Lie symmetries and exact solutions of Benney-Roskes/Zakharov-Rubenchik system

Abstract

Many physical phenomena in our lives are modeled using ordinary differential equations (ODE) and partial differential equations (PDE). Unlike PDEs, ODEs can be solved using more familiar and straightforward techniques. Partial differential equations are widely utilized in scientific fields that place on mathematics, such as physics and engineering. A wide range of partial differential equation types has been derived as a result of the diversity of the sources. Many methods have been developed to deal with the resulting individual equations. One of these methods used to solve PDEs is Lie symmetry analysis. Lie groups and Lie algebras are useful tools for reducing the number of independent variables in a PDE by using the reduction method. Lie's method leads to group-invariant solutions and conservation laws for partial differential equations. PDEs can be classified into equivalence classes and new solutions can be derived from existing ones by taking advantage of their symmetry. The first step in the method is finding the determining equations for the system's symmetries. By solving the determining equations, the vector field that generates the transformation group of the equation is obtained, which is called the infinitesimal generator of the symmetry group. In other words, we find the infinitesimal generators of the transformation groups, which will leave the solution of the system invariant. From this generator, the Lie algebra structure of the system emerges. However, applying Lie group methods to systems of equations takes a lot of time and effort. Even solving elementary differential equations is prone to mistakes if we do it with a pen and paper. All of that has changed thanks to the accessibility of computer algebra systems like Mathematica and Maple. Some of the calculations in this thesis were done using these programs. This thesis investigates the Lie symmetry algebra of the Benney--Roskes/ Zakharov--Rubenchik (BR/ZR) system and presents exact solutions to this system of equations. BR/ZR system includes the well-known Davey-Stewartson (DS) system and Zakharov system in the limiting case. Although the first appearance of the BR system dates back a few decades, it is seen that the research on qualitative and numerical analysis of the system finds a place in the recent literature. As this literature lacks the results on Lie symmetries and solitary-type analytic solutions of the system, it has been this work's main purpose to fulfill this gap. In Chapter 1, the problem statement of the thesis and the literature review of the problem are given. In Chapter 2, the fundamental definitions and notations for the Lie symmetry analysis of differential equations are provided. In Chapter 3, (2+1)-dimensional BR/ZR system and in Chapter 4, (3+1) BR/ZR system are investigated by the tools of Lie group analysis. The symmetry algebra of the (2+1)-dimensional BR/ZR system is obtained as an infinite dimensional Lie algebra. We found that the symmetry algebra is as not rich as the symmetry algebra of the DS system, which is one of the integrable equations in (2+1) dimensions. We succeeded in finding solutions in the forms of a line soliton and hyperbolic type. We also discovered the Lie symmetry algebra of the (3+1) BR/ZR system. The invariance algebra of the system turns out to be infinite-dimensional. Concentrating on traveling solutions, we found wave components of sech-tanh type, which proceed as line solitons and kinks in two-dimensional cross-sections in space. With this thesis, we have added original results to the literature on group-theoretical properties and exact solutions of the BR/ZR system. We believe that these results will serve as a source for future numerical and qualitative studies on this system.