A Study of Solitary-Wave Solution for the Extended Benjamin-Bona-Mahony Equation

The Regularized Long-Wave equation, also known as the Benjamin-Bona-Mahony (BBM)-equation was first studied as a model for small-amplitude long waves that propagate on the free surface of a perfect fluid. As an alternative to the Korteweg-de Vries equation, it features a balance between nonlinearity and a frequency dispersion term that allows the existence of traveling waves that are smooth and symmetric about their maximum. Such waves decay rapidly to zero on their outskirts and, because of their tendency to travel alone, are known as `solitary waves'. We investigate the behavior of solitary-wave solutions for the Extended BBM (EBBM)-equation which is the BBM-equation, but with two power nonlinearities in general, gradient form. We prove that this model is globally well-posed, which provides a rigorous foundation for the stability theory of their solitary-wave solutions. Since solitary-wave solutions of the EBBM-equation are not known analytically, they are generated and investigated numerically. It transpires that there can be as many as three stability regimes for the EBBM-solitary waves. We present numerical simulations of the formation and long-time evolution of solitary waves, the behavior of solitary waves under amplitude perturbations and the interaction of solitary waves. Minimal perturbations necessary to force a solitary wave to change stability regimes are also determined.