Symmetries, Differential Equations, and Applications: Galois Bicentenary
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Wave Equations in Bianchi SpaceTimes
Abstract
We investigate the wave equation in Bianchi type III spacetime. We construct a Lagrangian of the model, calculate and classify the Noether symmetry generators, and construct corresponding conserved forms. A reduction of the underlying equations is performed to obtain invariant solutions.
1. Introduction
The study of partial differential equations (PDEs) in terms of Lie point symmetries is well known and well established [1–5], where these symmetries can be used to obtain, inter alia, exact analytic solutions of the PDEs. In addition, Noether symmetries are also widely investigated and are associated with PDEs that possess a Lagrangian. Noether [6] discovered the interesting link between symmetries and conservation laws showing that for every infinitesimal transformation admitted by the action integral of a system there exists a conservation law. Investigations have been devoted to understand Noether symmetries of Lagrangians that arise from certain pseudoRiemannian metrics of interest [7, 8]. Recently, a study was aimed at understanding the effect of gravity on the solutions of the wave equation by solving the wave equation in various spacetime geometries [9].
In [10], the Bianchi universes were investigated using Noether symmetries. The authors of [11] studied the Noether symmetries of Bianchi type I and III spacetimes in scalar coupled theories. Therein, they obtained the exact solutions for potential functions, scalar field, and the scale factors, see also [12].
We pursue an investigation of the symmetries of the wave equation in Bianchi III spacetime. We construct solutions of these equations and find conservation laws associated with Noether symmetries. The plan of the paper is as follows.
In Section 2, we discuss the procedure to obtain an expression representing Noether symmetries and conservation laws. In Section 3, we derive and classify strict Noether symmetries of the Bianchi III spacetime. Also in Section 3, we briefly describe the relation of Noether symmetries to conservation laws. We then illustrate the reduction of the wave equation and obtain invariant solutions.
2. Definitions and Notation
We briefly outline the notation and pertinent results used in this work. In this regard, the reader is referred to [13].
The convention that repeated indices imply summation is used. Let be independent variable with , and let be the dependent variable with coordinates . Furthermore, let be the projection map . Also, suppose that is a smooth map such that , where is the identity map on . The jet bundle is given by the equivalence classes of sections of . The coordinates on are denoted by , where and corresponds to the partial derivatives of with respect to . The partial derivatives of with respect to are connected by the operator of total differentiation as The collection of all firstorder derivatives will be denoted by . Similarly, the collections of all higher order derivatives will be denoted by .
The jet bundle on U will be written as . We now review the space of differential forms on . To this end, let be the vector space of differential forms on with differential . A smooth differential form on is given by where each component . Note that for differential functions , where is the total differential or the total exterior derivative. Moreover, the total exterior derivative of is and by invoking (2.4) one has The total differential has properties analogous to the algebraic properties of the usual exterior derivative : for a form and an form and . Also, it is known that if , then is a locally exact form, that is, for some form , [14].
2.1. Action of Symmetries
Consider an thorder system of partial differential equations of independent variables and dependent variables:
Definition 2.1. A conserved form of (2.8) is a differential ()form, defined on if is satisfied on the surface given by (2.8).
Remark 2.2. When Definition 2.1 is satisfied, (2.10) is called a conservation law for (2.8).
It is clear that (2.10) evaluated on the surface (2.8) implies that
on the surface given by (2.8), which is also referred to as a conservation law of (2.8). The tuple , , , is called a conserved vector of (2.8).
We now review some definitions and results relating to EulerLagrange, Lie, and Noether operators ([15, 16] and references therein).
Let for some . Then is the universal space of differential functions of finite orders.
Consider a symmetry operator given by the infinite formal sum: where , , and the additional coefficients are determined uniquely by the prolongation formulae In (2.13), is the Lie characteristic function given by In particular, a symmetry operator of the form , where , is called a canonical or evolutionary representation of , and is called its characteristic.
An operator is said to be a Noether symmetry corresponding to a Lagrangian , if there exists a vector , such that If , then is referred to as a strict Noether symmetry corresponding to a Lagrangian . This case is also obtained by setting the Lie derivative on the form in the direction of to zero, that is, where is the Lie derivative operator.
In view of the above discussions and definitions, the Noether theorem [6] is formulated as follows.
Noether’s Theorem
For any Noether symmetry corresponding to a given Lagrangian , there corresponds a vector , defined by
which is a conserved current of the EulerLagrange equations , , where is the EulerLagrange operator given by
and the Noether operator associated with the operator is given by
in which the EulerLagrange operators with respect to derivatives of are obtained from (2.18) by replacing by the corresponding derivatives, for example,
3. Bianchi III SpaceTime
Consider the Bianchi III metric: The wave equation in (3.1) takes the form [17]
In [17], some aspects of the wave equation on the Bianchi metric were studied. The multiplier method [1] was adopted to determine some of the conserved densities. This lengthy procedure ultimately leads to the construction of only three symmetries and its associated conserved vectors.
In this paper, we investigate the wave equation on the Bianchi III metric using Noether’s theorem and the method of differential forms. We obtain a wide range of results and also perform symmetry reductions of the wave equation for some cases to obtain invariant solutions. For the purposes of Sections 3.1 and 3.2, we denote the Lagrangian by .
3.1. The Strict Noether Symmetries of (3.2)
We classify the cases that yield strict Noether symmetries (gauge is zero) of (3.2), via the Lagrangian Many of the calculations have been left out as they are tedious—the details are available to the reader in a number of texts that have been cited here.
The principle Noether algebra is Furthermore, specific cases of and give rise to the symmetries to from above, and some additional symmetries.
Case 1 (, ). The additional symmetries are,
Case 2 (, ). The additional symmetries are,
Case 3 (, ). The additional symmetries are,
Table 1 contains the conserved forms corresponding to each Noether symmetry , that is, it lists the three form . The four form is , and vanishes on the solutions of (3.2). Thus so that in (3.2).
