Group Analysis Methods 1. Classical Method for Symmetry Reductions

S.7. Group Analysis Methods S.7.1. Classical Method for Symmetry Reductions

The group analysis methods (also referred to as Lie group methods) suggest a regular procedure for identifying symmetries of an equation and allow us to find the following: The group analysis methods (also referred to as Lie group methods) suggest a regular procedure for identifying symmetries of an equation and allow us to find the following:

Remark. The methods of group analysis may be regarded as a wide generalization of the similarity methods described in Section S.3.

S.7.1-1. Local one parameter Lie group of transformations. Invariance condition. We will consider transformations of the following second-order partial differential equation:

where x = (x 1 , ...,x n ) are independent variables and w is a dependent variable (unknown function). Consider a set of transformations of the ( n + 1)-dimensional Euclidean space

x ¯ i = ϕ i ( x, w, ε), x ¯ i | ε=0 = x i ,

w = ψ(x, w, ε), ¯

w| ¯ ε=0 = w,

where the ϕ i and ψ are smooth functions of their arguments and ε is a real parameter. This set of transformations is called a one-parameter continuous point Lie group of transformations,

G, if for all ε 1 and ε 2 , we have T ε 1 ◦T ε 2 =T ε 1 + ε 2 , i.e., the successive application of two transformations of the form (1) with parameters ε 1 and ε 2 is equivalent to a single transformation of the same form with parameter ε 1 + ε 2 . Let

G be a group of transformations of a set M in the (n + 1)-dimensional Euclidean space, and let u = (x, w) be a point of that set. The set G(u) formed by all images Tu, as T ranges within the entire group

G, is called the orbit of the point u. The set M is called invariant under a group of transformations if the orbit of each point u of M belongs to M , i.e., G(M ) = M . In other words, any point of an invariant set remains in that set under arbitrary transformations of the group, i.e., the set is mapped into itself.

Below, we consider local one-parameter continuous point Lie groups of transformations (briefly, point groups) that correspond to the infinitesimal transformation (2) as ε → 0. Expanding the functions ¯ x and ¯ w from (2) into the Taylor series in powers of the parameter ε about the point ε = 0 and neglecting the second- and higher-order terms, we obtain

(3) where

x ¯ i ≃x i + ξ i ( x, w)ε, w ≃ w + ζ(x, w)ε, ¯

∂ϕ i ( x, w, ε)

∂ψ(x, w, ε)

ξ i ( x, w) =

ζ(x, w) =

The vector ( ξ, ζ) is tangent (at the point (x, w)) to the curve formed by the transformed points (¯ x, ¯ w) .

The first-order linear differential operator

X= ξ i ( x, w)

+ ζ(x, w)

corresponding to the infinitesimal transformation (3), is called the infinitesimal operator (or in- finitesimal generator) of the group (here and in what follows, summation over repeated indices is assumed).

function I 0 ( x, w) satisfying the condition I 0 (¯ x, ¯ w) = I 0 ( x, w). The expansion in powers of the small

parameter ε yields the following linear partial differential equation for I 0 :

ζ(x, w)

It follows that the group (2) and the operator (4) have n functionally independent universal invariants. This means that any function

F (x, w) which is invariant under the group (2) can be represented as a function of n invariants, which play the role of new variables. In the new variables (2), the first and second derivatives take the form

Here, the coordinates of the first and second prolongations ζ i and ζ ij are defined by

∂w ∂ 2 w where the following brief notation is used for the partial derivatives: p i =

, q ij = ; ∂x i

∂x i ∂x j ∂

+ · · · is the operator of total differentiation with respect to x i . ∂x i

Let us prove the first set of formulas (6) for the coordinates of the first prolongation. For simplicity, consider the case of two independent variables x and y. Then formulas (3) can be written as x≃x+ξ ¯ 1 ( x, y, w)ε, y≃y+ξ ¯ 2 ( x, y, w)ε, w ≃ w + ζ(x, y, w)ε. ¯

Obviously,

w ¯ x =¯ w x ¯ x ¯ x +¯ w y ¯ y ¯ x , w ¯ y =¯ w x ¯ x ¯ y +¯ w y ¯ y ¯ y .

