3 which corresponds to the infinitesimal transformation 15.8.1.2, is called an infinitesimal operator.* THEOREMLIE.. In this book, we call an operator 15.8.1.3 an infinitesimal operator
Trang 1Remark In the special case of transformations in the plane, the functions ϕ1 and ϕ2 in (15.8.1.1) are
independent of w, and ψ = w (i.e., w¯= w).
The expansion of (15.8.1.1) into truncated Taylor series in ε about ε = 0 to linear terms gives
¯x x + εξ(x, y, w), ¯y y + εη(x, y, w), w ¯ w + εζ(x, y, w), (15 8 1 2 ) where
ξ (x, y, w) = ∂ϕ1
∂ε
ε=0, η (x, y, w) =
∂ϕ2
∂ε
ε=0, ζ (x, y, w) =
∂ψ
∂ε
ε=0.
The linear first-order differential operator
X = ξ(x, y, w) ∂
∂x + η(x, y, w) ∂
∂y + ζ(x, y, w) ∂
∂w , (15 8 1 3 )
which corresponds to the infinitesimal transformation (15.8.1.2), is called an infinitesimal
operator.*
THEOREM(LIE) Suppose the coordinates ξ(x, y, w), η(x, y, w), ζ(x, y, z) of the
in-finitesimal operator (15.8.1.3) are known Then the transformation (15.8.1.1), having the group property, can be completely recovered by solving the Lie equations
dϕ1
dε = ξ(ϕ1, ϕ2, ψ), dϕ2
dε = η(ϕ1, ϕ2, ψ), dψ
dε = ζ(ϕ1, ϕ2, ψ)
with the initial conditions
ϕ1|ε=0= x, ϕ2|ε=0= y, ψ |ε=0= w.
15.8.1-2 Invariant of an infinitesimal operator Transformations in the plane.
An invariant of the operator (15.8.1.3) is a function I(x, y, w) that satisfies the condition
I ( ¯x, ¯y, ¯ w ) = I(x, y, w).
Let us expand this equation in Taylor series in the small parameter ε, divide the resulting relation by ε, and the proceed to the limit as ε → 0 to obtain a linear partial differential
equation for I:
XI = ξ(x, y, w) ∂I
∂x + η(x, y, w) ∂I
∂y + ζ(x, y, w) ∂I
∂w = 0 (15 8 1 4 ) Let the associated characteristic system of ordinary differential equations (see Paragraph 13.1.1-3)
dx
ξ (x, y, w) =
dy
η (x, y, w) =
dw
ζ (x, y, w) (15. 8 1 5 )
* In the literature, it is also known as an infinitesimal generator or a group generator In this book, we call
an operator (15.8.1.3) an infinitesimal operator, a group generator, or, for short, an operator.
Trang 2TABLE 15.7 One-parameter transformations in the plane
Name Transformation Transformation Invariant
Translation in the x-axis ¯x = x + ε, ¯y = y X = ∂x ∂ I1= y
Translation along the straight line
ax + by =0 ¯x = x + bε, ¯y = y – aε X = b ∂x ∂ – a ∂y ∂ I1= ax + by
Rotation ¯x = x cos ε + y sin ε,
¯y = y cos ε – x sin ε X = y ∂x ∂ – x ∂y ∂ I1= x2+ y2
Galileo transformation ¯x = x + εy, ¯y = y X = y ∂x ∂ I1= y
Lorentz transformation ¯x = x cosh ε + y sinh ε,
¯y = y cosh ε + x sinh ε X = y ∂x ∂ + x ∂y ∂ I1= y2– x2
Uniform extension ¯x = xe ε, ¯y = ye ε X = x ∂x ∂ + y ∂y ∂ I1= y/x
Nonuniform extension ¯x = xe aε, ¯y = ye bε X = ax ∂x ∂ + by ∂y ∂ I1 =|y|a|x|–b
have the functionally independent integrals
I1(x, y, w) = C1, I2(x, y, w) = C2, (15. 8 1 6 )
where C1and C2are arbitrary constants Then the general solution of equation (15.8.1.4) has the from
I = Ψ(I1, I2), (15. 8 1 7 ) where Ψ(I1, I2) is an arbitrary function of two arguments, I1= I1(x, y, w) and I2= I2(x, y, w).
This means that the operator (15.8.1.3) has two functionally independent invariants,
I1and I2, and any function Φ(x, y, w) invariant under the operator (15.8.1.3) can be
repre-sented as a function of the two invariants.
