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4. Symmetries of the nonlinear di¬usion equation
The (3 + 1)-nonlinear di¬usion equation is given by
∆(up+1 ) + kuq = ut , (1.62)
where u = u(x, y, z, t), ∆ = ‚ 2 /‚x2 + ‚ 2 /‚y 2 + ‚ 2 /‚z 2 , p, k, q ∈ Q, and
p = ’1.
We shall state the results for the Lie algebras of symmetries for all
distinct values of p, k, q.
First of all we derived that there are no contact symmetries, i.e., the
coe¬cients of any symmetry V ,
‚ ‚ ‚ ‚ ‚
V =Vx +Vy +Vz +Vt +Vu ,
‚x ‚y ‚z ‚t ‚u
V x , V y , V z , V t , V u depend on x, y, z, t, u only.
Remark 1.12. Such symmetries are called point symmetries contrary
to general contact symmetries whose coe¬cients at ‚/‚xi and ‚/‚u may
depend on coordinates in J 1 (π) (see Theorem 1.12 (ii)).
Secondly, for any value of p, k, q, equation (1.62) admits the following
seven symmetries:
‚ ‚ ‚ ‚
V1 = , V2 = , V3 = , V4 = ,
‚x ‚y ‚z ‚t
‚ ‚ ‚ ‚ ‚ ‚
’ x , V6 = z ’ x , V7 = z ’y .
V5 = y (1.63)
‚x ‚y ‚x ‚z ‚y ‚z
4. SYMMETRIES OF THE NONLINEAR DIFFUSION EQUATION 35

We now summarize the ¬nal results, while the complete Lie algebras are
given for all the cases that should be distinguished.

4.1. Case 1: p = 0, k = 0. The complete Lie algebra of symmetries of
the equation
∆(u) = ut (1.64)
is spanned by the vector ¬elds V1 , . . . , V7 given in (1.63) and

V8 = u ,
‚u
‚ ‚
’ xu ,
V9 = 2t
‚x ‚u
‚ ‚
’ yu ,
V10 = 2t
‚y ‚u
‚ ‚
’ zu ,
V11 = 2t
‚z ‚u
‚ ‚ ‚ ‚
V12 = x +y +z + 2t ,
‚x ‚y ‚z ‚t
‚ ‚ ‚ ‚ 1 ‚
+ t2 + u(’x2 ’ y 2 ’ z 2 ’ 6t)
V13 = xt + yt + zt (1.65)
‚x ‚y ‚z ‚t 4 ‚u
together with the continuous part F (x, y, z, t)‚/‚u, where F (x, y, z, t) is an
arbitrary function which has to satisfy (1.64). In fact, all linear equations
possess symmetries of this type.

4.2. Case 2: p = 0, k = 0, q = 1. The complete Lie algebra of
symmetries of the equation
∆(u) + ku = ut (1.66)
is spanned by the ¬elds V1 , . . . , V7 given in (1.63) and

V8 = u ,
‚u
‚ ‚
’ xu ,
V9 = 2t
‚x ‚u
‚ ‚
’ yu ,
V10 = 2t
‚y ‚u
‚ ‚
’ zu ,
V11 = 2t
‚z ‚u
‚ ‚ ‚ ‚ ‚
V12 = x +y +z + 2t + 2kut ,
‚x ‚y ‚z ‚t ‚u
‚ ‚ ‚ ‚ 1 ‚
+ t2 + u(4kt2 ’ x2 ’ y 2 ’ z 2 ’ 6t) .
V13 = xt + yt + zt
‚x ‚y ‚z ‚t 4 ‚u
(1.67)
36 1. CLASSICAL SYMMETRIES

