| Abstract: |
| Exponential asymptotics can be used to calculate exponentially small asymptotic contributions in singularly-perturbed problems. I will apply these techniques to determine the behaviour of breathers in the discrete nonlinear Schr\odinger equation. I will begin by demonstrating how these ideas predict the well-known feature of the existence of two types of fixed points, namely site-centered and inter-site-centered. I will then show that the exponentially small contributions to the solution can be used to calculate the asymptotic scaling and precise value of the exponentially small eigenvalues in the system associated with site-centered (stable) and inter-site-centered (unstable) configurations. I will then explain how this method can be extended to study related problems, such as the dynamics of kink solutions and solutions with long-range dependence. This method paves the way for such an analysis in a wide range of lattice nonlinear dynamical equation models. |
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