Table 15.7 lists the most common transformations in the plane and the corresponding operator (15.8.1.3) and invariants; only one invariant is specified, with the other being the
same: I2= w.
15.8.1-3 Formulas for derivatives Coordinates of the first and second prolongations.
In the new variables (15.8.1.1), the first derivatives become
∂ w ¯
∂ ¯x
∂w
∂x + εζ1, ∂ w ¯
∂ ¯y
∂w
∂y + εζ2 (15 8 1 8 )
Here, ζ1and ζ2are the coordinates of the first prolongation, which are found as
ζ1= Dx(ζ) – wxDx(ξ) – wyDx(η) = ζx+ (ζw– ξx)wx– ηxwy– ξww2x– ηwwxwy,
ζ2= Dy(ζ) – wxDy(ξ) – wyDy(η) = ζy– ξywx+ (ζw– ηy)wy – ξwwxwy– ηww2y,
(15 8 1 9 )
where Dxand Dyare the total differential operators with respect to x and y:
Dx= ∂x ∂ + wx∂w ∂ + wxx∂w ∂
x + wxy
∂
∂wy + · · · ,
Dy = ∂y ∂ + wy∂w ∂ + wxy∂w ∂
y + wyy
∂
∂wy + · · ·
(15 8 1 10 )
Trang 3Let us verify that the first formula in (15.8.1.8) holds Obviously,
¯
w x=w¯x¯¯x x+w¯y¯¯y x, w¯y=w¯x¯¯x y+w¯y¯¯y y (15.8.1.11)
Differentiating (15.8.1.2) with respect to x and y and retaining terms to the first order of ε, we have
¯x x=1+ εD x , ¯x y = εD y ,
¯y x = εD x η, ¯y y=1+ εD y η,
¯
w x = w x + εD x , w¯y = w y + εD y
(15.8.1.12)
In order to calculatew¯x¯, let us eliminatew¯¯yfrom (15.8.1.11) and then substitute the derivatives ¯x x, ¯x y, ¯y x,
¯y y,w¯x,w¯yfor their respective expression from (15.8.1.12) to obtain
¯
w¯x= w x + ε(D x + w x D y η – w y D x η ) + ε
2(D x ζD y η – D x ηD y )
1+ ε(D x + D y η ) + ε2(D x ξD y η – D x ηD y ) .
Expanding into a series in ε, we have
¯
w x¯ w x + εζ1, ζ1= D x – w x D x – w y D x η,
which was to be proved The coordinate ζ2is calculated likewise
The second derivatives in the new variables (15.8.1.1) are calculated as
∂2w ¯
∂ ¯x2 ∂2w
∂x2 + εζ11,
∂2w ¯
∂ ¯x∂¯y
∂2w
∂x∂y + εζ12, ∂2w ¯
∂ ¯y2 ∂2w
∂y2 + εζ22. (15. 8 1 13 )
Here, the ζij are the coordinates of the second prolongation and are found as
ζ11= Dx(ζ1) – wxxDx(ξ) – wxyDx(η),
ζ12= Dy(ζ1) – wxxDy(ξ) – wxyDy(η),
ζ22= Dy(ζ2) – wxyDy(ξ) – wyyDy(η),
or, in detailed form,
ζ11 = ζxx+ ( 2ζwx– ξxx)wx– ηxxwy+ (ζww– 2ξwx)w2x– 2ηwxwxwy–
– ξwww3
x– ηwww2
xwy + (ζw– 2ξx– 3ξwwx– ηwwy)wxx– 2 (ηx+ ηwwx)wxy,
ζ12 = ζxy+ (ζwy– ξxy)wx+ (ζwx– ηxy)wy– ξwyw2
x–
– (ζww– ξwx– ηwy)wxwy– ηwxw2
y– ξwww2
xwy– ηwwwxw2
y–
– (ξy+ ξwwy)wxx+ (ζw– ξx– ηy– 2ξwwx– 2ηwwy)wxy– (ηx+ ηwwx)wyy,
ζ22 = ζyy– ξyywx+ ( 2ζwy– ηyy)wy– 2ξwywxwy+ (ζww– 2ηwy)w2y–
– ξwwwxw2y– ηwww3y– 2 (ξy+ ξwwy)wxy + (ζw– 2ηy– ξwwx– 3ηwwy)wyy.
(15 8 1 14 ) The above formulas for the coordinates of the first and second prolongation, (15.8.1.9) and (15.8.1.14), will be required later for the analysis of differential equations.
