U.U.D.M. Project Report 2020:43

Examensarbete i matematik, 15 hpHandledare: Malte Litsgård Ämnesgranskare: Kaj NyströmExaminator: Jörgen ÖstenssonSeptember 2020

Department of MathematicsUppsala University

Lie Groups and PDE

Carl Öhrnell

Abstract

We introduce some central concepts of Lie groups and algebras to a reader with-out a background in topology and differentiable manifolds. Presenting the jetspace and prolongation of vector fields, we then find Lie groups of transforma-tions that leave differential equations invariant, called symmetries. These areused to find solutions using coordinate changes, transformations of known solu-tions and invariant solutions. A non-linear ODE and the heat equation (PDE)are used to exemplify the methods.

Contents

1 Introduction 2

2 Lie Theory 4

2.1 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 The Lie Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Differential Equations and the Application of Lie Groups 14

3.1 Geometrical Setting . . . . . . . . . . . . . . . . . . . . . . . . . 143.2 Symmetry Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 153.3 Solving ODE and PDE Using Symmetries . . . . . . . . . . . . . 18

3.3.1 Canonical Coordinates . . . . . . . . . . . . . . . . . . . . 183.3.2 Transformation of a Known Solution . . . . . . . . . . . . 193.3.3 Group-invariant Solutions . . . . . . . . . . . . . . . . . . 19

4 Examples 20

4.1 A Non-linear ODE . . . . . . . . . . . . . . . . . . . . . . . . . . 204.2 The Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 224.3 The Kolmogorov Equation . . . . . . . . . . . . . . . . . . . . . . 26

1

Chapter 1

Introduction

Differential equations are equations that contain one or more derivatives of anunknown function. They appear in just about every field of science, wherever astate or quantity can be related to its rate of change. The mathematical studyof these equations removed from direct applicability is also a thriving field inand of itself. The diversity of different equations naturally leads to classificationattempts to inform on what properties to expect and how to attempt to solvethem.

The first distinction is made by considering the number of variables theunknown function is dependent on: only one gives an ordinary differential equa-tion (ODE), while more than one results in a partial differential equation (PDE).Generally speaking, the latter are more difficult to solve. Complexity usuallyalso increases with the order of the equation, given by the highest order deriva-tive present. Another source of complexity is the presence of non-linearity inthe equation.

The natural progression in the curriculum follows this increase in difficulty,usually beginning with linear first order ODE, to successively progress to higherorder linear ODE, the occasional non-linear ODE, then onto linear PDE, etc.The methods taught are often tailored to the specific type of equation, and insome instances the underlying principles are somewhat obscure.

This paper is dedicated to present methods that have many merits: they areapplicable to a wide variety of differential equations and they have a geomet-ric foundation making them intuitive. They build on the theory of Lie groups,so the ambition is to introduce some concepts central to this theory while si-multaneously giving an intuition for its geometry and later its application todifferential equations.

In Chapter 2 we introduce the concept of a smooth manifold, Lie groupsas manifolds with group operations and their action on manifolds. To findlinear models we introduce vector fields and their flows which correspond toone-parameter subgroups. We conclude by defining the Lie algebra correspond-ing to a Lie group and its application in finding symmetry groups of equations.Throughout the chapter we exemplify using the circle group.

Chapter 3 describes how the introduced Lie group machinery can be used

2

to solve or simplify differential equations. We start by introducing the geomet-ric extensions needed to apply Lie group actions to differential equations, mostimportantly the jet space and the prolongation of group actions generating vec-tor fields. We then present specific methods: using canonical variables to solveODE, finding new solutions to PDE by transforming a known solution and fi-nally finding group-invariant solutions of a PDE.

In Chapter 4 we give two extensive examples to demonstrate the methodsin practice. The first is a non-linear ODE to which we find a symmetry, aninvariant to that symmetry and finally canonical coordinates which let us solvethe equation by quadrature. The second is the heat equation, a PDE in twodimensions. We solve the determining equation to find the Lie algebra of sym-metries, transform the constant solution using one of the found symmetries, andfinally find translation invariant travelling wave solutions. We then conclude bygiving a brief history of the Kolmogorov equation, a PDE still under theoreticalinvestigation, finding the translations and dilations that define the Lie groupstructure needed for further study of the equation.

3

Chapter 2

Lie Theory

Most of the definitions in this chapter are adapted from the first chapter ofOlver [11], which gives an introduction to Lie groups to then focus on applica-tions in later chapters, as well as his lectures [12]. For a more comprehensiveintroduction to differentiable manifolds, vector fields and Lie groups, the firstfour chapters of Boothby [2] are recommended.

2.1 Manifolds

The natural setting for our development is on m-dimensional smooth (C∞)manifolds, which can be thought of as smooth deformations of Rm with noself-intersections. Thus m = 1 gives a curve, m = 2 a surface, and m ≥ 3 ahyper-surface.

Definition 2.1. An m-dimensional manifold is a set M , together with a count-able collection of subsets Uα ⊂ M , called coordinate charts, and injective func-tions χα : Uα → Vα onto connected open subsets Vα of Rm called local coordinatemaps, which satisfy the following properties:

1. The coordinate charts cover M :

∪αUα =M.

2. On the overlap of any pair of coordinate charts Uα ∩ Uβ the compositemap

χβ χ−1α : χα(Uα ∩ Uβ) → χβ(Uα ∩ Uβ)

is a smooth function.

3. If x ∈ Uα, x ∈ Uβ are distinct points of M , then there exist open subsets

W of χα(x) in Vα and W of χβ(x) in Vβ such that

χ−1α (W ) ∩ χ−1

β (W ) = ∅.

Example 2.2. The most obvious example of an m-dimensional manifold is Rm

itself, or any open subset of Rm, using the identity map as coordinate map.

4

Example 2.3. An example of a one-dimensional manifold that we are familiarwith is the unit circle

S1 = (x, y) ∈ R2 : x2 + y2 = 1.We cover it with the charts

U1 = S1\(0, 1), U2 = S1\(0,−1),i.e. the unit circle with deleted north and south pole, respectively. We then letthe coordinate maps

χ1 : U1 → R, χ2 : U2 → R,

be the stereographic projections from the north respectively south pole,

χ1(x, y) =( x

1− y

)

, χ2(x, y) =( x

1 + y

)

.

We then check that the composite map on the overlap,

χ1 χ−12 : R\(0) → R\(0),

is a smooth map. Solving for t in χ2(x, y) = t using that x2 + y2 = 1 on theunit circle we get

χ−12 (t) =

( 2t

t2 + 1,1− t2

t2 + 1

)

,

which composed with χ1 gives us

χ1 χ−12 (t) =

1

t,

which is clearly a smooth map away from the origin. The third criterion con-cerning the separation of points, called the Hausdorff separation property, isinherited from R2.

Note that this collection of charts and maps is not unique. Another commonatlas for S1 uses four charts where the maps are projections onto the x and yaxes.

