Coupled mode reductions and rotating wave approximations in nonlinear equations

In this thesis, we investigate the applicability of coupled mode theory in the cubic-quintic nonlinear Schrodinger/Gross Pitaevskii (NLS/GP) equation with a linear double-well potential and study justifications of the rotating wave approximations in lattice systems. First, we study the long-time dynamics near a symmetry breaking bifurcation point of the cubic-quintic NLS/GP with symmetric double-well potentials. We investigate the stability of the solutions of NLS/GP and analyze the error for the finite dimensional ansatz. Next, we consider a class of discrete nonlinear Klein-Gordon equations with damping and parametric drive. Using small amplitude ansatzs, one usually approximates the equations using a damped, driven discrete nonlinear Schrodinger type equation. Here, we show for the first time the justification of this approximation by finding the error bound using energy estimates. Additionally, we prove the local and global existence of the solutions of Schrodinger equation. Numerical comparisons of discrete breathers obtained from the original nonlinear equation and the discrete nonlinear Schrodinger equation are presented describing the analytical results. Finally, we consider a damped, externally driven nonlinear Klein-Gordon equation and justify the small amplitude ansatz yields a discrete nonlinear Schrodinger equation with damping and external drive. The same problems as the Klein-Gordon equation with damping and parametric drive are addressed.