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.