Symmetries of the Eikonal equation

doi:10.1016/j.cnsns.2018.01.007

Communications in Nonlinear Science and Numerical Simulation

Volume 60, July 2018, Pages 137-144

https://doi.org/10.1016/j.cnsns.2018.01.007 Get rights and content

Highlights

•
The symmetry Lie algebra of the Eikonal Equation is being found.
•
The Optimal system of the Lie algebra is being classified.
•
Invariant solutions have been found.

Abstract

The infinitesimal algebra of Lie symmetries of the Eikonal equation is shown to be isomorphic to $o (n + 1, 2)$ when there are n independent variables. An explicit basis is found that is aligned with the standard basis coming from the standard matrix representation of $o (n + 1, 2)$ thereby making it possible to read off inequivalent one-dimensional symmetry vector fields. The symmetries are used to construct various solutions of the Eikonal equation.

Introduction

In this paper we consider the problem of classifying the Lie symmetries of the Eikonal equation with an arbitrary number of independent variables. The Eikonal equation arises in the study of geometric optics and electro-magnetism and is discussed in [5], for example. It provides a very nice example of a nonlinear equation that still has a lot of underlying symmetry. Nonetheless it is too complicated to be able to find a closed form general solution in terms of an arbitrary function; indeed such a property is the prerogative of quasilinear equations, when it comes to first order PDE’s. Since it is hard to find solutions, the approach of finding Lie symmetries to map solutions to solutions becomes particularly attractive. It is also a matter of practical importance to find satisfactory numerical solutions [7] and [6].

Here is an outline of the paper. In Section 2 we derive the PDE system that determines an infinitesimal symmetry of the Eikonal equation for an arbitrary number n of independent variables. In Section 3 we show that the determining functions of the symmetry are quadratic and give the general solution of the PDE. We find that the resulting Lie symmetry algebra is isomorphic to $o (n + 1, 2)$ . In Section 4 we discuss briefly the pseudo-orthogonal Lie algebra o(p, q) in general and exhibit, for the case $n = 2,$ a basis of symmetries for the Eikonal equation that accords with a standard basis for o(3, 2). In view of the precise correspondence between the vector field and matrix representations of o(3, 2), in Section 5 we consider the adjoint representation of o(3, 2) and in Section 6 we are able to determine inequivalent one-dimensional subalgebras of vector fields that correspond to so called optimal systems, for which the reader is referred to, for example, [1] and [2]. In Section 7 we finally turn to the issue of finding solutions of the Eikonal equation by applying Lie symmetries.

The Einstein summation is applied for the most part in the paper but at certain points it is convenient to suspend its usage.

Section snippets

Derivation of the PDE determining the symmetry algebra

We shall write the Eikonal equation as $\sum_{i = 1 . n} p_{i}^{2} = 1$ where $p_{i} = \frac{\partial z}{\partial x^{i}}$ . We shall also find it convenient to use $p_{i} p_{i} = 1$ where the summation convention is in force without necessarily having to have the sum run over a subscript and superscript.

Define $D_{i} = \frac{\partial}{\partial x^{i}}$ and $X = a^{i} (x^{j}, z) \frac{\partial}{\partial x^{i}} + c (x^{j}, z) \frac{\partial}{\partial z}$ . The first prolongation of X is given by $\tilde{X} = X + (D_{i} c - p_{j} (D_{i} a^{j})) \frac{\partial}{\partial p_{i}} .$

Now we apply $\tilde{X}$ to 2.1 so as to make X a symmetry and we find that $(\frac{\partial c}{\partial x_{i}} + p_{i} \frac{\partial c}{\partial z} - (\frac{\partial a^{j}}{\partial x^{i}} + p_{i} \frac{\partial a^{j}}{\partial z}) p_{j}) p_{i} = 0 .$ Using 2.1 and rearranging once we obtain $\frac{\partial c}{\partial x_{i}} p_{i} + \partial$

Solving the pde

We are going to write the possible third order derivatives of aⁱ as follows: $\begin{matrix} \frac{\partial^{3} a^{i}}{\partial x^{i} \partial x^{j} \partial x^{k}}, \frac{\partial^{3} a^{i}}{\partial x^{i} \partial x^{j} \partial z}, \frac{\partial^{3} a^{i}}{\partial x^{i} \partial z \partial z}, \frac{\partial^{3} a^{i}}{\partial z \partial z \partial z}, \frac{\partial^{3} a^{i}}{\partial x^{j} \partial x^{k} \partial x^{m}}, \frac{\partial^{3} a^{i}}{\partial x^{j} \partial x^{k} \partial z}, \frac{\partial^{3} a^{i}}{\partial x^{j} \partial z \partial z} \end{matrix}$ where i ≠ j but there are no other restrictions on the values of i, j, k, m.

From 2.9 we have, where i ≠ j, $\frac{\partial c}{\partial x^{i} \partial x^{j}} - \frac{\partial a^{i}}{\partial z \partial x^{j}} = 0$ and from 2.11 $\frac{\partial a^{i}}{\partial x^{j} \partial z} + \frac{\partial a^{j}}{\partial x^{i} \partial z} = 0 (i \neq j) .$ It follows that $\frac{\partial a^{i}}{\partial x^{j} \partial z} = 0 (i \neq j)$ and $\frac{\partial c}{\partial x^{i} \partial x^{j}} = 0 (i \neq j) .$ It follows that the second, sixth and seventh types of derivatives in 3.1 are zero.

