The Benjamin Equation is a nonlinear dispersive wave equation that admits solitary waves with oscillatory tails. The equation sports two dispersive terms, one a KdV-like term and another nonlocal term and it is this competition in balance with the nonlinearity that leads to its oscillatory yet permanent shape features. I will first derive the equation in a physical setting and describe its physical regime of validity. I will then recount how we proved existence and stability of the solutions, and approximated them via numerical means.
I will end with some numerical experiments that suggest that the equation supports bounded state solutions. Bounded state solutions in dispersive waves describe solutions that support collisional interacting waves that are bounded by a much larger structure in the solution. I will argue that this is a good problem to investigate since there is no theory behind these bounded states. Bounded states should be ubiquitous for equations with a certain type of competing dispersive terms.