15.8.2 Symmetries of Nonlinear Second-Order Equations.
Invariance Condition
15.8.2-1 Invariance condition Splitting in derivatives.
We will consider second-order partial differential equations in two independent variables
F
x , y, w, ∂w
∂x , ∂w
∂y , ∂
2w
∂x2,
∂2w
∂x∂y , ∂
2w
∂y2
= 0 (15 8 2 1 )
Trang 4The procedure for finding symmetries* of equation (15.8.2.1) consists of several stages.
At the first stage, one requires that equation (15.8.2.1) must be invariant (preserve its form) under transformations (15.8.1.1), so that
F
¯x, ¯y, ¯ w , ∂ w ¯
∂ ¯x ,
∂ w ¯
∂ ¯y ,
∂2w ¯
∂ ¯x2,
∂2w ¯
∂ ¯x∂¯y ,
∂2w ¯
∂ ¯y2
= 0 (15 8 2 2 )
Let us expand this equation into a series in ε about ε = 0 , taking into account that the leading term vanishes, according to (15.8.2.1) Using formulas (15.8.1.2), (15.8.1.8), (15.8.1.13)
and retaining the terms to the first-order of ε, we obtain
X
2F
x , y, w, ∂w
∂x , ∂w
∂y , ∂
2w
∂x2,
∂2w
∂x∂y , ∂
2w
∂y2
F =0 = 0 , (15 8 2 3 ) where
X
2F = ξ ∂F
∂x +η ∂F
∂y +ζ ∂F
∂w +ζ1 ∂F
∂wx+ζ2
∂F
∂wy +ζ11
∂F
∂wxx+ζ12
∂F
∂wxy+ζ22
∂F
∂wyy (15. 8 2 4 )
The coordinates of the first, ζi, and the second, ζij, prolongation are defined by formulas
(15.8.1.9) and (15.8.1.14) Relation (15.8.2.3) is called the invariance condition, and the
operator X
2 is called the second prolongation of the operator (15.8.1.3).
At the second stage, either the derivative ∂ ∂y2w2 or ∂ ∂x2w2 is eliminated from (15.8.2.3) using equation (15.8.2.1) The resulting relation is then written as a polynomial in the “inde-pendent variables,” the various combinations of the products of the derivatives (involving
various powers of wx, wy, wxx, and wxy):
Ak1 k2k3k4(wx)k1(wy)k2(wxx)k3(wxy)k4 = 0 , (15 8 2 5 )
where the functional coefficients Ak1 k2 k3k4 are dependent on x, y, w, ξ, η, ζ and the deriva-tives of ξ, η, ζ only and are independent of the derivaderiva-tives of w Equation (15.8.2.5) is satisfied if all Ak1k2k3 k4 are zero Thus, the invariance condition is split to an overdeter-mined determining system, resulting from equating all functional coefficients of the various
products of the remaining derivatives to zero (recall that the unknowns ξ, η, and ζ are independent of wx, wy, wxx, and wxy).
At the third stage, one solves the determining system and finds admissible coordinates
ξ , η, and ζ of the infinitesimal operator (15.8.1.3).
Remark 1 It should be noted that the functional coefficients A k1k2k3k4and the determining system are
linear in the unknowns ξ, η, and ζ.
Remark 2 An invariant I that is a solution of equation (15.8.1.4) is also a solution of the equation X
2I=0
The procedure for finding symmetries of differential equations is illustrated below by specific examples.
15.8.2-2 Examples of finding symmetries of nonlinear equations.
Example 1 Consider the two-dimensional stationary heat equation with a nonlinear source
∂2w
∂x2 +∂
2w
The corresponding left-hand side of equation (15.8.2.1) is F = w xx + w yy – f (w).
* A symmetry of an equation is a transformation that preserves its form
Trang 5Infinitesimal operators X admitted by the equations are sought in the form (15.8.1.4), where the coordinates
ξ = ξ(x, y, w), η = η(x, y, w), ζ = ζ(x, y, w) are yet unknown and are to be determined in the subsequent analysis.