In the following, we will often use the Cartesian product of two mani-folds to create new manifolds. If M is an m-dimensional manifold and N isan n-dimensional manifold, then M × N is an (m + n)-dimensional manifold.E.g., two open subsets of Rn and Rm, respectively, gives an open subset ofRn × Rm = Rm+n, and the Cartesian product of two circles, S1 × S1, is thetwo-dimensional torus T 2, which has the shape of a doughnut.

We conclude with two useful definitions regarding maps between manifolds.

Definition 2.4. The rank of a map F : M → N at a point x ∈ M is definedto be the rank of the n×m Jacobian matrix (∂F i/∂xj) of any local coordinateexpression for F at the point x. The map F is called regular if its rank isconstant.

Definition 2.5. A smooth n-dimensional immersed submanifold of a manifoldM is a subset N ⊂M parametrized by a smooth, one-to-one map F : N → N ⊂M whose domain N , the parameter space, is a smooth n-dimensional manifold,and such that F is everywhere regular, of maximal rank n.

Remark 2.6. In the following we will assume that M is a smooth manifold ofdimension m, all manifolds are connected, and maps are regular.

5

2.2 Lie Groups

We begin by recalling the properties and some examples of abstract groups.

Definition 2.7. A group is a set G with an operation G×G→ G, (x, y) 7→ xysuch that

1. The set G is closed under the operation: x, y ∈ G =⇒ xy ∈ G,

2. The operation is associative: (xy)z = x(yz) for all x, y, z ∈ G,

3. There exists an identity element e such that xe = ex = x for all x ∈ G,

4. All elements have inverses: ∀x ∈ G ∃x−1 ∈ G such that x−1x = xx−1 = e.

Example 2.8. Some examples of groups are

1. Z, Q, R, C or Rn with addition,

2. Q∗, R∗ or C∗ with multiplication,

3. The circle group, i.e. S1 = z ∈ C : |z| = 1 with multiplication.

Now, merging the algebraic properties of a group with the topological anddifferentiable properties of a smooth manifold, we get a Lie group.

Definition 2.9. A Lie group is a smooth manifold G with a group structuresuch that the operation µ : G×G→ G, (x, y) 7→ xy and the inversion ι : G→ G,x 7→ x−1 are smooth maps.

It turns out that many of the examples of algebraic groups listed above arealso examples of Lie groups, specifically all involving some set of R or C.

Example 2.10. An interesting example is the general linear group in n dimen-sions, GL(n,R). It can be identified with n×n invertible matrices under matrixmultiplication as group operation. The set of such matrices form an open subsetof the space Mn×n of all n × n matrices which is isomorphic to Rn2

. Further,matrix multiplication is smooth since it is polynomial in the elements, and sincethe determinant is non-zero so is the inverse. Thus GL(n,R) is a Lie group.

We will primarily be interested in Lie groups acting as transformations onmanifolds, sometimes only locally. As such the group action is not always definedfor the entire manifold or all the elements of the group.

Definition 2.11. Let M be a smooth manifold. A local group of transforma-tions acting on M is given by a (local) Lie group G, an open subset U , with

e ×M ⊂ U ⊂ G×M,

which is the domain of definition of the group action, and a smooth mapψ : U →M , denoted ψ(g, x) = g · x, with the following properties:

(a) If (h, x) ∈ U , (g, h · x) ∈ U , and also (gh, x) ∈ U , then

g · (h · x) = (gh) · x.

6

(b) For all x ∈M ,e · x = x.

(c) If (g, x) ∈ U , then (g−1, g · x) ∈ U and

g−1 · (g · x) = x.

Remark 2.12. In the following we will not always distinguish between globaland local groups of transformations. Statements will concern the group elementsfor which the action is defined.

Example 2.13. We mentioned the circle group as an example of a Lie group.Regarded as a transformation group acting on R2 it is called the special or-thogonal group, SO(2), or the rotation group, since its elements rotate pointsaround the origin by an angle ε,

gε ·(

xy

)

=

(

cos ε − sin εsin ε cos ε

)(

xy

)

=

(

x cos ε− y sin εx sin ε+ y cos ε

)

.

We include the matrix form to give a glimpse of the fact that SO(2) consists oforthogonal matrices with determinant 1, and is a subgroup of GL(2).

Definition 2.14. Starting at a point x in a manifold M , the set of points thatcan be reached by sequences of group transformations is called the orbit of Gthrough x. For a regular group action, the orbits are submanifolds of M .

In the example above, since the transformation depends on one parameter,ε, the orbits are curves in R2. Specifically, starting with a point x a distance cfrom the origin, the orbit will be the circle x2 + y2 = c. The circle as a set willremain unchanged by all the rotations of the group; we say that it is invariantunder the group SO(2). Defining such a circle as the solution set of a real-valuedfunction f(x, y) = x2 + y2 − c, we have an example of the following definition.

Definition 2.15. Let G be a group acting on M , and f :M → R. The group Gis called a symmetry (group) of an equation f(x) = 0 if and only if the solutionset of f

Sf =

x : f(x) = 0

is invariant under G, that is g · x ∈ Sf whenever g ∈ G and x ∈ Sf .

Given that the equation is of maximal rank at all points satisfying f(x) = 0,the solution set is a submanifold of M called the solution surface. Thus thetransformations of G move points along the solution surface, meaning it mapssolutions of f to other solutions. This property will be essential when we studysymmetry groups of differential equations in the next chapter.

We could also make a similar definition regarding all the values of a function,not only its solution set.

Definition 2.16. A function f :M → R is an invariant of a group of transfor-mations G if for all x ∈M and all g ∈ G,

f(g · x) = f(x). (2.1)

Put differently, the function f is constant on all the orbits of G.

7

2.3 Vector Fields

A very useful property of connected Lie groups is that the whole group can berecovered from the tangent space at the identity element. To show this we needto introduce the notion of tangents and vector fields on manifolds. One way isto define the tangent of a curve lying on the manifold.

Definition 2.17. Let C be a smooth curve on a manifold M , parametrized byγ : I → M , where I ⊂ R. In local coordinates x = (x1, . . . , xm), C is given bym smooth functions γ(ε) = (γ1(ε), . . . , γm(ε)) of the real variable ε. At eachpoint x = γ(ε) of C the curve has a tangent vector, namely the componentwisederivative γ(ε) = dγ/dε. We write

γ(ε) = γ1(ε)∂

∂x1+ . . .+ γm(ε)

∂

∂xm

for the tangent vector to C at x = γ(ε).

Example 2.18. One parametrization of the unit circle in R2,

γ(ε) = (cos ε, sin ε),

with coordinates (x, y), has tangent vector

γ(ε) = − sin ε∂

∂x+ cos ε

∂

∂y= −y ∂

∂x+ x

∂

∂y

at the point (x, y) = γ(ε) = (cos ε, sin ε).

The collection of all such possible tangent vectors forms the tangent spaceto M at the point x, denoted TxM . For each point the tangent space is a vectorspace of the same dimension as M . Choosing one tangent from TxM for eachx ∈M we get a vector field.

Definition 2.19. A smooth vector field X is an assignment of a tangent vectorXx ∈ TxM to each point x in M , varying smoothly. In local coordinates

X = ξ1(x)∂

∂x1+ . . . + ξm(x)

∂

∂xm,

where the ξi(x) are smooth functions.