Differentiating relations (8) with respect to x and y, we obtain

In order to calculate ¯ w x ¯ , we eliminate ¯ w y ¯ from (9) and then replace the derivatives ¯ x x ,¯ x y ,¯ y x ,¯ y y ,¯ w x ,¯ w y by the corresponding expressions from (10) to obtain

w x + ε(D x ζ+w x D y ξ 2 − w y D x ξ 2 )+ ε ( D x ζD y ξ 2 − D x ξ 2 D y ζ) 2 1+ . ε(D x ξ 1 + D y ξ 2 )+ ε ( D x ξ 1 D y ξ 2 − D x ξ 2 D y ξ 1 )

Using the expansion in powers of ε, we find that

w ¯ x ¯ ≃w x + ζ 1 ε,

ζ 1 = D x ζ−w x D x ξ 1 − w y D x ξ 2 ,

as required. In a similar way, one can calculate ζ 2 and the coordinates of the second prolongation ζ ij .

Let us require that equation (1) be invariant (i.e., preserve its form) under the transformations in question,

Let us expand this expression into a series in powers of the small parameter ε → 0. Taking into account that the leading term of the expansion (1) is zero, using (3) and (6), and retaining only the first-order terms, we obtain

where X 2 is the twice prolonged operator,

X 2 = ξ i ( x, w)

+ ζ(x, w)

Relation (11) is called the invariance condition. Remark. The invariant I 0 , which is a solution of equation (5), also satisfies the equation X 2 I 0 = 0.

S.7.1-2. Group analysis of second-order nonlinear equations in two independent variables. Consider the second-order equation in two independent variables

∂x ∂y ∂x ∂x∂y

In this case, the infinitesimal operator (4) has the form

X= ξ(x, y, w)

+ η(x, y, w)

+ ζ(x, y, w)

where we have used the notation ξ=ξ 1 and η=ξ 2 .

The coordinates of the first prolongation are given by

ζ 1 = D x ( ζ) − w x D x ( ξ) − w y D x ( η), ζ 2 = D y ( ζ) − w x D y ( ξ) − w y D y ( η),

which, after suitable calculations, become ζ 1 = ζ x +( ζ w − ξ x ) w x − η x w y − ξ w w 2 x − η w w x w y ,

2 (13) ζ 2 = ζ y − ξ y w x +( ζ w − η y ) w y − ξ w w x w y − η w w y .

The coordinates of the second prolongation are expressed as

ζ 11 = D x ( ζ 1 )− w xx D x ( ξ) − w xy D x ( η), ζ 12 = D y ( ζ 1 )− w xx D y ( ξ) − w xy D y ( η),

ζ 22 = D y ( ζ 2 )− w xy D y ( ξ) − w yy D y ( η),

or, after calculations, ζ 11 = ζ xx + (2

ξ ww w x w y η ww w x w y (14) −( ξ y + ξ w w y ) w xx +( ζ w − ξ x − η y −2 ξ w w x −2 η w w y ) w xy −( η x + η w w x ) w yy ,

ζ 2 ww ξ wx η wy w x w y η wx w

ζ 22 = ζ yy − ξ yy w x + (2 ζ wy − η yy ) w y −2 ξ wy w x w y +( ζ ww −2 η wy ) w 2 y

− ξ w x w 2 ww y − η ww w 3 y − 2( ξ y + ξ w w y ) w xy +( ζ w −2 η y − ξ w w x −3 η w w y ) w yy . The invariance condition (11) for equation (12) reads

∂w xy and in the expressions (13) and (14) of the coordinates of the first and second prolongations, ζ i and ζ ij ,

H, in accordance with equation (12). The resulting equation can be rewritten as a polynomial in the “independent variables” represented by the remaining derivatives ( w x , w y , w xx , and w xy in our case):

the derivative ∂ 2 ∂y w 2 should be replaced by the function

1 2 3 4 k 1 2 3 A 4 k k k k ( w x ) ( w y ) k ( w xx ) k ( w xy ) k = 0,

(16) where the functional coefficients A k 1 k 2 k 3 k 4 depend only on x, y, w, ξ, η, ζ and the derivatives of the

functions ξ, η, ζ and are independent of the derivatives of w. Relation (16) holds if all A k 1 k 2 k 3 k 4 = 0. Thus, the invariance condition is split and can be rewritten as an overdetermined determining system, which is obtained by equating to zero the functional coefficients of the “independent variables” represented by the remaining derivatives w x , w y , w xx , w xy , of which the unknown functions ξ, η, ζ are independent.

with respect to the desired quantities ξ, η, ζ. Below we illustrate the above procedure by examples. Example 1. Consider the two-dimensional stationary heat equation with a nonlinear source

which corresponds to the right-hand side

H = f (w) − w xx of equation (12).

