutions are found both with linear and nonlinear phases through direct integration. In order to seek polar solutions, the nonlocal nonlinear Schrödinger equation is expressed in cylindrical coordinate and the resulting equation, containing additional terms inversely proportional to the radius r is not easily solvable. The type of solutions of this equation is predicted by the investigation of new equilibrium points and their stability. The analytical cylindrical compactons and pulse solitons as solutions of this equation are provided by using the Rayleigh–Ritz variational method, usually used for symmetric radial solitons to approximate pulse solitons in polar coordinate, and the stability of these solutions is checked through numerical investigations.