# Lie Symmetries of Differential Equations: Classical Results and Recent Contributions

## Abstract

**:**

**MSC**22E70; 34A05; 35A30; 58J70; 58J72

## 1. Introduction

The older examinations on ordinary differential equations as found in standard books are not systematic. The writers developed special integration theories for homogeneous differential equations, for linear differential equations, and other special integrable forms of differential equations.However, the mathematicians did not realize that these special theories are all contained in the term infinitesimal transformations, which is closely connected with the term of a one parametric group.

^{®}and Mathematica

^{®}(commercial), Maxima and Reduce (open source), and the needed algebraic manipulations can be rapidly done automatically [15,20,21,22,23,24]. In fact, many specific packages for performing symmetry analysis of differential equations are currently available in the literature.

## 2. Basic Theory of Lie Groups of Transformations

**Definition 1**

**(Groups**

**of**

**transformations)**

- 1.
- For each value of the parameter $a\in S$ the transformations are one–to–one onto D;
- 2.
- S with the law of composition μ is a group with identity e;
- 3.
- $\mathbf{Z}(\mathbf{z};e)=\mathbf{z}$, $\forall \phantom{\rule{0.166667em}{0ex}}\mathbf{z}\in D$;
- 4.
- $\mathbf{Z}(\mathbf{Z}(\mathbf{z};a);b)=\mathbf{Z}(\mathbf{z};\mu (a,b))$, $\forall \phantom{\rule{0.166667em}{0ex}}\mathbf{z}\in D$, $\forall \phantom{\rule{0.166667em}{0ex}}a,b\in S$.

**Definition 2**

**(Lie**

**group**

**of**

**transformations)**

- 1.
- a is a continuous parameter, i.e. , S is an interval in $\mathbb{R}$;
- 2.
- $\mathbf{Z}$ is ${C}^{\infty}$ with respect to $\mathbf{z}$ in D and an analytic function of a in S.
- 3.
- $\mu (a,b)$ is an analytic function of a and b, $\forall \phantom{\rule{0.166667em}{0ex}}a,b\in S$.

**Theorem 1**

**(First**

**Fundamental**

**Theorem**

**of**

**Lie)**

**Theorem**

**2**

**Theorem**

**3**

## 3. Lie Groups of Differential Equations

- a transformation of the group maps any solution of $\mathcal{E}$ into another solution of $\mathcal{E}$;
- a transformation of the group leaves $\mathcal{E}$ invariant, say, $\mathcal{E}$ reads the same in terms of the variables $(\mathbf{x},\mathbf{u})$ and in terms of the transformed variables $({\mathbf{x}}^{\star},{\mathbf{u}}^{\star})$.

- the first order prolongation$${\Xi}^{(1)}=\Xi +\sum _{A=1}^{m}\sum _{i=1}^{n}{\eta}_{\left[i\right]}^{A}(\mathbf{x},\mathbf{u},{\mathbf{u}}^{(1)})\frac{\partial}{\partial {u}_{i}^{A}},\phantom{\rule{2.em}{0ex}}{u}_{i}^{A}=\frac{\partial {u}^{A}}{\partial {x}_{i}}$$$${\eta}_{\left[i\right]}^{A}(\mathbf{x},\mathbf{u},{\mathbf{u}}^{(1)})=\frac{D{\eta}^{A}}{D{x}_{i}}-\frac{D{\xi}_{j}}{D{x}_{i}}\frac{\partial {u}^{A}}{\partial {x}_{j}}$$
- the general kth–order prolongation recursively defined by$${\Xi}^{(k)}={\Xi}^{(k-1)}+\sum _{A=1}^{m}\sum _{{i}_{1}=1}^{n}\dots \sum _{{i}_{k}=1}^{n}{\eta}_{\left[{i}_{1}\dots {i}_{k}\right]}^{A}\frac{\partial}{\partial {u}_{{i}_{1}\dots {i}_{k}}^{A}},\phantom{\rule{2.em}{0ex}}{u}_{{i}_{1}\dots {i}_{k}}^{A}=\frac{{\partial}^{k}{u}^{A}}{\partial {x}_{{i}_{1}}\dots \partial {x}_{{i}_{k}}}$$$${\eta}_{\left[{i}_{1}\dots {i}_{k}\right]}^{A}=\frac{D{\eta}_{\left[{i}_{1}\dots {i}_{k-1}\right]}^{A}}{D{x}_{{i}_{k}}}-{u}_{{i}_{1}\dots {i}_{k-1}j}^{A}\frac{D{\xi}_{j}}{D{x}_{{i}_{k}}}$$

**Theorem 4**

**(Infinitesimal**

**Criterion**

**for**

**differential**

**equations)**

## 4. Use of Lie Symmetries of Differential Equations

- to lower the order or eventually reduce the equation to quadrature, in the case of ordinary differential equations;
- to determine particular solutions, called invariant solutions, or generate new solutions, once a special solution is known, in the case of ordinary or partial differential equations.