In view of identity F = w xx + w yy – f (w), the invariance condition (15.8.2.3)–(15.8.2.4) is written as
ζ22+ ζ11 – ζf (w) =0 Substituting here the expressions of the coordinates of the second prolongation (15.8.1.14) and then replacing
w yy by f (w) – w xx, which follows from equation (15.8.2.6), we obtain
–2ξ w w x w xx+2η w w y w xx–2η w w x w xy–2ξ w w y w xy–2(ξ x – η y )w xx–2(ξ y + η x )w xy–
– ξ ww w3x – η ww w x2w y – ξ ww w x w y2– η ww w3y + (ζ ww–2ξ xw )w x2–2(ξ yw + η xw )w x w y+
+ (ζ ww–2η yw )w2y+ (2ζ xw – ξ xx – ξ yy – f ξ w )w x+ (2ζ yw – η xx – η yy–3f η w )w y+
+ ζ xx + ζ yy + f (ζ w–2η y ) – ζf =0,
where f = f (w) and f = df /dw Equating the coefficients of all combinations of the derivatives to zero, we
have the system
w x w xx: ξ w=0,
w y w xx: η w=0,
w xx: ξ – η y=0,
w xy: ξ + η x=0,
w x2: ζ ww–2ξ wx=0,
w x w y: η wx + ξ wy=0,
w x: 2ζ wx – ξ xx – ξ yy – ξ w f (w) =0,
w2y: ζ ww–2η wy=0,
w y: 2ζ wy – η xx – η yy–3η w f (w) =0,
1: ζ xx + ζ yy – f (w)ζ + f (w)(ζ w–2η y) =0
(15.8.2.7)
Here, the left column indicates a combination of the derivatives and the right column gives the associated
coefficient The coefficients of w y w xy , w x w xy , w3x , w x2w y , w x w2y , w3yeither are among those already listed
or are their differential consequences, and therefore they are omitted It follows from the first, second, and fifth equations and their consequences that
ξ = ξ(x, y), η = η(x, y), ζ = a(x, y)w + b(x, y). (15.8.2.8) The third and fourth equations of system (15.8.2.7) give
ξ xx + ξ yy=0, η xx + η yy=0 (15.8.2.9) Substituting (15.8.2.8) into the seventh and ninth equations of (15.8.2.7) and using (15.8.2.9), we find that
a x = a y=0, whence
In view of (15.8.2.8) and (15.8.2.10), system (15.8.2.7) becomes
ξ – η y=0,
ξ + η x=0,
b xx + b yy – awf (w) – bf (w) + f (w)(a –2η y) =0
(15.8.2.11)
For arbitrary f , it follows that a = b = η y=0, and then ξ = C1 y + C2, η = –C1 x + C3, and ζ =0 By setting one
of the constants to unity and the others to zero, we establish that the original equation admits three different operators:
X1= ∂ x (C2=1, C1= C3=0);
X2= ∂ y (C3=1, C1= C2=0);
X3= y∂ x – x∂ y (C1=1, C2 = C3=0)
(15.8.2.12)
The first two operators define a translation along the x- and y-axis The third operator represents a rotational
symmetry
Let us dwell on the third equation of system (15.8.2.11) If the relation
(aw + b)f (w) – f (w)(a –2η y) =0 (15.8.2.13)
Trang 6holds, there may exist other solutions of system (15.8.2.11) that will result in operators other than (15.8.2.12).
Let us study two cases: a≠ 0and a =0
Case 1 If a≠ 0, the solution of equation (15.8.2.13) gives
f (w) = C(aw + b)1 2a γ,
where γ = η y = const and b = const Therefore, for f (w) = w k, equation (15.8.2.6) admits another operator,
X4= x∂ x + y∂ y+ 2
1– k w∂ w,
that defines a nonuniform extension
Case 2 If a =0, we have
f (w) = Ce λw,
where λ = const Then b = –2η y /λ , and the functions ξ and η satisfy the first two equations (15.8.2.11), which
coincide with the Cauchy–Riemann conditions for analytic functions The real and the imaginary part of any analytic function,Φ(z) = ξ(x, y) + iη(x, y), of the complex variable z = x + iy satisfies the Cauchy–Riemann conditions In particular, if b = const and f (w) = e w, the equation admits another operator,
X4= x∂ x + y∂ y–2∂ w,
which corresponds to extension in x and y with simultaneous translation in w.