Definition 2.20. An integral curve of a vector fieldX is a smooth parametrizedcurve x = γ(ε) whose tangent vector is the same as the vector field X at everypoint:

γ(ε) = Xγ(ε)

for all ε.Thus in local coordinates, x = γ(ε) must be a solution to the autonomous

system of ordinary differential equations

dxi

dε= ξi(x), i = 1, . . . ,m, (2.2)

where the ξi(x) are the coefficients of X at x. For a smooth vector field theexistence and uniqueness theorems for systems of ODE guarantee there is aunique solution to the system for each set of initial data γ(0) = x0. So througheach x in M passes a unique, maximal integral curve.

8

Example 2.21. In example 2.18 we found the tangents of a parametrized circle.Conversely, given the vector field X = −y∂x+x∂y, and initial condition γ(0) =(1, 0), we find the integral curve by solving the system

dx

dε= −y, dy

dε= x.

The solution is as expected the curve γ(ε) = (cos ε, sin ε), see figure 2.1.

-1 0 1

-1

0

1

Figure 2.1: The vector field X = −y∂x + x∂y and a (non-maximal) integralcurve with γ(0) = (1, 0). (The vectors are scaled for visibility)

We can picture the vector field X as the velocity field of a fluid, carryingparticles along in its flow. Starting at x for ε = 0, letting the particle follow theflow for a time ε, the trajectory of the particle is an integral curve γ(ε).

When viewed as tangent vectors, the ∂/∂xi are seen as basis vectors in thetangent space. Another way to define vector fields is as derivations on real-valued functions f :M → R. This reflects our notation in a natural way, giving

X(f) =m∑

i=1

ξi(x)∂f

∂xi,

which is itself a smooth function given any smooth f ∈ C∞(M). Since we havelocal coordinates in Rm with basis vectors ei, we can think of a derivation as adirectional derivative in the direction

∑mi=1 ξ

iei.

For maps between manifolds, and Lie groups in particular, we want to knowhow tangent vectors get mapped between the tangent spaces. This is describedby the induced linear map, the differential.

Definition 2.22. Let F : M → N be a smooth map, x = γ(ε) a curve in Mand F (x) = F (γ(ε)) ∈ N . The differential dF of F is a linear map between thetangent spaces, dF : TxM → TF (x)N , where

dF( d

dεγ(ε)

)

=d

dεF(

γ(ε))

.

In local coordinates, for Xx =∑n

i=1 ξi(x) ∂/∂xi,

dF (Xx) =

n∑

i=1

(

m∑

i=1

ξi∂F j

∂xi(x)

) ∂

∂yj=

n∑

i=1

X(F j(x))∂

∂yj.

9

So, when F maps a smooth curve γ ∈ M to smooth curve F (γ) ∈ N , thedifferential dF maps the tangent to γ at x = γ(ε) to the tangent of F (γ) atF (x) = F (γ(ε)). In local coordinates, dF is the Jacobian matrix of F at x.

Definition 2.23. Let X and Y be vector fields on a manifold M , then theirLie bracket or commutator [X,Y ] is the unique vector field satisfying

[X,Y ](f) = X(Y (f)− Y (X(f))

for all smooth functions f :M → R. In local coordinates, if

X =m∑

i=1

ξi(x)∂

∂xi, Y =

m∑

i=1

ηi(x)∂

∂xi,

then

[X,Y ] =

m∑

i=1

(

X(ηi)− Y (ξi)) ∂

∂xi=

m∑

i=1

m∑

j=1

(

ξj∂ηi

∂xj− ηj

∂ξi

∂xj

)

∂

∂xi. (2.3)

Proposition 2.24. The Lie bracket has the following properties:

1. Bilinearity. For constants a, b,

[aX + bY, Z] = a[X,Z] + b[Y,Z], [X, aY + bZ] = a[X,Y ] + b[X,Z],

2. Skew-symmetry[X,Y ] = −[Y,X],

3. Jacobi identity

[Z, [X,Y ]] + [Y, [Z,X]] + [X, [Y,Z]] = 0.

It is convenient to collect the commutators in a commutator table, where theintersection of the Xi row and the Xj column is the commutator [Xi, Xj ]. Sincethe commutator is anti-symmetric, we only need the cells above the diagonal.We illustrate with an example. Note that we will write ∂x to mean ∂/∂x whenmotivated to save space.

Example 2.25. In the next chapter we will investigate vector fields associatedwith the heat equation, ut = uxx. Six vector fields will be of special importancesince they generate the symmetry groups of the equation:

X1 = ∂t, X2 = x∂x + 2t∂t X3 = 4xt∂x + 4t2∂t − (x2 + 2t)u∂u,

X4 = ∂x, X5 = 2t∂x − xu∂u, X6 = u∂u.

The commutators are given in the following table:

X1 X2 X3 X4 X5 X6

X1 0 2X1 4X2 − 2X6 0 2X4 0X2 0 2X3 X4 X5 0X3 0 2X5 0 0X4 0 −X6 0X5 0 0X6 0

10

For instructive purposes we calculate one of the table entries using (2.3):

[X4, X6] =(

(x∂x + 2t∂t)(4xt)− (4xt∂x + 4t2∂t − (x2 + 2t)u∂u)(x))

∂x

+(

(x∂x + 2t∂t)(4t2)− (4xt∂x + 4t2∂t − (x2 + 2t)u∂u)(2t)

)

∂t

+(

(x∂x + 2t∂t)(−x2u− 2tu)− (4xt∂x + 4t2∂t − (x2 + 2t)u∂u)(0))

∂u

=(

(4xt+ 8xt)− 4xt)

∂x +(

16t2 − 8t2)

∂t +(

(−2x2u− 4tu)− 0)

∂u

= 2(

4xt∂x + 4t2∂t − (x2 − 2t)u∂u

)

= 2X6.

Note that we consider (x, t, u) as independent differential variables, and therefore∂xu = ∂tu = 0.

2.4 The Lie Algebra

In order to work with a linear algebra instead of the full complexity of a Liegroup, we now wish to establish a correspondence between vector fields and Liegroups. We start by defining a property of vector fields called left invariance.

Definition 2.26. For a Lie group (G, ·) and any element x ∈ G we define theleft translation

Lx : G→ G, Lx(y) = x · y,

which is a diffeomorphism with inverse

(Lx)−1(y) = Lx−1(y) = x−1 · y.

Definition 2.27. A vector field X is called left invariant if it is preserved underleft translations, meaning that

dLx(X) = X,

for all x ∈ G. Equivalently, if we consider the value of X at two distinct pointsy ∈ G and z = xy ∈ G, they must be related by the linear map dLx : TyG→ TzG

Xz = Xxy = dLx(Xy).

This means that a left invariant vector field is completely determined by itsvalue at the identity, Xe, since the value of the tangent at any point x ∈ G isobtained by a left translation

Xx = Xxe = dLx(Xe). (2.4)

Now, linear combinations of left invariant vector fields are also left invariant,meaning the set of them form a vector space. Using the above observation, wecan identify this vector space with the tangent space at the identity. We statethis as a theorem, a proof of which is found in van den Ban [13, p. 15].