#### 4.1. Ordinary Differential Equations

#### 4.2. Invariant Solutions of Partial Differential Equations

_{1}, called invariant surface conditions, have the form

_{2}, a reduced system of differential equations involving $(n-1)$ independent variables (called similarity variables) is obtained. The name similarity variables is due to the fact that the scaling invariance, i.e., the invariance under similarity transformations, was one of the first examples where this procedure has been used systematically.

**Example**

**1**

- take a generator of a subgroup (say, $\Xi $, written in terms of the variables involved in the system $\mathbf{\Delta}=\mathbf{0}$) and build the associated similarity reduction;
- write the original system of differential equations $\mathbf{\Delta}=\mathbf{0}$ in terms of the similarity variables and similarity functions, thus obtaining the reduced system $\widehat{\mathbf{\Delta}}=\mathbf{0}$;
- if a further reduction is wanted, set $\mathbf{\Delta}=\widehat{\mathbf{\Delta}}$, and go to step 1.

**Example**

**2**

**Example**

**3**

#### 4.3. H–Invariant Solutions and Factor Systems

**Definition**

**3**

**Definition**

**4**

**Theorem**

**5**

**Theorem**

**6**

- any nonsingular H–invariant solution $\mathbf{u}=\mathbf{\Theta}(\mathbf{x})$ can be defined from the functional equations$$\mathbf{w}(\mathbf{x},\mathbf{u}(\mathbf{x}))=\mathbf{W}(\mathbf{t}(\mathbf{x},\mathbf{\Theta}(\mathbf{x})))$$
- if a solution $\mathbf{W}(\mathbf{t})$ of the system $\mathcal{E}/H$ produces a manifold in the space ${\mathbb{R}}^{n}(\mathbf{x})\times {\mathbb{R}}^{m}(\mathbf{u})$$$\mathbf{w}(\mathbf{x},\mathbf{u})=\mathbf{W}(\mathbf{t}(\mathbf{x},\mathbf{u}))$$

- Determine $m+n-{r}_{\star}=m+\sigma $ independent invariants ${I}_{k}={I}_{k}(\mathbf{x},\mathbf{u})$ ($k=1,\dots ,m+\sigma $) of the group H. The rank of the Jacobian matrix$$\frac{\partial ({I}_{1},\dots ,{I}_{m+n-{r}_{\star}})}{\partial ({u}^{1},\dots ,{u}^{m})}$$$$\mathrm{rank}\left(\frac{\partial ({I}_{1},\dots ,{I}_{m})}{\partial ({u}^{1},\dots ,{u}^{m})}\right)=m$$
- Assume the first m invariants ${I}_{k}$ ($k=1,\dots ,m$) to depend on the remaining invariants ${I}_{k}$ ($k=m+1,\dots ,m+\sigma $) , i.e.,$${I}_{k}(\mathbf{x},\mathbf{u})={W}^{k}({I}_{m+1},\dots ,{I}_{m+\sigma}),\phantom{\rule{1.em}{0ex}}(k=1,\dots ,m)$$
- Substitute the representation of the functions ${u}^{A}$ into the initial system of partial differential equations to obtain the system of equations for the unknown functions ${W}^{k}$ ( $k=1,\dots ,m$), i.e., the factor system $\mathcal{E}/H$.

**Example**

**4**

**Definition**

**5**

**Definition**

**6**

**([7,10,18,35])**Two Lie subalgebras ${L}^{\prime}$ and ${L}^{\u2033}$ of a Lie algebra L are similar if there exists an inner automorphism $\varphi \in \mathrm{Int}(L)$ such that $\varphi ({L}^{\prime})={L}^{\u2033}$. Since the similarity between Lie subalgebras is a relation of equivalence, all subalgebras of the given Lie algebra L are decomposed into classes of similar algebras. A set of the representatives of each class is called an optimal system of subalgebras.