Let us insert H = f (w) − w xx into the invariance condition (15), taking into account the expressions (13) and (14) for the coordinates of the first and second prolongations. Now, replacing w yy by

f (w) − w xx [a consequence of equation (17)] and equating the coefficients of the remaining derivatives to zero, we obtain the following system:

ζ xx + ζ yy − f ′ ( w)ζ + f (w)(ζ w −2 η y ) = 0.

Here, the first column contains combinations of derivatives and the second column contains the corresponding coefficients (up to constant factors); the coefficients of w y w xy , w x w xy , w 3 x , w 2 x w y , w x w 2 y , and w 3 y are omitted, since these coincide with some of the equations of the system or are their differential consequences. Using the first, the second, and the fifth equations, we find that ξ = ξ(x, y), η = η(x, y), ζ = aw + b(x, y), and a = const. Ultimately, the system becomes

ξ x − η y = 0, ξ y + η x = 0,

b xx + b yy − awf ′ ( w) − bf ′ ( w) + f (w)(a − 2η y ) = 0.

Obviously, for an arbitrary function f , we have a = b = η y = 0, and therefore, ξ=C 1 y+C 2 , η = −C 1 x+C 3 , and ζ = 0. Successively, taking one of the constants equal to unity and the others equal to zero, we find that the original equation admits three operators

X 1 = ∂ x , X 2 = ∂ y , X 3 = y∂ x − x∂ y .

The first two operators correspond to all possible translations along the axes x and y, and the third operator corresponds to a rotation.

Consider more closely the third equation of system (18). If

( aw + b)f ′ ( w) − f (w)(a − 2η y ) = 0,

then there may exist other solutions of system (18) which lead to operators other than (19). We should investigate two cases: a ≠ 0 and a = 0.

Case 1 . Solving equation (20) for

a ≠ 0, we get

1− 2 f (w) = C(aw + b) γ a ,

where γ=η y = const and b = const. Therefore, for f (w) = w k equation (17) admits an additional operator

which describes nonuniform scaling. Case 2 . For

a = 0, the solution has the form

f (w) = Ce λw ,

where λ = const. Then b = −2η y /λ and the functions ξ and η satisfy the first two equations in (18), which coincide with the Cauchy–Riemann equations for analytic functions. These conditions hold for the real and the imaginary parts of any analytic function

f (z) = ξ(x, y) + iη(x, y) of the complex variable z = x + iy. In particular, for b = const and f (w) = e w , the following additional operator is admitted:

X 4 = x∂ x + y∂ y −2 ∂ w ,

which corresponds to scaling in x and y combined with a translation in w.

The invariance condition is obtained by applying the operator X 2 = ξ∂ x + η∂ t + ζ∂ w + ζ 1 ∂ w x + ζ 2 ∂ w t + ζ 11 ∂ w xx to the equation

w 2 t − f (w)w xx − f ′ ( w)(w x ) = 0.

Using the expressions (13) and (14) for the coordinates of the first and the second prolongations ζ 1 and ζ 11 for y = t, and replacing w t in the invariance condition by the right-hand side of equation (21), let us equate to zero the coefficients of different powers of the remaining derivatives. We obtain the following system:

f (w)(2ξ x − η t ) = 0,

w x w xt :

f (w)η w = 0,

w xt :

f (w)η x = 0,

f ′ ( w)η w +

f (w)η ww = 0,

2[ f ′ (

w)] 2 η x +

f (w)ξ ww + f ′ ( w)ξ w +2

f (w)f ′ ( w)η wx = 0,

f (w)f ′ ( w)η ww = 0, w x :

f (w)ζ ww + f ′′ ( w)ζ − 2f (w)ξ wx − f ′ ( w)(2ξ x − η t )+ f ′ ( w)ζ w −

2 f (w)ζ wx +2 f ′ ( w)ζ x −

f (w)ξ xx + ξ t = 0,

f (w)ζ xx = 0.

Here, the first column lists combinations of derivatives and the second column contains the corresponding functional coefficients (up to a constant factor); identical expressions and those obtained by differentiation are omitted. Since

f (w) ✽ 0, the third and the fourth equations of the system imply that η = η(t). Then, from the first and the second equations we have

f (w)(2ξ − η

ξ = ξ(x, t),

f ′ ( w)

Taking into account the relations obtained above, we can rewrite the system in the form

(the equations have been divided by common factors which are always nonzero). In the general case, for arbitrary

f (w), the first equation implies that 2 ξ x − η t = 0, and the second equation implies that ξ t = 0. From the third equation, we get

ξ=C 1 + C 2 x, and therefore, η = 2C 2 t+C 3 . It follows that for arbitrary

f (w), equation (21) admits three operators:

X 3 =2 t∂ t + x∂ x .