Since (1.66) is a linear equation, it also possesses symmetries of the form
F (x, y, z, t)‚/‚u, where
∆(F ) + kF = Ft . (1.68)
4.3. Case 3: p = 0, k = 0, q = 1. The complete Lie algebra of
symmetries of the equation
∆(u) + kuq = ut (1.69)
is spanned by V1 , . . . , V7 given in (1.63) and the ¬eld
‚ ‚ ‚ ‚ 2 ‚
+ 2t ’
V8 = x +y +z u. (1.70)
‚t q ’ 1 ‚u
‚x ‚y ‚z
4.4. Case 4: p = ’4/5, k = 0. The complete Lie algebra of symmetries
of
∆(u1/5 ) = ut (1.71)
is spanned by V1 , . . . , V7 given in (1.63) together with the ¬elds
‚ ‚
V8 = 4t + 5u ,
‚t ‚u
‚ ‚ ‚ ‚
’ 5u ,
V9 = 2x + 2y + 2z
‚x ‚y ‚z ‚u
‚ ‚ ‚ ‚
V10 = (x2 ’ y 2 ’ z 2 ) ’ 5xu ,
+ 2xy + 2xz
‚x ‚y ‚z ‚u
‚ ‚ ‚ ‚
+ (’x2 + y 2 ’ z 2 ) ’ 5yu ,
V11 = 2xy + 2yz
‚x ‚y ‚z ‚u
‚ ‚ ‚ ‚
+ (’x2 ’ y 2 + z 2 ) ’ 5zu .
V12 = 2xz + 2yz (1.72)
‚x ‚y ‚z ‚u
4.5. Case 5: p = ’4/5, p = 0, k = 0. The complete Lie algebra of
symmetries of the equation
∆(up+1 ) = ut (1.73)
is spanned by V1 , . . . , V7 given in (1.63) and two additional vector ¬elds
‚ ‚
V8 = ’pt +u ,
‚t ‚u
‚ ‚ ‚ ‚
V9 = px + py + pz + 2u . (1.74)
‚x ‚y ‚z ‚u
4.6. Case 6: p = ’4/5, k = 0, q = 1. The complete Lie algebra of
symmetries of the equation
∆(u1/5 ) + ku = ut (1.75)
is spanned by V1 , . . . , V7 given in (1.63) and
‚ 4kt ‚
4kt
V8 = e + kue 5 ,
5
‚t ‚u
5. THE NONLINEAR DIRAC EQUATIONS 37

‚ ‚ ‚ ‚
’ 5u ,
V9 = 2x + 2y + 2z
‚x ‚y ‚z ‚u
‚ ‚ ‚ ‚
= (x2 ’ y 2 ’ z 2 ) ’ 5xu ,
V10 + 2xy + 2xz
‚x ‚y ‚z ‚u
‚ ‚ ‚ ‚
+ (’x2 + y 2 ’ z 2 ) ’ 5yu ,
V11 = 2xy + 2yz
‚x ‚y ‚z ‚u
‚ ‚ ‚ ‚
+ (’x2 ’ y 2 + z 2 ) ’ 5zu .
V12 = 2xz + 2yz (1.76)
‚x ‚y ‚z ‚u
4.7. Case 7: p = 0, p = ’4/5, k = 0, q = 1. The complete Lie algebra
of symmetries of the equation
∆(up+1 ) + ku = ut (1.77)
is spanned by V1 , . . . , V7 given in (1.63) and by
‚ ‚
V8 = e’pkt + 4ku ,
‚t ‚u
‚ ‚ ‚ ‚
V9 = px + py + pz + 2u . (1.78)
‚x ‚y ‚z ‚u
4.8. Case 8: p = 0, p = ’4/5, q = p + 1. The complete Lie algebra of
symmetries of the equation
∆(up+1 ) + kup+1 = ut (1.79)
is spanned by V1 , . . . , V7 given in (1.63) and by the ¬eld
‚ ‚
’u .
V8 = pt (1.80)
‚t ‚u
4.9. Case 9: p = 0, p = ’4/5, q = 1, q = p + 1. The complete Lie
algebra of symmetries of the equation
∆(up+1 ) + kuq = ut (1.81)
is spanned by V1 , . . . , V7 given in (1.63) and by the ¬eld
‚ ‚ ‚ ‚ ‚
V8 = (’p + q ’ 1) x + 2(q ’ 1)t ’ 2u .
+y +z (1.82)
‚x ‚y ‚z ‚t ‚u
The results in these nine cases are a generalization of the results of
other authors [13]. We leave to the reader to describe the corresponding Lie
algebra structures in the cases above.

5. The nonlinear Dirac equations
In this section, we consider the nonlinear Dirac equations and compute
their classical symmetries [33]. Symmetry classi¬cation of these equations
leads to four di¬erent cases: linear Dirac equations with vanishing and non-
vanishing rest mass, nonlinear Dirac equation with vanishing rest mass, and
38 1. CLASSICAL SYMMETRIES

