Analytical and Approximate Methods in Rogue Wave Theory

Abstract

Rogue waves are defined as localized structures in both space and time. They can be described as isolated large-amplitude waves that unexpectedly appear on a relatively calm background. They exist in various areas of physics such as plasmas and 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 (NLSE). This equation is a generic model for wave propagation in dispersive media with weak nonlinearity. This thesis is devoted to the investigation of rogue waves in various nonlinear systems, including water waves, plasma, and optical fibers. In particular, modeling rogue waves using other equations has been considered. Rational solutions have been found for the Gardner equation which is known to be an accurate model for the propagation of internal waves in multi-layer fluids. The first four exact rational solutions to the Gardner equation are given as a mathematical description for internal rogue waves. These are the lowest-order solutions of the corresponding hierarchies of rogue-wave solutions of this equation. Integral relations for the rogue waves of the Gardner equation are presented. It is shown that conserved quantities in these relations tend to be integers. Another subject covered in the thesis is the formation of rogue waves in shallow water. Namely, the three lowest-order exact rational solutions to the complex-valued Korteweg-de Vries (KdV) equation are found. These solutions can serve as a likely model for shallow water rogue waves. It is found that the amplitude amplification factor of such waves is much larger than the amplitude amplification factor of rogue waves occurring in deep water. Further progress is reached in extending the complex-valued KdV equation to a more general equation that contains an infinite number of high-order terms with arbitrary real coefficients. This equation is shown to be integrable allowing us to find a general form of soliton and rogue wave solutions for this equation. Three first integrals of motion have been found for new solutions. Rogue waves in dusty plasma is another application of the theory developed in the thesis. Namely, a long wave-length model in a plasma with cold and heavy dust particles at thermal equilibrium has been considered. The detailed derivation leads to the Gardner equation which is an accurate model for weakly nonlinear and dispersive wave propagation in the regime of small amplitude waves. A discrete hierarchy of rational solutions describing rogue waves in dusty plasma shows that a considerable amount of energy (or electric charge) could become concentrated in a relatively small area of plasma at certain instance of time. When exact solutions cannot be found, approximate methods are used for modelling rogue waves. In this thesis, we use a Lagrangian approach for finding rogue wave solutions of the extended NLSE. The higher-order terms in the equation distort the rogue wave solutions leaving the main features such as localisation in space and time intact. When the equation with higher-order terms is integrable, the approximate wave forms naturally coincide with the exact rogue wave solutions. This is demonstrated in several well-known cases such as the Hirota equation, and the Kundu- Echkaus equation. However, the main advantage of using the Lagrangian approach is achieved in cases of non-integrable equations when finding exact solutions is impossible. In particular, we used this technique to evaluate the effects of fourth and sixth order dispersion on rogue waves. We also considered the influence of Raman delay as an example of treating the nonlinear terms. In order to confirm the applicability of the technique, the results are compared with numerical simulations of the original equation. Good agreements have been found confirming the usefulness of the approach. Show more