Theorem 2.28. The space of all left invariant vector fields on a Lie group Gis isomorphic to the tangent space at the identity TeG.

11

Recalling that an algebra is a vector field with a multiplication operation,we define the Lie algebra corresponding to a Lie group using the Lie bracketfrom definition 2.23 as multiplication.

Definition 2.29. The Lie algebra g of a Lie group G is the space of all leftinvariant vector fields on G equipped with the Lie bracket.

Returning to the flow generated by a vector field, we will establish a cor-respondence between one-parameter subgroups of a Lie group, and the left-invariant vector fields of the Lie algebra.

Definition 2.30. The integral curve that passes through x ∈M at ε = 0, givenby the flow generated by the vector field X, is written γX(ε) = exp(εX)x.

Proposition 2.31. The integral curve in definition 2.30 satisfies the followingproperties:

1. exp(εX)x = x for ε = 0

2. exp(εX) exp(δX)x = exp((

ε+ δ)

X)x

3. exp(εX)−1x = exp(−εX)x

4. ddε

exp(εX)x = Xexp(εX)x

Proposition 2.32. Let X be a left invariant vector field on a Lie group G.Then the flow generated by X through the identity

gε = exp(εX)e ≡ exp(εX)

is defined for all ε ∈ R and forms a one-parameter subgroup of G, with

gε+δ = gε · gδ, g0 = e, g−1ε = g−ε.

Conversely, any connected one-dimensional subgroup of G is generated by sucha left invariant vector field.

Proof. A more general statement is proved in Olver [11, p. 45].

We now have a correspondence between transformation groups and vectorfields. A one-parameter group of transformations on a manifold can be gen-erated by a vector field, obtained by using the fourth property of proposition2.31 setting ε = 0. Conversely, given a vector field X we find the transfor-mation group by solving the system (2.2). We will therefore refer to X as the(infinitesimal) generator of the corresponding group.

In definition 2.15 we introduced a condition for being a symmetry of anequation. In practice, finding a symmetry group is much simpler on the level ofgenerating vector fields using the following theorem.

Theorem 2.33. A connected Lie group G is a symmetry group of the regularequation f(x) = 0 if and only if

X(f(x)) = 0, when f(x) = 0, (2.5)

for every infinitesimal generator X ∈ g of G.

12

Proof. A proof can be found in Olver [11, p. 80].

This result is central in that it allows us to find the vector fields and thus theone-parameter groups under which an equation is invariant. In the next chapterwe will introduce geometrical constructs that allow us to apply the theorem todifferential equations.

We finish the chapter by defining properties of a Lie algebra that can beleveraged when solving or reducing the order of differential equations [9].

Definition 2.34. Let g be a Lie algebra. A subspace h ⊂ g is called a subalgebraof the Lie algebra g if it is closed under commutation:

[X,Y ] ∈ h for all X,Y ∈ h.

The subalgebra h ⊂ g is called an ideal of g if

[X,Y ] ∈ h for all X ∈ h, Y ∈ g.

A finite Lie algebra g is said to be solvable if there is a sequence of subalgebras

g1 ⊂ . . . ⊂ gm−1 ⊂ gm = g

of dimensions 1, . . . ,m−1,m such that gk is an ideal in gk+1 for k = 1, . . . ,m−1.

13

Chapter 3

Differential Equations and the

Application of Lie Groups

3.1 Geometrical Setting

The geometrical interpretation of differential equations explored in this sectionis adapted from Cicogna [4] and Olver [12].

Consider an equation f(x) = 0 for x in Rn, as discussed in section 2.2. For asmooth function f : Rn → R of maximal rank the solution set Sf = x : f(x) =0 is a submanifold of Rn. We want to regard solutions of differential equationsin an analogous way: as the solution set defined by the vanishing of functionsin the space of all variables as well as relevant derivatives, the jet space. Forclarity we will only consider a single scalar differential equation at a time, butthe extension to vector valued and systems of equations is straightforward.

A general n-th order differential equation

F (x, u(n)) = 0 (3.1)

is a relation between p independent variables x = (x1, . . . , xp), the dependentvariable u and partial derivatives up to order n, here collectively denoted u(n).The differential equation is thus defined by the vanishing of a differential func-tion F :M (n) → R defined on the n-th jet space M (n), defined as follows:

Definition 3.1. Let x ∈ Ω ⊂ Rp, u ∈ U ⊂ R, and let Uk be the space of allk-th order derivatives of u with respect to xi, i.e. ∂u/∂xi, ∂2u/∂xj∂xi, etc., upto order k, for i, j = 1, ..., p. Then the jet space of order n is

M (n) = Ω× U × U1 × · · · × Un,

also called the n-th prolongation of the space of variables M ≃ Ω× U .

The jet space is locally isomorphic to a real space with one important dis-tinction: there is a contact structure in it that sets the relations among thederivatives. This is best illustrated by example.

14

Example 3.2. If we consider the case with one independent variable x andone dependent variable y, then we have Ω ≃ R, U ≃ R, M ≃ R2 and thefirst jet space M (1) ≃ R3 with coordinates (x, y, yx). Now, if we have a curveγ ∈M corresponding to a function y(x) with derivative dy/dx = yx(x), and wethen consider its prolongation γ(1) ∈ M (1) given by (x, y(x), yx(x)), then γ(1)

is not an arbitrary curve in R3, for it must at all points satisfy the conditionyx(x) = dy(x)/dx.

So, just like the solution set of a smooth function f : Rn → R defines amanifold in Rn, the solution of a differential equation defines a submanifold

SF = (x, u(n)) : F (x, u(n)) = 0 (3.2)

in the jet space M (n), called the solution manifold of the equation.

Given a function u = f(x), f : Ω → U , we define its graph Γf in M

Γf = (x, u) ∈M : u = f(x) ⊂M.

Since we know all the partial derivatives of a given f up to order n, we candefine its n-th prolongation f (n), and identify it with its graph

Γf(n) = (x, u(n)) ∈M (n) : u(n) = f (n)(x) ⊂M (n).

So, for u = f(x) to be a solution to the differential equation (3.1), thegraph of f (n) must be entirely contained in the solution manifold: Γf(n) ⊂ SF .This geometric reformulation is equivalent to the classical requirement that asolution must satisfy F (x, f (n)) = 0, but it will enable us to find solutionsthrough prolonged groups of transformations acting on submanifolds of M (n).

3.2 Symmetry Groups

This section is based on material from Hydon [6] and Ibragimov [8], both ofwhich present a large number of symmetry methods for ODE and PDE, as wellas Olver [11, 12].

In the previous chapter we introduced Lie groups and their action as (local)groups of transformations. We also introduced the correspondence between Liegroups and vector fields, giving rise to the associated Lie algebra. We shall nowapply these concepts to manipulate and solve differential equations.

