#### Date of Award

Spring 1988

#### Project Type

Dissertation

#### Program or Major

Physics

#### Degree Name

Doctor of Philosophy

#### First Advisor

Harvey Shepard

#### Abstract

We analytically study the kink-antikink (K-K) collisions in the classical one spatial dimension and time phi-fourth field theory as an example of inelastic collisions between solitary waves. We use the linear eigenvalue collective coordinate approach to describe the system in terms of the separation distance between the kink and the antikink and the amplitude of shape vibrations generated on each kink as a result of the collision. By calculating the energy given to the shape vibrations as a function of the incoming velocity, we find the critical value of the initial velocity above which the two colliding kinks always separate after the collision. A model previously proposed to explain the two-bounce collisions in terms of a resonant energy exchange between the orbital frequency of the bound K-K pair and the frequency of shape vibrations is modified using our analytical results. We derive a (data-free) formula that predicts the values of the initial velocities for which resonance occurs. A generalized version of this modified model is shown to give good results when it is applied to K-K collisions in other similar field theories. In the Appendices Nonlinear Klein Gordon equations with solitary (travelling) wave solutions are reviewed and solved for particular cases. The solutions are related to soliton solutions of the sine-Gordon equation. Also the phi-fourth equation perturbed with a constant force and dissipation is solved, and finally, we present new kink-bearing integro-differential and nonlinear differential equations.

#### Recommended Citation

Papadopoulos, Panagiotis Andreas, "Solitary waves and nonlinear Klein-Gordon Equations" (1988). *Doctoral Dissertations*. 1543.

https://scholars.unh.edu/dissertation/1543