general nonlinear Dirac equation (with nonvanishing rest mass). We con-
tinue to study the last case in the next chapter (Subsection 2.2) and compute
there conservation laws associated to some symmetries.
We shall only give here a short idea of the solution procedure, since all
computations follow to standard lines. The Dirac equations are of the form
[11]:
3
‚(γk ψ) ‚(γ4 ψ)
’i + m0 cψ + n0 ψ(ψψ) = 0, (1.83)
‚xk ‚x4
k=1
where
x4 = ct,
ψ = (ψ1 , ψ2 , ψ3 , ψ4 )T ,
— — — —
ψ = (ψ1 , ψ2 , ’ψ3 , ’ψ4 ), (1.84)
T stands for transposition, — is complex conjugate and γ1 , γ2 , γ3 , γ4 are
4 — 4-matrices de¬ned by
«  « 
0 0 0 ’i 0 0 0 ’1
¬0 0 ’i 0 · ¬0 0 1 0·
¬ ·, γ2 = ¬ ·
γ1 =   0 1 0 0 ,
0
0i 0
’1 0 0 0
i00 0
«  « 
0 0 ’i 0 10 0 0
¬0 0 0 i· ¬0 1 0 0·
γ3 = ¬ ·, γ4 = ¬ ·
0 0 ’1 0  . (1.85)
i 0 0 0
0 ’i 0 0 0 0 0 ’1
After introduction of the parameter

»= , (1.86)
m0 c
we obtain
3
‚ ‚
(γ4 ψ) + ψ + »3 ψ(ψψ) = 0.
(γk ψ) ’ »i
» (1.87)
‚xk ‚x4
k=1
In computation of the symmetry algebra of (1.87) we have to distinguish
the following cases:
1. = 0, »’1 = 0: Dirac equations with vanishing rest mass,
2. = 0, »’1 = 0: Dirac equations with nonvanishing rest mass,
3. = 0, »’1 = 0: nonlinear Dirac equations with vanishing rest mass,
4. = 0, »’1 = 0: nonlinear Dirac equations.
These cases are equivalent to the respective choices of m0 and n0 in (1.83):
e.g., = 0, »’1 = 0 is the same as m0 = n0 = 0, etc.
We put ψj = uj + iv j , j = 1, . . . , 4, and obtain a system of eight coupled
partial di¬erential equations
»v1 ’ »u4 + »v3 + »v4 + (1 + »3 K)u1 = 0,
4 3 1
2
5. THE NONLINEAR DIRAC EQUATIONS 39

»v1 + »u3 ’ »v3 + »v4 + (1 + »3 K)u2 = 0,
3 4 2
2
’»v1 + »u2 ’ »v3 ’ »v4 + (1 + »3 K)u3 = 0,
2 1 3
2
’»v1 ’ »u1 + »v3 ’ »v4 + (1 + »3 K)u4 = 0,
1 2 4
2
’»u4 ’ »v2 ’ »u3 ’ »u1 + (1 + »3 K)v 1 = 0,
4
1 3 4
’»u3 + »v2 + »u4 ’ »u2 + (1 + »3 K)v 2 = 0,
3
1 3 4
»u2 + »v2 + »u1 + »u3 + (1 + »3 K)v 3 = 0,
2
1 3 4
»u1 ’ »v2 ’ »u2 + »u4 + (1 + »3 K)v 4 = 0,
1
(1.88)
1 3 4
where
‚uj ‚v j
uj = j
, vk = , j, k = 1, . . . , 4,
k ‚xk ‚xk
and
K = (u1 )2 + (u2 )2 ’ (u3 )2 ’ (u4 )2 + (v 1 )2 + (v 2 )2 ’ (v 3 )2 ’ (v 4 )2 . (1.89)
Thus (1.87) is a determined system E ‚ J 1 (π) in the trivial bundle π : R8 —
R4 ’ R 4 .
Using relations (1.34) and symmetry conditions (1.37), we construct the
overdetermined system of partial di¬erential equations for the coe¬cients of
the vector ¬eld V
‚ ‚ 1‚ 4‚
V = F x1 + · · · + F x4 + Fu + ··· + Fv . (1.90)
‚u1 ‚v 4
‚x1 ‚x4
From the resulting overdetermined system of partial di¬erential equations
we derive in a straightforward way the following intermediate result:
F x1 , . . . , F x4 are independent of u1 , . . . , v 4 ,
1:
F x1 , . . . , F x4 are polynomials of degree 3 in x1 , . . . , x4 ,
2:
1 4
Fu ,...,Fv are linear with respect to u1 , . . . , v 4 .
3: (1.91)
Combination of this intermediate result (1.91) with the remaining system of
partial di¬erential equations leads to the following description of symmetry
algebras in the four speci¬c cases.
= 0, »’1 = 0. The complete Lie algebra of classical
5.1. Case 1:
symmetries for the Dirac equations with vanishing rest mass is spanned by 23
generators. In addition, there is a continuous part generated by functions
1 4
F u , . . . , F v dependent on x1 , . . . , x4 and satisfying the Dirac equations
(1.88) due to the linearity of these equations. The Lie algebra contains the
¬fteen in¬nitesimal generators of the conformal group X1 , . . . , X15 and eight
vertical vector ¬elds X16 , . . . , X23 :
‚ ‚ ‚ ‚
X1 = , X2 = , X3 = , X4 = ,
‚x1 ‚x2 ‚x3 ‚x4
‚ ‚ ‚ ‚ ‚ ‚
’ v1 1 + v2 2 ’ v3 3 + v4 4
’ 2x1
X5 = 2x2
‚x1 ‚x2 ‚u ‚u ‚u ‚u
40 1. CLASSICAL SYMMETRIES