^{®}package, SymboLie, has been developed in order to determine optimal systems of Lie subalgebras.

#### 4.4. New Solutions from A Known Solution

_{1},

_{2}thus obtaining

**Theorem**

**7**

**Example**

**5**

#### 4.5. Conservation Laws and Noether’s Theorem

**Definition**

**7**

**Lemma**

**1**

**Theorem 8**

**(Noether’s**

**theorem)**

- 1.
- the identity$${\tilde{\eta}}^{A}\left[\mathbf{u}\right]{E}_{{u}^{A}}(\mathcal{L}[\mathbf{x},\mathbf{u}])=-{D}_{i}\left({\xi}_{i}(\mathbf{x}\mathbf{u})\mathcal{L}[\mathbf{x},\mathbf{u}]+{W}^{i}[\mathbf{x},\mathbf{u},\tilde{\eta}\left[\mathbf{u}\right]]\right)$$
- 2.
- the local conservation law$${D}_{i}\left({\xi}_{i}(\mathbf{x},\mathbf{u})\mathcal{L}\left[\mathbf{u}\right]+{W}^{i}[\mathbf{x},\mathbf{u},\tilde{\eta}\left[\mathbf{u}\right]]\right)=0$$

**Example**

**6**

**(Klein–Gordon equation (see [19]))**The class of Klein–Gordon wave equations,

- to derive a set of determining equations to find all sets of multipliers of the system (84) yielding its nontrivial conservation laws;
- to establish the conditions for which all nontrivial conservation laws arise from sets of local multipliers and viceversa;
- to construct the fluxes of a conservation law arising from a given set of multipliers.

**Theorem**

**9**

**(See [19])**A set of non–singular multipliers $\left\{{\mathrm{\Lambda}}_{\nu}[\mathbf{x},\mathbf{U}]:\nu =1,\dots ,q\right\}$ yields a local conservation law for the system $\left\{{\mathrm{\Delta}}_{\nu}\left[\mathbf{U}\right]=0:\nu =1,\dots ,q\right\}$ if and only if the set of identities

## 5. Transformation to Autonomous Form

**Theorem**

**10**

#### 5.1. Reduction to Autonomous Form: Applications

**Example 7**

**(Cylindrical**

**KdV**

**Equation)**

**Example 8**

**(Axi–symmetric**

**gas**

**dynamics)**

## 6. Transformation to Linear Form

- both the source and target system of partial differential equations are first order systems;
- the mapping is a one–to–one transformation;
- the target system of partial differential equations is linear.

**Theorem 11**

**([61])**

#### 6.1. Transformation to Linear Form: Applications

**Example 9**

**(Hodograph**

**transformation)**

**Example 10**

**(Nonlinear**

**diffusion**

**equation)**

**Example 11**

**(Monge–Ampère**

**equation**

**[64])**

## 7. Reduction of First Order Quasilinear Systems to Homogeneous Form

**Theorem 12**

**([71])**

**Theorem 13**

**([72])**

- determine the Lie algebra L of point symmetries of system (144);
- if $\mathrm{dim}(L)\ge n+2$:
- determine the $(n+2)$–dimensional Lie subalgebras (an optimal system suffices), for instance by means of the Mathematica
^{®}package SymboLie [40]; - check if among the Lie subalgebras there is a Lie algebra having the required structure;
- find the canonical variables of the symmetries, and reduce the system to homogeneous and autonomous form.

#### 7.1. Reduction to Homogeneous Form: Applications

**Example 12**

**(Nonlinear**

**Hyperbolic**

**Heat**

**Equation)**

**Example 13**

**(Rate–Type**

**Materials)**

## 8. Nonlocal Symmetries

**Example**

**14**

**Example 15**

**([32])**

## 9. Conclusions

- the equivalence transformations [7,31,92] that are continuous transformations mapping a specific class of differential equations involving arbitrary functions or parameters into the same class, and therefore are more general than symmetries (i.e., an equation belonging to a class is mapped to another equation of the same class); equivalence transformations provide useful when group classification of differential equations containing arbitrary functions is needed [93] and also to linearize differential equations;
- the approximate Lie symmetries [29,30], useful to apply suitably Lie’s machinery to differential equations containing small terms; approximate Lie symmetries allow to build approximate invariant solutions to partial differential equations, or approximate reduction of order of ordinary differential equations, or approximate conservation laws.

## Acknowledgments

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