Conserved vectors, symmetry reductions and solutions for some nonlinear partial differential equations

Abstract

In this thesis, Lie group analysis is utilized to examine a number of higher-order nonlinear partial differential equations (NPDEs). Additionally, conserved vectors for these NPDEs are derived. The equations addressed come from various scientific fields and have a wide range of real world applications. These models are taken from a variety of fields, including fluid mechanics, engineering sciences, economics, and social sciences. The equations considered are the Yu-Toda-Sasa-Fukuyama equation of plasma physics; a 2D generalized shallow water wave equation; a 2D breaking solition equation of fluid mechanics; a 2D KdV equation; a 3D KdV equation; a generalised 2D equal-with of engineering and a derivative nonlinear Schrödinger equation. Using the symmetry analysis approach, we explored the 2D Yu-Toda-Sasa-Fukuyama equation. To begin with, all Lie point symmetries were computed and used to execute symmetry reductions on this model. As a result, several nonlinear ordinary differential equations (NODE) were derived. Furthermore, we used a variety of approaches to solving these ordinary differential equations, including direct integration and the power series expansion method. Finally, the conserved vectors that represent energy, mass, and momentum, among other things, were calculated using the multiplier approach and the classical Noether's theorem in the second part. We consider the generalized 2D shallow water wave equation. We calculated its Lie symmetries, performed symmetry reductions, and reduced it to a NODE. The NODE is then solved using direct integration, with the result represented in terms of the incomplete elliptic integral. Furthermore, Kudrayshov's approach was used to find the NODE solution. Finally, the conservation laws were derived using the multiplier approach and Noether's theorem. The multiplier approach produced eight multipliers, resulting in eight local conservation laws for the equation, whereas

Noether's method produced six local conservation laws. The NPDE proposed by the Italian scientists Calogero and Degasperis [60, 61] is known as a 2D breaking soliton equation. The 2D interchange of a Riemann wave propagating along the y-axis with a long wave propagating along the x-axis is described by this mathematical model. This system will be studied using the Lie symmetry approach, and conserved vectors will be found. Furthermore, a power series solution method is used to find explicit solutions. To get conserved quantities, we'll use Noether's method. We investigate a constant coefficients 2D KdV equation which was recently introduced in the literature. This model is integrable employing the Painlevé test. It is vital for us to study such equations since they are used to describe real world problems since they are more realistic models of natural and man-made phenomena. Moreover, we find travelling waves group-invariant solutions. We then use multipliers to derive conserved vectors of the equation. We analyze a generalized 2D equal-width equation which arises in various fields of sciences. With the aid of different methods which include Lie symmetry analysis, power series expansion and the Weierstrass method we produce closed-form solutions of this model. Moreover, we give graphical representation of the obtained solutions using certain parametric values. Furthermore, conserved vectors that represent among other laws of motion charge, momentum, etc., are constructed with the aid of the multiplier method and Noether's theorem. The derivative nonlinear Schrödinger equation (α = ±1) has many physical applications in different fields of science, most importantly in plasma physics and in nonlinear optics. It is known to be the governing equation of the evolution of small-amplitude nonlinear Alfvén waves propagating quasiparallel with respect to the background magnetic field. We study this equation by finding its first integrals and thereafter use them to compute its most general traveling wave solution. Moreover, we find the conservation laws for the system that is equivalent to this equation using the multiplier method.