Soliton evolution and radiation loss for the nonlinear Schrödinger equation

The transient evolution of general initial pulses into solitons for the nonlinear Schr\"odinger (NLS) equation is considered. By employing a trial function which consists of a solitonlike pulse with variable parameters plus a linear dispersive term in an averaged Lagrangian, ordinary differential equations (ODE's) are derived which approximate this evolution. These approximate equations take into account the effect of the generated dispersive radiation upon the pulse evolution. Specifically, in the approximate ODE's the radiation acts as a damping which causes the pulse to decay to a steady soliton. The solutions of the approximate ODE's are compared with numerical solutions of the NLS equation and are found to be in very good agreement. In addition, the potential implications for obtaining improved approximate ODE models for soliton propagation in optical fibers and other devices governed by NLS-type equations, such as soliton logic gates, are discussed.

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