We consider smooth, invertible transformations gε :M →M ,

(x, u) 7→ gε · (x, u) = (x, u) =(

x(x, u, ε), u(x, u, ε))

, (3.3)

where ε is a real parameter. For transformations such that

1. g0 is the identity transformation, i.e. (x, u) = (x, u) for ε = 0,

2. g−1ε = g−ε, and

15

3. gεgδ = gε+δ for every ε, δ sufficiently close to zero,

gε is a one-parameter Lie group.

We write the infinitesimal generator, the vector field that generates a flowequal to the transformations,

X =

p∑

i=1

ξi(x, u)∂

∂xi+ η(x, u)

∂

∂u, (3.4)

where

ξi(x, u) =∂xi(x, u, ε)

∂ε, η(x, u) =

∂u(x, u, ε)

∂ε.

At the heart of applying Lie theory to the solution of differential equations isfinding symmetries, transformations that take solutions to solutions. We coulddefine a symmetry only in terms of this property as follows.

Definition 3.3. A group of transformations G acting on M is a symmetry ofthe differential equation (3.1) if whenever u = f(x) is a solution, then u = f(x)is also a solution.

However, we want to make use of our introduced geometrical setting for adefinition from which we can construct solutions. In definition 2.15 we defineda symmetry for the solutions of a function in Rn. Now that we have definedthe jet space M (n), we can adapt this definition to the solution manifold of adifferential equation.

Definition 3.4. Let G be a group acting on M , and F : M (n) → R. Then Gis a symmetry of F (x, u(n)) = 0 if the solution manifold

SF =

(x, u(n)) : F (x, u(n)) = 0

⊂M (n)

is invariant under the action of the prolonged actions of G(n) on M (n), which isgiven in terms of its generator in the following (see proposition 3.6).

One question arises: a given group of transformations only defines an actionon the independent and dependent variables, but how are the derivative variablesthat we have introduced affected? We need to prolong the transformations toact on the jet space, so that we extend our transformations from being mapsg : M → M to being prolonged maps g(n) : M (n) → M (n). To this endwe use the total derivative, which treats u and its derivatives as functions ofthe independent variables. We emphasise that we consider ui = ∂u(x)/∂xi,uij = ∂u(x)/∂xj∂xi, etc., as variables and coordinates in the jet space M (n).

Definition 3.5. For a differential function F (x, u(n)) :M (n) → R,

DiF =∂F

∂xi+ ui

∂F

∂u+

p∑

j=1

uij∂F

∂uj+ . . .

is the total derivative of F with respect to xi.

The total derivative treats u and all its derivatives as functions of xi, andso captures all of F ’s dependence on xi. We can now use it to deduce how thetransformations on M act on the derivatives.

16

Proposition 3.6. For a one-parameter group of transformations G acting onM ≃ Rp × R, with generator

X =

p∑

i=1

ξi(x, u)∂

∂xi+ η(x, u)

∂

∂u,

the first prolongation of the generator of G(1), acting on M (1), is

X(1) = X +

p∑

i=1

ηi(x, u(1))∂

∂ui,

the second prolongation of the generator of G(2), acting on M (2), is

X(2) = X(1) +

p∑

j=1

p∑

i=1

ηij(x, u(2))∂

∂uij,

and so on. The ηi and ηij are found recursively by the formulas

ηi = Di(η)−p

∑

k=1

ukDi(ξk), (3.5)

ηij = Dj(ηi)−

p∑

k=1

uikDj(ξk). (3.6)

Proof. A proof of a more general proposition can be found in [11, p. 110].

Now that we have introduced the machinery of prolongations we can applytheorem 3.7 to differential equations. We reformulate it in this context.

Theorem 3.7. A connected Lie group G is a symmetry group of an n-th orderdifferential equation F (x, u(n)) = 0 if and only if

X(n)(F(

x, u(n))

) = 0, whenever F (x, u(n)) = 0, (3.7)

for every infinitesimal generator X ∈ g of G.

Geometrically we want to find a transformation that leaves the solutionmanifold invariant. If we imagine our solution set as a manifold embedded inthe jet space, we seek a transformation whose prolonged generating vector fieldis always parallel to it.

Equation (3.7) leads to a system of linear, over-determined system of PDEsfor the coefficients ξi and η of X. These are called the determining equations,and they can often be explicitly solved.

Example 3.8. Let us consider a first order ODE on the form y′(x) = f(x, y).Viewed in the jet space M (1), with coordinates (x, y, y′), the solution surfaceis defined by F (x, y, y′) = f(x, y)− y′ = 0. To find a group of transformationsG acting on the (x, y)-plane with generator X = ξ(x, y)∂x + η(x, y)∂y that is asymmetry of the ODE, we consider its first prolongation,

X(1) = ξ(x, y)∂

∂x+ η(x, y)

∂

∂y+ ηx(x, y, y′)

∂

∂y′,

17

to find its effect on the derivative coordinate y′. By (3.5), we have

ηx = Dx(η)− yxDx(ξ) = ηx + y′ηy − y′(ξx + y′ξy), (3.8)

and so by theorem 3.7, G is a symmetry of the ODE when

X(1)(F ) = ξ(x, y)∂F

∂x+ η(x, y)

∂F

∂y+ ηx(x, y, y′)

∂F

∂y′

= ξ(x, y)∂f

∂x+ η(x, y)

∂f

∂y− ηx(x, y, y′)

= ξfx + ηfy −(

ηx + y′(ηy − ξx) + (y′)2ξy

)

= 0,

where the last equality follows from (3.8). Since y′(x) = f(x, y), we can substi-tute f for y′ above to get

ξfx + ηfy −(

ηx + f(ηy − ξx) + (f)2ξy)

= 0, (3.9)

which for a given function f(x, y) is a system of equations to be solved for ξand η. See chapter 4.1 for a concrete example of an ODE on this form.

3.3 Solving ODE and PDE Using Symmetries

Solving the determining equations we get a number of generators for one-parameter groups that are symmetries of our differential equation. How theycan be used in solving or exploring it depends on the type of equation. Wegive but a brief summary of three methods. For a more thorough treatment werecommend the accessible textbooks by Hydon [6] and Ibragimov [8].

3.3.1 Canonical Coordinates

A first order ODE with one symmetry can be solved by quadrature (integration).We will present the method of finding invariants of the symmetry and using themto find a suitable change of coordinates.

A symmetry is a bijective transformation that maps solutions to other solu-tions. The generating vector field in general depends on all variables, but withthe right invertible coordinate change r = r(x, y), s = s(x, y), called canonicalcoordinates, we transform X(x,y) = ξ(x, y)∂x+η(x, y)∂y into the constant vectorfield X(r,s) = ∂s. The flow it generates is a translation in s, and so the ODEwill not depend on s but be of the form s′(r) = f(r), which can be integrated.

Proposition 3.9. We find these canonical coordinates by solving

X(r) = ξ(x, y)∂r

∂x+ η(x, y)

∂r

∂y= 0, (3.10)

X(s) = ξ(x, y)∂s

∂x+ η(x, y)

∂s

∂y= 1. (3.11)

Proof. This is immediate by the chain rule, since in the new coordinates

X(r(x,y),s(x,y)) = ξ(x, y)( ∂r

∂x

∂

∂r+∂r

∂y

∂

∂r

)

+ η(x, y)( ∂s

∂x

∂

∂s+∂s

∂y

∂

∂s

)

= X(r)∂

∂r+X(s)

∂

∂s.