Example 2 Consider the nonlinear nonstationary heat equation
∂w
∂t = ∂
∂x
*
f (w) ∂w
∂x
+
In the invariance condition (15.8.2.3)–(15.8.2.4), one should set
y = t, F = w t – f (w)w xx – f (w)w x2, ζ12= ζ22=0
The coordinates of the first and second prolongations, ζ1, ζ2, and ζ11, are expressed by (15.8.1.9) and (15.8.1.14)
with y = t In the resulting equation, one should replace w twith the right-hand side of equation (15.8.2.14) and equate the coefficients of the various combinations of the remaining derivatives to zero, thus arriving at the system of equations
w x w xx: 2f (w)[η wx f (w) + ξ w ] + f (w)η x=0,
w xx: ζf (w) – f2(w)η xx – f (w)(2ξ – η t) =0,
w x w xt: f (w)η w=0,
w xt: f (w)η x=0,
w4x: f (w)η w + f (w)η ww=0,
w3x: 2[f (w)]2η x + f (w)ξ ww + f (w)ξ w+2f (w)f (w)η wx=0,
w2x: f (w)ζ ww + f (w)ζ –2f (w)ξ wx – f (w)(2ξ – η t ) + f (w)ζ w – f (w)f (w)η ww=0,
w x: 2f (w)ζ wx+2f (w)ζ x – f (w)ξ xx + ξ t=0,
1: ζ t – f (w)ζ xx=0
Here, the left column indicates a combination of the derivatives, and the right column gives the associated
equation (to a constant factor); identities and differential consequences are omitted Since f (w)0, it follows
from the third and fourth equations of the system that η = η(t) Then from the first and second equations, we
have
ξ = ξ(x, t), ζ= f (w)(2ξ – η t)
f (w) .
With these relations, the remaining equations of the system become
[f f f – f (f )2+ (f )2f ](2ξ – η t) =0,
f[4f f –7(f )2]ξ xx – (f )2ξ t=0,
2f ξ xxx–2ξ xt + η tt=0;
Trang 7the equations have been canceled by nonzero factors In the general case, for arbitrary function f , from the
first equation it follows that2ξ – η t=0and from the second it follows that ξ t=0 The third equation gives
ξ = C1 + C2 x , and then η =2C2t + C3 Therefore, for arbitrary f , equation (15.8.2.14) admits three operators
X1= ∂ x (C1=1, C2 = C3=0);
X2= ∂ t (C3=1, C1= C2=0);
X3=2t∂ t + x∂ x (C2=1, C1 = C3=0)
Similarly, it can be shown that the following special forms of f result in additional operators:
1 f = e w: X4= x∂ x+2∂ w;
2 f = w k , k≠ 0, –4/3: X4= kx∂ x+2w∂ w;
3 f = w–4/3: X4=2x∂ x–3w∂ w, X5 = x2∂ x–3xw∂ w
Example 3 Consider now the nonlinear wave equation
∂2w
∂t2 = ∂
∂x
*
f (w) ∂w
∂x
+
In the invariance condition (15.8.2.3)–(15.8.2.4), one should set
y = t, F = w tt – f (w)w xx – f (w)w2x, ζ2= ζ12=0,
and use the coordinates of the first and second prolongations, ζ1 , ζ11, and ζ22, expressed by (15.8.1.9) and (15.8.1.14) with y = t In the resulting equation, one should replace w ttwith the right-hand side of equation (15.8.2.15) and equate the coefficients of the various combinations of the remaining derivatives to zero, thus arriving at the system of equations
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,
w3x: f (w)ξ w + f (w)ξ ww =0,
w2x w t: f (w)η ww – f (w)η w=0,
w2x: f (w)ζ ww + f (w)ζ w + f (w)ζ –2f (w)ξ wx–2f (w)(ξ x – η t) =0,
w x w t: 2f (w)η x+2f (w)η wx–2ξ wt =0,
w x: 2f (w)ζ x – f (w)ξ xx+2f (w)ζ wx + ξ tt=0,
w t2: ζ ww–2η wt =0,
w t: f (w)η xx+2ζ wt – η tt=0,
1: ζ tt – f (w)ζ xx=0
Identities and differential consequences have been omitted Since f (w)≠const, it follows from the first two
equations that ξ = ξ(x, t) and η = η(x, t) The tenth equation of the system becomes ζ ww=0, thus giving
ζ = a(x, t)w + b(x, t) As a result, the system becomes
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,
2f (w)(a x w + b x ) – f (w)ξ xx+2f (w)a x=0,
2a t – η tt=0,
a tt w + b tt – f (w)(a xx w + b xx) =0
For arbitrary function f (w), we have a = b =0, ξ xx=0, η tt=0, and ξ x – η t =0 Integrating yields three operators:
X1= ∂ x, X2= ∂ t, X3= x∂ x + t∂ t