Likewise, it can be shown that for the following specific f there arise additional operators:

1. f=e w : X 4 = x∂ x +2 ∂ w . 2. f=w k , k ≠ 0, −4/3: X 4 = kx∂ x +2 w∂ w . 3. f=w −4 /3 : X 4 =2 x∂ x −3 w∂ w , X 5 = x 2 ∂ x −3 xw∂ w .

Example 3. Consider the nonlinear wave equation

Let us use the invariance condition (15) for y = t and H = f (w)w xx + f ′ ( w)(w x ) 2 . We substitute the expressions (13) and (14) of the coordinates of the first and the second prolongations, at y = t, and replace w tt in the invariance condition by the right-hand side of equation (22), and then equate the coefficients of different powers of the remaining derivatives to zero. Thus, we obtain the following system (identical expressions and those obtained by differentiation are omitted):

w x w xx :

f (w)ξ w = 0,

w t w xx :

f (w)η w = 0,

w xx :

f ′ ( w)ζ + 2f (w)(η t − ξ x ) = 0,

w xt :

f (w)η x − ξ t = 0,

f ′ ( w)ξ w +

f (w)ξ ww = 0,

f (w)η ww − f ′ ( w)η w = 0,

w 2 x : f (w)ζ ww + f ′ ( w)ζ w + f ′′ ( w)ζ − 2f (w)ξ wx −2 f ′ ( w)(ξ x − η t ) = 0, w x w t :

2 f ′ ( w)η x +2 f (w)η wx −2 ξ wt = 0,

2 f ′ ( w)ζ x −

f (w)ξ xx +2 f (w)ζ wx + ξ tt = 0,

ζ ww −2 η wt = 0,

f (w)η xx +2 ζ wt − η tt = 0,

ζ tt −

f (w)ζ xx = 0.

the form ζ ww = 0 and we obtain the expression ζ = a(x, t)w + b(x, t). As a result, there remain the following equations of the system:

wf ′ ( w)a(x, y) + f ′ ( w)b(x, y) + 2f (w)(η t − ξ x ) = 0,

f ′ ( w)a(x, y) + wf ′′ ( w)a(x, y) + f ′′ ( w)b(x, y) − 2f ′ ( w)(ξ x − η t ) = 0,

2 f ′ ( w)(a x w+b x )−

f (w)ξ xx +2 f (w)a x = 0,

2 a t − η tt = 0, a tt w+b tt −

f (w)(a xx w+b xx ) = 0.

For an arbitrary function f (w), we obtain a = b = 0, η tt = 0, and ξ x − η t = 0. The integration yields three operators:

X 3 = x∂ x + t∂ t .

Likewise, it can be shown that for the following specific f , there are additional operators:

1. f=e w : X 4 = x∂ x +2 ∂ w .

2. f=w k , k ≠ 0, −4/3, −4: X 4 = kx∂ x +2

S.7.1-3. Finding exact solutions with the help of an admissible group. Invariant solutions.

1 ◦ . Suppose that we know a solution w of an equation under investigation. Then every admissible group generates a one-parameter family of solutions, namely the orbit T w, except for the case in which the solution is transformed into itself under the action of the group transformations (see Item 2 ◦ ).

2 ◦ . A solution w = w(x, y) of equation (12) is called invariant under a group G if the corresponding orbit T w is an invariant set. Let

G be a one-parameter group admitted by equation (12) and let I 1 = I 1 ( x, y) and I 2 = I 2 ( x, y, w)

be two functionally independent invariants of the group G. Invariant solutions are sought in the form

(23) where Φ is a function to be determined. Solving (23) for w and substituting the result into (12), we

I 2 = Φ( I 1 ),

obtain an ordinary differential equation for the function Φ.

A well-known and very important special class of invariant solutions is represented by self- similar solutions which are constructed on the basis of invariants of extension groups. For the sake of illustration, the general scheme of the construction of invariant solutions of second-order evolution equations is represented in Figure 3. Here, we omit the first-order partial differential equation for the determination of the group invariants (because we can proceed directly to the corresponding characteristic system of ordinary differential equations).

Example 4. Again, consider the stationary heat equation with a nonlinear source

f (w).

∂x

∂y

1 ◦ . Let us examine the case f=w k , in which the equation admits an additional operator (see Example 1):

In order to find invariants of this operator, one should consider the linear first-order partial differential equation X 4 I=0 which can be written out in complete form as

The corresponding characteristic system of ordinary differential equations

admits the first integrals