18

Equation (3.10) dictates that r(x, y) is an invariant of the symmetry, con-stant on the orbits ofX. It is found by the method of characteristics. Essentially,we solve (2.2) to get the group action, and then eliminate the parameter andsolve for the constant of integration to get an invariant function ψ(x, y) = C,also called a first integral. We then set r = ψ(x, y), or a function thereof.

This can also be done by solving the parameter independent characteristicequation

dx

ξ(x, y)=

dy

η(x, y).

Similarly, we find s = s(x, y) that satisfies (3.11) by solving

dx

ξ(x, y)=

dy

η(x, y)= ds.

See chapter 4.1 for an example.This technique can also be used on higher order ODE. For each one-parameter

group the order can be lowered by one as long as the Lie algebra is solvable, seee.g. Hydon [6, Ch. 4.1] and Olver [11, Ch. 2.5].

3.3.2 Transformation of a Known Solution

In the case of PDE, we can use the knowledge of one solution to generate wholefamilies of solutions using the symmetries we have found. A one-parametergroup of transformations gives rise to a one-parameter family of solutions. Soif u = f(x) is a solution to a PDE, then given a symmetry (x, u) = gε · (x, u)

u = gε · f(x)

is also a solution for those ε where it is defined. For an example of this, see inchapter 4.2.

3.3.3 Group-invariant Solutions

This method uses invariants of a group G to reformulate the differential equationin terms of these, which will result in an equation with one less independentvariable. Say we have a PDE F (x, t, u(n)) = 0 in two independent variables xand t with a symmetry generated by

X = ξ(x, t, u)∂

∂x+ τ(x, t, u)

∂

∂t+ η(x, t, u)

∂

∂u,

with corresponding characteristic equations

dx

ξ=dt

τ=du

η.

We get two invariants r(x, t, u) and s(x, t, u) (in general one less than the numberof variables [8]). Letting one play the role of dependent variable, s = f(r), wedetermine the derivatives of u in terms of s with respect to r (and perhaps x ort which will later be redundant). Substituting back into the PDE, the result isan ODE, which we solve to find the solutions that are invariant under the givengroup action. An example is given at the end of chapter 4.2.

19

Chapter 4

Examples

4.1 A Non-linear ODE

Let us consider the ODE

y′(x) =x2y − x+ y3 − y

x3 + xy2 − x+ y. (4.1)

Seeking a symmetry we use equation (3.7),

X(1)(F ) = ξ(x, y)∂F

∂x+ η(x, y)

∂F

∂y+ ηx(x, y, y′)

∂F

∂y′= 0,

where F = x2y−x+y3−y

x3+xy2−x+y− y′(x), to determine ξ and η. Using the prolongation

formula (3.8) for ηx and (4.1) to replace y′, after simplification we get

ξ (−x4y + 2x3 − 2x2y3 + 2x2y + 2xy2 − y5 + 2y3 − 2y)

+ η (x5 + 2x3y2 − 2x3 + 2x2y + xy4 − 2xy2 + 2x+ 2y3)

+ ηx (−x6 − 2x4y2 + 2x4 − 2x3y − x2y4 + 2x2y2 − x2 − 2xy3 + 2xy − y2)

+ (ηy − ξx) (−x5y + x4 − 2x3y3 + 2x3y − x2 − xy5 + 2xy3 − y4 + y2)

+ ξy (x4y2 − 2x3y + 2x2y4 − 2x2y2 + x2 − 2xy3 + 2xy + y6 − 2y4 + y2) = 0.

To solve this unwieldy equation we make an ansatz that the coefficient functionsare polynomial. Constant polynomials ξ(x, y) = c1 and η(x, y) = c2 give nonon-trivial solutions, and so the ODE has no translation symmetry in the (x, y)-plane. Using first degree polynomials ξ(x, y) = c1x + c2y + c3 and η(x, y) =c4x + c5y + c6 gives non-trivial solutions. We get an over-determined systemof equations, with one equation for each monomial, of which only 5 linearlyindependent, namely

c3 = 0

c6 = 0

2(c1 + c5) = 0

c1 + c2 + c4 − c5 = 0

2(−c1 + c2 + c4 + c5) = 0.

20

This gives a one-dimensional space of solutions, namely c2 = −c4 with all otherci = 0. Choosing c2 = −1, c4 = 1 gives us the generator X = −y∂x +x∂y of therotation group SO(2).

Next we wish to use this symmetry to find canonical coordinates in whichthe ODE is integrable by quadrature. Since the transformation under whichthe equation is invariant is rotation around the origin, it might be clear thatpolar coordinates are the right choice. Generally, the choice of coordinates isnot obvious and we therefore demonstrate how to find them using the methodof characteristics.

We use (3.10) to determine the independent variable r:

X(r) = −y ∂r∂x

+ x∂r

∂y= 0,

which corresponds to the characteristic equation

dx

−y =dy

x.

It is integrated to give the first integral x2+y2 = c1, and so the general solutionis r = F (x2 + y2). We choose r =

√

x2 + y2. The dependent variable is foundusing (3.11),

X(s) = −y ∂s∂x

+ x∂s

∂y= 1,

which corresponds to the characteristic equations

dx

−y =dy

x= ds.

For x > 0 we substitute x =√

r2 − y2 giving

ds =dy

√

r2 − y2,

with solution s = arcsin(y/r) + c2. Using arcsin(y/r) = arctan(y/√

r2 − y2),and substituting back for r we get s = arctan(y/x) + c2. We can thereforeconfirm that polar coordinates

r =√

x2 + y2, s = arctan(y/x),

are canonical coordinates for (4.1) and will render it translation invariant andthus solvable by quadrature. Using the chain rule and the inverse coordinatechange x = r cos s, y = r sin s we get

ds

dr=sx + syyxrx + ryyx

= ... =1

√

x2 + y2(1− x2 − y2)=

1

r(1− r2),

which can be integrated (using partial fractions) to yield

s(r) =

∫

dr

r(1− r2)=

1

2ln∣

∣

∣

r2

1− r2

∣

∣

∣+ C. (4.2)

21

Changing back to (x, y)-coordinates we get an implicit expression for the solu-tions where x > 0, namely

arctan(y

x) = ln

∣

∣

∣

x2 + y2

1− x2 − y2

∣

∣

∣+ C.

The solution (4.2) in polar coordinates is more illuminating, revealing e.g. thatlimr→∞ s(r) = C, i.e. the constant C is an asymptote for the angle s. See figure4.1 for the graph of solutions for two choices of the constant C. It is clear tosee that one solution is attained from the other by rotating it π radians, or,equivalently, following the flow of the vector field X(r,s) = ∂s for ε = π.

Figure 4.1: Two solutions of (4.1) for C1 = 0, C2 = π.

