Simultaneous solitary-wave solutions in a nonlinear parametric waveguide

We introduce a simple topological classification for the solitary-wave solutions of the coupled equations describing ${\mathrm{\chi }}^{\mathrm{}\left(2\right)\mathrm{}}$ parametric waveguides. Both temporal and spatial cases are studied. Simultaneous solitary-wave solutions at two different frequencies, which we term ‘‘simultons,’’ are shown to exist using topological arguments. In each case, families of general one-parameter simultons are given numerically. Each family consists of both fundamental and higher-order simultons of a topological or nontopological nature. There are several possible ways to combine dispersions at different frequencies. Thus there are more possible ways of maintaining temporal simultons in 1+1 space-time than for purely spatial simultons in two space dimensions. © 1996 The American Physical Society.