Exact solitary-wave solutions of ${\chi }^{\mathrm{}\left(2\right)\mathrm{}}$ Ginzburg-Landau equations

A family of exact temporal solitary-wave solutions (dissipative solitons) to the equations governing second-harmonic generation in quadratically nonlinear optical waveguides, in the presence of linear bandwidth-limited gain at the fundamental harmonic and linear loss at the second harmonic, is found, and the existence domain for the solutions is delineated. Direct numerical simulations of the solitons demonstrate that, as well as the classical pulse solutions to the cubic Ginzburg-Landau equation, the dissipative solitons can propagate robustly over a considerable distance before the model’s intrinsic instability leads to onset of “turbulence.” Two-soliton bound states are also predicted and then found in the direct simulations. We estimate real values of the physical parameters necessary for the existence of the solitons predicted, and conclude that they can be observed experimentally. A promising application for the solitons is their use in closed-loop cavities.