4.2 The Heat Equation

Let us consider the heat equation in one space dimension,

ut = uxx, (4.3)

elaborating the treatment given in Olver [11, Examples 2.41 and 3.3], filling insome computational details.

We seek generating vector fields acting on the space of independent anddependent variables, M ≃ X × U , of the form

X = ξ(x, t, u)∂

∂x+ τ(x, t, u)

∂

∂t+ η(x, t, u)

∂

∂u. (4.4)

Since the equation involves uxx the solution manifold lies in the second jet spaceM (2) ≃ X ×U (2) ≃ R2 ×R6, and we therefore require the second prolongation,

X(2) = X + ηx∂

∂ux+ ηt

∂

∂ut+ ηxx

∂

∂uxx+ ηxt

∂

∂uxt+ ηtt

∂

∂utt,

to account for the action on all derivatives involved. A generator gives a symme-try transformation when the vector field is everywhere tangent to the solution

22

manifold, by theorem 3.7 when

X(2)(F ) = ηt − ηxx = 0, (4.5)

where F = ut−uxx. The coefficients ηt and ηxx are found using (3.5) and (3.6):

ηt = Dt(η)− uxDt(ξ)− utDt(τ)

= (ηt + utηu)− ux(ξt − utξu)− ut(τt − utτu)

= ηt + ξtux + (ηu − τt)ut − ξuuxut − τuu2t ,

ηx = Dx(η)− uxDx(ξ)− utDx(τ)

= (ηx + uxηu)− ux(ξx − uxξu)− ut(τx − uxτu)

= ηx + (ηu − ξx)ux − τxut − ξuu2x − τuuxut,

ηxx = Dx(ηx)− uxxDx(ξ)− uxtDx(τ)

= Dx(ηx + (ηu − ξx)ux − τxut − ξuu2x − τuuxut)− uxxDx(ξ)− uxtDx(τ)

= ηxx + (2ηxu − ξxx)ux − τxxut + (ηuu − 2ξxu)u2x − 2τxuuxut − ξuuu

3x

− τuuu2xut + (ηu − 2ξx)uxx+ 2τxuxt − 3ξuuxuxx − τuutuxx − 2τuuxuxt.

Substituting the above expressions into equation (4.5) and substituting uxx forut we get the determining equations, one equation for each monomial (ux, ut, u

2x,

etc.),

uxuxt 0 = −2τu (4.6)

uxt 0 = −2τx (4.7)

u2xx −τu = −τu (4.8)

u2xuxx 0 = −τuu (4.9)

uxuxx −ξu = −2τxu − 3ξu (4.10)

uxx ηu − τt = −τxx + ηu (4.11)

u3x 0 = −ξuu (4.12)

u2x 0 = ηuu − 2ξxu (4.13)

ux −ξt = 2ηxu − ξxx (4.14)

1 ηt = ηxx (4.15)

Equations (4.6) and (4.7) imply that τ = τ(t), i.e. τ is a function only of t,

and then (4.10) implies that ξ = ξ(x, t). We can then use (4.11) to conclude

ξ(x, t) = 12τt(t)x+ σ(t), (4.16)

where σ is an arbitrary function of t. Equation (4.13) is now ηuu = 0, so η islinear in u, i.e.

η(x, t, u) = β(x, t)u+ α(x, t),

for arbitrary functions α and β. Now (4.14) gives 2βx(x, t) = 2ηxu = −ξt =− 1

2τtt(t)x+ σt(t), so

β(x, t) = 18τtt(t)x

2 − 12σt(t)x+ ρ(t). (4.17)

23

Also, (4.15) requires βt(x, t)u + αt(x, t) = βxx(x, t)u + αxx(x, t), so αt(x, t) =αxx(x, t) which means α(x, t) must be a solution of the heat equation. Secondly,together with (4.17), we get

− 18τttt(t)x

2 − 12σtt(t)x+ ρt(t) = − 1

4τtt(t) (4.18)

so τ(t)ttt = 0, which implies τ is a second degree polynomial in t, i.e.

τ(t) = c1 + c2t+ c3t2. (4.19)

Equation (4.18) also implies ρ(t) = − 14τt and σ(t)tt = 0, dictating that ρ(t) =

− 14 (2c3t+ c6) and σ(t) = c4 + c5t, so

β(x, t) = − 182c3x

2 − 12c5x− 1

4 (c6 + 2c3t). (4.20)

Then by (4.16) and (4.19) we conclude

ξ(x, t) = 12

(

c2 + 2c3t)

x+ c4 + c5t, (4.21)

and finally, using (4.20),

η(x, t, u) = (− 14c6 − 1

2c5x− 12c3t− 1

4c3x2)u+ α(x, t). (4.22)

So we have six arbitrary constants and degrees of freedom. By setting all ci butone to zero, recalling the form of our vector field from (4.4), we arrive at thefollowing six vector fields spanning the Lie algebra:

X1 = ∂t (c1)

X2 = x∂x + 2t∂t (c2)

X3 = 4xt∂x + 4t2∂t − (x2 + 2t)u∂u (c3)

X4 = ∂x (c4)

X5 = 2t∂x − xu∂u (c5)

X6 = u∂u (c6)

and an infinite-dimensional subalgebra Xα = α(x, t)∂u. The commutator ta-ble is given in example 2.25. Exponentiating these give us the most generalsymmetry group of (4.3), and the corresponding group actions are, given as(x, t, u) = exp(εXi)(x, t, u):

G1 : (x, t+ ε, u)

G2 : (eεx, e2εt, u)

G3 : (x

1− 4εt,

t

1− 4εt, u

√1− 4εt exp

( −εx21− 4εt

)

)

G4 : (x+ ε, t, u)

G5 : (x+ 2εt, t, u exp(−εx− εt2))

G6 : (x, t, eεu)

Gα : (x, t, u+ εα(x, t))

Some of these symmetries are easily interpreted: G1 and G4 show that the equa-tion is time- and space-invariant, and Gα and G6 show it is linear; we can add

24

solutions and multiply them by constants. These are also quite easily deducedfrom the equation (4.3) itself. The symmetry group G5 on the other hand is notat all obvious, and we use it below to find an important solution.

To find solutions to (4.3), one method is to transform a known solution. Theconstant solution u = c, can be transforming it using the symmetry G3 to getthe solution

u =c√

1 + 4εtexp

( −εx21 + 4εt

)

. (4.23)

So for each c ∈ R, we get a one-parameter family of solutions depending on ε.Setting c =

√

ε/π,

u =

√ε

√

π(1 + 4εt)exp

( −εx21 + 4εt

)

,

and translating in t with −1/4ε using G1, we get

u =1√4πt

exp(−x2

4t

)

,

which is the so called fundamental solution. It can be used to solve initial valueproblems of (4.3) using convolution, see e.g. Evans [5, Chapter 2.3.1].

Using the method of group-invariant solutions, we can choose a linear com-bination of the translation symmetries X1 and X4, i.e. we set X = X1 + cX4

giving the generatorX = ∂t + c∂x, c ∈ R.