From 2.11 we have $\frac{\partial a^{i}}{\partial x^{i} \partial x^{j} \partial x^{k}} = - \frac{\partial a^{j}}{\partial}$

The Lie algebra o(p, q)

If we put $g = [\begin{matrix} I_{p} & 0 \\ 0 & - I_{q} \end{matrix}]$ then A ∈ O(p, q) if and only if $A^{t} g A = g$ and a matrix R is in o(p, q) if and only if $g R + {(g R)}^{t} = 0 .$ It follows that the Lie algebra o(p, q) is the set of matrices of the form $[\begin{matrix} a & b \\ b^{t} & c \end{matrix}]$ where a is p × p, $a^{t} = - a,$ the matrix c is q × q and $c^{t} = - c$ and b is p × q arbitrary. We are going to consider a canonical basis for o(p, q) as follows: F_ab is the $(p + q) \times (p + q)$ matrix whose only non-zero entries are 1 in the (a, b)-th and $- 1$ in the (b, a)-th positions, respectively, and 1 ≤ a < b ≤ p; H_ij

The adjoint representation of $o (n + 1, 2)$

The Lie group O(p, q) acts on its Lie algebra o(p, q) by conjugation giving the adjoint representation of O(p, q). Recall that the Lie algebra o(p, q) is defined in 4.4. From 4.3 it follows from that $R = G S$ where $G = g^{- 1}$ is non-singular symmetric and S is skew-symmetric. Suppose that λ is an eigenvalue of R so that $\det (G S - λ I) = 0 .$ Since G is non-singular 5.2 is equivalent to $\det (S - λ g) = 0 .$ Taking the transpose in 5.3, since $S^{t} = - S$ and $g^{t} = g,$ we find that $\det (S + λ g) = 0 .$ Thus if λ is an eigenvalue of R so too is

Optimal systems

The classification of the subalgebras of o(p, q) coming from the adjoint representation of O(p, q) is altogether more involved than is the case for $p = n, q = 0$ . Indeed, for the orthogonal group proper, elements of o(n) are semi-simple matrices and the eigenvalues are either zero, or pure imaginary occurring in conjugate pairs. The normal form for matrices under the action of the classical groups is treated systematically in [4]. In the case of o(3, 2) any matrix is singular and the normal forms are

Solutions of the Eikonal equation

It is too much to be able to integrate the vector fields appearing at the end of the last section in complete generality so in this Section we shall use symmetries to find some solutions of the Eikonal equation.

Acknowledgment

The authors thank the Qatar Foundation and Virginia Commonwealth University in Qatar for funding this project.

References (7)

S. Ali et al.
Embedding algorithms and applications to differential equations
J Symb Comput
(2017)
H. Azad et al.
Symmetry analysis of wave equation on static spherically symmetric spacetimes with higher symmetries
Jour Math Phys
(2013)
M.d. Carmo
Differential geometry of curves and surfaces
(1976)

There are more references available in the full text version of this article.

Cited by (4)

Nonlinear self-adjointness, conserved quantities and Lie symmetry of dust size distribution on a shock wave in quantum dusty plasma
2022, Communications in Nonlinear Science and Numerical Simulation
The current exploration explains new traveling wave profiles by the Lie symmetry approach Here, we derive the ( $3 + 1$ )-dimensional Zakharov–Kuznetsov burgers equation for the shock wave phenomena in plasma with dust-charged particles having quantum effects. The structures of this model are made by Abelian algebra. The considered equation is reduced to an ordinary differential equation by the use of this Abelian algebra. The complete derivation of the classical symmetries of the considered model and reduction corresponding to the first four symmetries has been added. The new auxiliary equation method is implemented for the formation of the new traveling wave structures. The graphical explanation of some obtained results are elaborated in detail. The nonlinear self-adjointness of the considered model is also part of this work. By this concept, we have evaluated the conservation laws of the model. This work is new and does not exist in literature.
Eikonal Equation 0.1
2022, arXiv
Lie symmetries of the canonical connection: codimension one Abelian nilradical case
2021, Journal of Nonlinear Mathematical Physics
Lie symmetries of the canonical connection: Codimension one abelian nilradical case
2021, arXiv

View full text

Research paperSymmetries of the Eikonal equation

Highlights

Abstract

Introduction

Section snippets

Derivation of the PDE determining the symmetry algebra

Solving the pde

The Lie algebra o(p, q)

The adjoint representation of o(n+1,2)

Optimal systems

Solutions of the Eikonal equation

Acknowledgment

Embedding algorithms and applications to differential equations

J Symb Comput

Symmetry analysis of wave equation on static spherically symmetric spacetimes with higher symmetries

Jour Math Phys

Differential geometry of curves and surfaces

Research paper
Symmetries of the Eikonal equation

The adjoint representation of $o (n + 1, 2)$