Asymptotic solutions for singularly perturbed Boussinesq equations

We consider a family of singularly perturbed Boussinesq equations. We obtain a rational weak solution to the classical Boussinesq equation and demonstrate that this solution can be used to construct perturbation solutions for singularly perturbed high-order Boussinesq equations. These solutions take the form of an algebraic function which behaves similarly to a peakon, and which decays as time becomes large. We show that approximate solutions obtained via perturbation for the singularly perturbed models are asymptotic to the true solutions as the residual errors rapidly decay away from the origin.