Soliton evolution and radiation loss for the sine-Gordon equation

An approximate method for describing the evolution of solitonlike initial c onditions to solitons for the sine-Gordon equation is developed. This metho d is based on using a solitonlike pulse with variable parameters in an aver aged Lagrangian for the sine-Gordon equation. This averaged Lagrangian is t hen used to determine ordinary differential equations governing the evoluti on of the pulse parameters. The pulse evolves to a steady soliton by sheddi ng dispersive radiation. The effect of this radiation is determined by exam ining the: linearized sine-Gordon equation and loss terms are added to the variational equations derived from the averaged Lagrangian by using the mom entum and energy conservation equations for the sine-Gordon equation. Solut ions of the resulting approximate equations, which include loss, are found to be in good agreement with full numerical solutions of the sine-Gordon eq uation. [S1053-651X(99)10508-7].