‚ ‚ ‚ ‚
+ u1 ’ u2 2 + u3 3 ’ u4 4 ,
‚v 1 ‚v ‚v ‚v
‚ ‚ ‚ ‚ ‚ ‚
’ u2 1 + u1 2 ’ u4 3 + u3 4
’ 2x1
X6 = 2x3
‚x1 ‚x3 ‚u ‚u ‚u ‚u
‚ ‚ ‚ ‚
’ v2 1 + v1 2 ’ v4 3 + v3 4 ,
‚v ‚v ‚v ‚v
‚ ‚ ‚ ‚ ‚ ‚
+ v2 1 + v1 2 + v4 3 + v3 4
= ’2x3
X7 + 2x2
‚x2 ‚x3 ‚u ‚u ‚u ‚u
‚ ‚ ‚ ‚
’ u2 1 ’ u1 2 ’ u4 3 ’ u3 4 ,
‚v ‚v ‚v ‚v
‚ ‚ ‚ ‚ ‚ ‚
+ u4 1 + u3 2 + u2 3 + u1 4
X8 = 2x4 + 2x1
‚x1 ‚x4 ‚u ‚u ‚u ‚u
‚ ‚ ‚ ‚
+ v4 1 + v3 2 + v2 3 + v1 4 ,
‚v ‚v ‚v ‚v
‚ ‚ ‚ ‚ ‚ ‚
+ v4 1 ’ v3 2 + v2 3 ’ v1 4
X9 = 2x4 + 2x2
‚x2 ‚x4 ‚u ‚u ‚u ‚u
‚ ‚ ‚ ‚
’ u4 1 + u3 2 ’ u2 3 + u1 4 ,
‚v ‚v ‚v ‚v
‚ ‚ ‚ ‚ ‚ ‚
+ u3 1 ’ u4 2 + u1 3 ’ u2 4
X10 = 2x4 + 2x3
‚x3 ‚x4 ‚u ‚u ‚u ‚u
‚ ‚ ‚ ‚
+ v3 1 ’ v4 2 + v1 3 ’ v2 4 ,
‚v ‚v ‚v ‚v
‚ ‚ ‚ ‚
X11 = x1 + x2 + x3 + x4 ,
‚x1 ‚x2 ‚x3 ‚x4
‚ ‚ ‚ ‚
= (x2 ’ x2 ’ x2 + x2 )
X12 + 2x1 x2 + 2x1 x3 + 2x1 x4
1 2 3 4
‚x1 ‚x2 ‚x3 ‚x4

’ (3x1 u1 ’ x2 v 1 ’ x3 u2 ’ x4 u4 ) 1
‚u

’ (3x1 u2 + x2 v 2 + x3 u1 ’ x4 u3 ) 2
‚u

’ (3x1 u3 ’ x2 v 3 ’ x3 u4 ’ x4 u2 ) 3
‚u

’ (3x1 u4 + x2 v 4 + x3 u3 ’ x4 u1 ) 4
‚u

’ (3x1 v 1 + x2 u1 ’ x3 v 2 ’ x4 v 4 ) 1
‚v

’ (3x1 v 2 ’ x2 u2 + x3 v 1 ’ x4 v 3 ) 2
‚v

’ (3x1 v 3 + x2 u3 ’ x3 v 4 ’ x4 v 2 ) 3
‚v

’ (3x1 v 4 ’ x2 u4 + x3 v 3 ’ x4 v 1 ) 4 ,
‚v
5. THE NONLINEAR DIRAC EQUATIONS 41

‚ ‚ ‚ ‚
’ (x2 ’ x2 + x2 ’ x2 )
X13 = 2x1 x2 + 2x2 x3 + 2x2 x4
1 2 3 4
‚x1 ‚x2 ‚x3 ‚x4

’ (3x2 u1 + x1 v 1 ’ x3 v 2 ’ x4 v 4 ) 1
‚u

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