This project concerns properties of solutions of nonlinear evolution equations and related ordinary differential equations that model the propagation of large-amplitude waves in physical media such as surface and internal water waves or electromagnetic waves in optical fibers. The equations studied are completely integrable systems, for which there are many more analytical techniques available than for more general equations. However, at the same time integrable equations are known to arise from more general equations in certain limiting cases. Each time that a new result is established for a completely integrable model that arises in this way, a version of that result immediately applies to all of the more general models. These properties make the study of integrable models both mathematically compelling and also of prime physical relevance. This project uncovers new solutions of integrable equations of recognized importance and studies their exact and asymptotic properties. Knowledge of the solutions obtained impacts several application areas such as marine engineering (prediction/properties of large-amplitude surface water waves or "freak waves" and deformations of the free surface between layers in a density-stratified ocean) and condensed matter physics (Bose-Einstein condensation). As part of the project, computer codes and a textbook are being produced for public consumption. Three PhD students are being partially supported to work on the project. Specifical