Bright and dark rogue internal waves: The Gardner equation approach

Abstract

We have found 'bright and dark' solutions of the Gardner equation which can model internal rogue waves in three-layer fluids. We provide the first four 'bright' and 'dark' exact rational solutions to the Gardner equation. These are the lowest-order solutions of the corresponding hierarchies of rogue-wave solutions of this equation. They have been obtained from the rogue-wave solutions of a modified Korteweg-de Vries equation by using a Lorentz-type transformation. The maximal (and minimal) amplitudes and the background levels of these solutions for arbitrary order are deduced, based on the lowest-order examples. These solutions can be useful for explanations of extremely large amplitude internal waves in the ocean, as well as for abnormally large-amplitude waves in other areas of nonlinear physics, such as optics and dusty plasmas.