Rogue waves are defined as localised structures in both space and time. They can be described as isolated large-amplitude waves that unexpectedly appear on a relatively calm background. In the ocean, they often cause injury and damage. Rogue waves are ubiquitous, being found in oceans and in the atmosphere.
They exist in various areas of physics such as optics. The mechanism of formation of these unusual waves can vary depending on the subject. Mathematically, such waves can be modelled by specific (rational) solutions of the nonlinear Schrodinger equation. This equation is a generic model for wave propagation in dispersive media with weak nonlinearity. When higher-order terms in the equation are involved, the solution becomes distorted, but its main features remain localised in space and time. Although exact solutions do not exist in all cases, approximations that describe the solutions with reasonable accuracy can be found.

In this seminar, I will introduce a modification of the Lagrangian approach for finding rogue wave solutions of the extended nonlinear Schrödinger equation. Several examples of the application of this technique for particular physically-relevant extended equations will be presented. Namely, the Lagrangian approach is used to evaluate the effects of third and fourth order dispersion terms as well as influence of the Raman intra-pulse effect on rogue waves.

In the second part of the talk, the technique called the bilinear Hirota method is used for finding the exact rational solutions of another evolution equation, known as the Gardner equation. These solutions describe the emergence of internal rogue waves in layered fluids.