Extensions of the General Solution to the Inverse Problem of the Calculus of Variations, and Variational, Perturbative and Reversible Systems Approaches to Regular and Embedded Solitary Waves

In the first part of this Dissertation, hierarchies of Lagrangians of degree two, three or four, each only partly determined by the choice of leading terms and with some coefficients remaining free, are derived. These have significantly greater freedom than the most general differential geometric criterion currently known for the existence of a Lagrangian and variational formulation since our existence conditions are for individual coefficients in the Lagrangian. For different choices of leading coefficients, the resulting variational equations could also represent traveling waves of various nonlinear evolution equations. Families of regular and embedded solitary waves are derived for some of these generalized variational ODEs in appropriate parameter regimes. In the second part, an earlier approach based on soliton perturbation theory is significantly generalized to obtain an analytical formula for the tail amplitudes of nonlocal solitary waves of a perturbed generalized fifth-order Korteweg-de Vries (FKdV) equation. On isolated curves in the parameter space, these tail amplitudes vanish, producing families of localized embedded solitons in large regions of the space. Off these curves, the tail amplitudes of the nonlocal waves are shown to be exponentially small in the small wavespeed limit. These seas of delocalized solitary waves are shown to be entirely distinct from those derived in that earlier work. These perturbative results are also discussed within the framework of known reversible systems results for various families of homoclinic orbits of the corresponding traveling-wave ordinary differential equation of our generalized FKdV equation. The third part considers a variety of dynamical behaviors in a multiparameter nonlinear Mathieu equation with distributed delay. A slow flow is derived using the method of averaging, and the predictions from that are then tested against direct numerical simulations of the nonlinear Mathieu system. Both areas of agreement and disagreement between the averaged and full numerical solutions are considered