Abstract

The real exponential series approach used to solve baseband nonlinear evolution equations is suitably modified to solve complex nonlinear evolution equations describing envelope propagation in a cubically nonlinear, dispersive optical fiber. The essence of the real exponential method is to build up the solution to the nonlinear (real) PDE from real exponential solutions to its linear dispersive part, and to match the boundary conditions at ξ = 0, where ξ is a moving frame of reference. The solution to the complex PDE is based on a similar concept; however, the solution is now expressed in schematic form: where u(z, t) is the dependent variable in the PDE and where ξ_a= τ– aζ, ξ = τ - bζ; τ and ζ are independent variables. The quantities a, b, k', k" have been assumed to be real, although the A_n terms can be complex. Each A_n is found recursively in terms of the respective A₁ in each region (greater than and less than zero), and continuity of the functions and their first derivatives demanded at the boundary. The method is useful in predicting solitary wave solutions in a nonlinear optical fiber at zero dispersion wavelength, which otherwise has no closed form solutions.

© 1989 Optical Society of America