Solitary wave interaction and evolution

Asymptotic theory is applied to examine solitary wave interaction for three higher-order model equations, which represent small perturbations to integrable equations. The higher-order equations considered are the higher-order Nonlinear Schr�dinger equation and the focusing and defocusing higher-order Hirota equations. The asymptotic theory, which involves a transformation, allows the straightforward determination of parameter choices, for which the higher-order equations are asymptotically integrable, and of the higher-order phase and coordinate shifts due to the collision, in the asymptotically integrable cases. For the higher-order Hirota equations, direct soliton perturbation theory is also used, to determine the details of the evolving solitary waves; in particular analytical expressions are found for the solitary wave tails. An important feature of the asymptotic and perturbation theories is that they allow cross-validation of the theoretical results and also allow families of asymptotic embedded solitons to be identified. Numerical solutions of the governing equations are also obtained. For solitary wave interaction, asymptotically elastic and inelastic cases are considered. When the higher-order coefficients satisfy the appropriate algebraic relationship then the numerical results confirm the prediction of the asymptotic theory. Numerical solutions for evolving solitary waves are also used to confirm the results of the soliton perturbation theory.