We then use the method of characteristics to find the invariants of X. Thecharacteristic equation

dx

c=dt

1

gives x − ct = C, so one invariant is r = x − ct. The other is clearly u, so weset s = u. Letting s be the dependent variable, the group-invariant solutionss = f(r) will be u = f(x − ct) which are waves travelling at speed c. Thederivatives ut and uxx can then be found in terms of s and r using the chainrule:

ut = st = srrt = −csr, ux = sx = srrx = sr, uxx = (sr)x = (sr)rrx = srr,

and these can then be substituted into the heat equation (4.3) reducing it tothe ODE

s′′(r) = −cs′(r).It can be solved by setting w = s′(r) making it an elementary system of firstorder linear equations; the solution is

s(r) = k1e−cy + k2,

where ki are constants. Substituting back to our original variables we finallyget the solutions

u(x, t) = k1e−c(x−ct) + k2,

a family of travelling waves of exponentials.

25

4.3 The Kolmogorov Equation

An equation still under theoretical investigation, and with applications rangingfrom the kinetic theory of gases to the pricing of options (models which arerelated by the random nature of the motion of particles as well as stock prices),is the Kolmogorov equation

Lu = ut − uxx − xuy = 0, (4.24)

which is a degenerate parabolic PDE for t > 0.The study of this equation spans the last century: in 1931 Kolmogorov

publishes the paper Über die analytischen Methoden in der Wahrscheinlichkeit-srechnung, on analytical methods of probability, and a couple of years laterZur Theorie der stetigen zufälligen Prozesse on continuous random processes.Following the methods presented he then provides a fundamental solution for(4.24) in a short paper on Brownian motion [10] in 1934.

In 1967 Hörmander uses (4.24) as an example in a central work [7] estab-lishing sufficient and necessary conditions for a differential operator to be hy-poelliptic. Essentially, hypoellipticity means that if the right hand side of ourequation is smooth, then so is the solution. Formally, a differential operator Lwith C∞(Ω) coefficients (Ω open subset of Rn) is said hypoelliptic in Ω if, for anyopen set Ω′ ∈ Ω and any distribution u ∈ D(Ω), Lu ∈ C∞(Ω) =⇒ u ∈ C∞.

A contemporary application is the price of "Asian options". They satisfyinga stochastic differential equation dS(t) = µ0S(t)dt+ σSdW , where the exercise(strike/purchase) price depends on an average on the history of the stock price.For certain conditions the price of the option can be shown to satisfy a PDEequivalent to the Kolmogorov equation (4.24) [3].

We will not solve the determining equations for all possible symmetries, butfocus on finding the translations and dilations need to define a (homogeneous)group structure on R3. See Anceschi [1] for a recent article presenting a surveyof results on Kolmogorov type equations.

We begin as before by prolonging a general vector field generator

X = ξ∂

∂x+ η

∂

∂y+ τ

∂

∂t+ ω

∂

∂u, (4.25)

with unknown coefficients ξ, η, τ and ω that are functions of (x, y, t, u), findingthe determining equations by imposing the invariant solution manifold conditionfrom theorem 3.7: X(2)(ut−uxx−xuy) = 0, substituting ut, and collecting termsfor each monomial on which the generator does not depend. See section 4.2 forthe details of the procedure for the similar example of the heat equation.

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The determining equations are

ξu = 0 (4.26)

ηu = 0 (4.27)

τu = 0 (4.28)

ωuu = 0 (4.29)

ηx = 0 (4.30)

τx = 0 (4.31)

−2ωxu − ξt + xξy + ξxx = 0 (4.32)

−τt + xτy + 2ξx = 0 (4.33)

−ξ − ηt + xηy − xτt + x2τy = 0 (4.34)

ωt − xωy − ωxx = 0 (4.35)

To find the dilations, we assume there exists a generator

Xδ = ax∂

∂x+ by

∂

∂y+ ct

∂

∂t. (4.36)

Using equations (4.33) and (4.34) and setting a = 1 (since the dilations areonly unique up to scaling), we get b = 3 and c = 2 corresponding to the dilationδε(x, y, t) = (eεx, e3εy, e2εt). Using a different parametrization to emphasise thescaling relationships we write

δλ(x, y, t) = (λx, λ3y, λ2t), λ > 0. (4.37)

For the translations, we begin with the ansätze

Xx =∂

∂x, Xy =

∂

∂y, Xt =

∂

∂t. (4.38)

We find that Xy and Xt satisfy all determining equations (4.26)-(4.35), but Xx

does not. Given our ansatz ξ = 1 for Xx, equation (4.34) demands ηt = −1;choosing the simplest solution η = −t results in Xx = ∂x − t∂y. In total, alinear combination gives the vector field

X = x0Xx + y0X

y + t0Xt = x0

∂

∂x+ (y0 − x0t)

∂

∂y+ t0

∂

∂t, (4.39)

corresponding to translations (x, y, t)(x0, y0, t0) = (x+x0, y+(y0−x0t), t+t0),for any (x, y, t) and (x0, y0, t0) ∈ R3.

We have thus found the symmetries needed to define the Lie group structure,which is crucial for further study of the Kolmogorov equation.

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Bibliography

[1] Anceschi, Francesca, & Polidoro, Sergio. 2019. A survey on the classicaltheory for Kolmogorov equation. arXiv:1907.05155 [math.AP].

[2] Boothby, William M. 1975. An Introduction to Differentiable Manifolds andRiemannian Geometry. Academic Press, Inc., New York.

[3] Bramanti, Marco. 2014. An Invitation to Hypoelliptic Operators and Hör-mander‘s Vector Fields. Springer International Publishing.

[4] Cicogna, Giampaolo, & Gaeta, Giuseppe. 1999. Symmetry and PerturbationTheory in Nonlinear Dynamics. Springer-Verlag Berlin Heidelberg.

[5] Evans, Lawrence C. 2010. Partial Differential Equations. Providence, R.I.:American Mathematical Society.

[6] Hydon, Peter E. 2000. Symmetry Methods for Differential Equations. Cam-bridge University Press, New York.

[7] Hörmander, Lars. 1967. Hypoelliptic second order differential equations.Acta Mathematica, 119, 147–171.

[8] Ibragimov, Nial H. 2009. A Practical Course In Differential Equations AndMathematical Modelling, A. World Scientific Publishing Co Pte Ltd.

[9] Ibragimov, Nial H. 2013. Transformation Groups and Lie Algebras. HigherEducation Press Limited Company, Beijing.

[10] Kolmogorov, Andrej. 1934. Zufällige Bewegungen (Zur Theorie derBrownschen Bewegung). Annals of Mathematics, 35(1), 116–117.https://www.jstor.org/stable/1968123. Accessed 2020-05-09.

[11] Olver, Peter J. 1986. Applications of Lie Groups to Differential Equations.Springer-Verlag.

[12] Olver, Peter J. 2012. Lectures on Lie Groups and Differential Equations.http://www-users.math.umn.edu/~olver/sm.html. Accessed 2020-06-12.

[13] van den Ban, Erik. 2010. Lecture Notes on Lie groups.https://www.staff.science.uu.nl/~ban00101/lecnot.html. Accessed2020-06-12.

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