Rogue waves can appear in nature, as well as in various physical, chemical and biological systems. Their mathematical description is based on partial differential equations that have solutions that are localized both in time and space. One example is the Peregrine solution of the nonlinear Schrodinger equation (NLSE). When higher-order terms are involved in the equation, the solution becomes distorted, but its main features remain localized in space and time. Although exact solutions are not obtained in all cases, approximations which describe the solutions with reasonable accuracy can be found.
So far in this project, we have used a Lagrangian approach to find rogue wave solutions of higher-order extensions of the NLSE. This approach had been successfully applied to find the soliton solutions but had never been examined for finding rogue wave solutions, due to its algebraic complexity. We have developed this technique in a study of rogue waves. For example, approximate solutions in the form of rogue waves are obtained and analysed for optical fibres with fourth and sixth order dispersions, as well as the Raman delay response. In order to confirm the applicability of the technique, we have tested it on a few integrable equations and compared the results with the known exact solutions. The other non-integrable equations do not have exact solutions in the form of a rogue wave. In each case, we have conducted numerical simulations confirming that the approximate solutions are sufficiently accurate.