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where the Galilei operator has the form
1
Ga = t?xa ? (8)
xa u?u .
2?
For operator (8), the function u is changed as follows:
t(?a )2
xa ?a
u > u = u exp ? ? (9)
,
2? 4?
Symmetry of equations with convection terms 173

Thus, the operators Qa and Ga are essentially di?erent representations of the Galilei
operator.
Let us now investigate the symmetry of systems including equation (2) and addi-
tional conditions for the potentials. Note that in [3], the authors ?nd a nontrivial
symmetry of the nonlinear Fokker–Planck equation by imposing the additional condi-
tions for coe?cient functions.
Let the additional conditions for the potentials Vk be the Euler equations. In other
words, consider the following system:
?u ?u
? ? u = Vk ,
?t ?xk
(10)
?Vk ?Vk
? ?1 Vl = 0, k = 1, n.
?t ?xl
Symmetry of the nonlinear system (10) essentially depends on the value of the
parameter ?1 . There are two di?erent cases.
The ?rst case. ?1 = 1. In this case, system (10) in the class of operators (3) is
invariant under the Lie algebra with the basis operators
Jab = xa ?xb ? xb ?xa + Va ?Vb ? Vb ?Va ,
P 0 = ?t , Pa = ?xa ,
Ga = t?xa ? ?Va , D = 2t?t + xk ?xk ? Vk ?Vk , (11)
A = t2 ?t + txk ?xk ? (xk + tVk )?Vk , Z1 = u?u , Z2 = ?u .
The Galilei operator Ga generates the following ?nite transformations:
t > t = t,
xb > xb = xb + t?b ?ab ,
(12)
Vb > V b = Vb ? ?b ?ab ,
u > u = u,
where we mean summation from 1 to n over the repeated index b.
Conclusion 1. Thus, the scalar function u, unlike the heat equation, is not changed
under the Galilei transformations.
The second case. ?1 = 1. In this case, the invariance algebra of system (10) is
essentially more restricted and does not include the Galilei operator and the projective
one. In other words, for ?1 = 1 in the class of operators (3), system (10) is invariant
under the Lie algebra with basis elements P0 , Pa , Jab , D, Z1 , Z2 of the form (11).
The ?rst case is essentially more interesting and important that the second one.
Therefore, in what follows, we consider system (10) in the case where ?1 = 1.
Consider now system (10), where the Euler equations have the right-hand sides of
?u
the form F (u) ?xk , i.e., the following nonlinear system:
?u ?u
? ? u = Vk ,
?t ?xk
(13)
?Vk ?Vk ?u
? Vl = F (u) , k = 1, n,
?t ?xl ?xk
where F (u) is a smooth function of u. Let us carry out symmetry classi?cation of
system (13), i.e., determine all classes of functions F (u), which admit a nontrivial
symmetry of system (13). We consider the following six cases:
174 W.I. Fushchych, Z.I. Symenoh

Case 1. F (u) is an arbitrary smooth function. System (13) is invariant under the
Galilei algebra
(14)
AG(1, n) = P0 , Pa , Jab , Ga ,
where the basis operators have the form (11).
Case 2. F = C exp(?u) (? and C are arbitrary constants, ? = 0, C = 0). In this
case, the symmetry of system (13) is more extended and includes algebra (14) and
the dilation operator
2
D(1) = 2t?t + xk ?xk ? Vk ?Vk ? ?u .
?
Case 3. F = Cu? (? and C are arbitrary constants, ? = 0, ? = 1, C = 0). In
this case, system (13) is invariant under the extended Galilei algebra (14) with the
dilation operator
2
D(2) = 2t?t + xk ?xk ? Vk ?Vk ? u?u .
?+1
C
Case 4. F = (C is an arbitrary constant, C = 0). The maximal invariance
u
algebra is
P0 , Pa , Jab , Ga , Z1 ,
where Z1 = u?u .
Case 5. F = C (C is an arbitrary constant, C = 0). The maximal invariance
algebra is
P0 , Pa , Jab , Ga , D(2) , Z2 ,
where Z2 = ?u . In this case, the dilation operator D(2) has the form
D(2) = 2t?t + xk ?xk ? Vk ?Vk ? 2u?u .
Case 6. F = 0. In this case, system (13) admits the widest invariance algebra,
namely,
P0 , Pa , Jab , Ga , D, A, Z1 , Z2 ,
where the dilation operator D and the projective operator A have the form (11).
Conclusion 2. It is important that system (13) is invariant under the Galilei transfor-
mations for an arbitrary smooth function F (u). It should be stressed once more that,
unlike the standard heat equation, the function u is not changed under the Galilei
transformations.
Consider other examples of systems of the equation of convection di?usion and
additional conditions for the potentials Vk .
Let the functions Vk satisfy the heat equation, i.e., we investigate the following
system:
?u ?u
? ? u = Vk ,
?t ?xk (15)
?Vk
? ?1 Vk = 0, k = 1, n,
?t
where ?1 = 0 is an arbitrary real parameter.
Symmetry of equations with convection terms 175

Theorem 2. System (14) in the class of operators (3) is invariant under the Lie
algebra with the basis operators
P0 , Pa , Jab , D, Z1 , Z2
of the form (11).
The case where the functions Vk satisfy the Laplace equation is more important:
?u ?u
? ? u = Vk ,
?t ?xk (16)
Vk = 0, k = 1, n.
Theorem 3. System of equations (16) in the class of operators (3) is invariant under
the in?nite-dimensional Lie algebra with the basis operators
QA , Qkl , Qa , Z1 , Z2
of the form (5).
Note that the symmetry of system (16) is the same as the symmetry of equa-
tion (2). In other words, the conditions Vk = 0 do not contract the symmetry of the
equation of convection di?usion.


2 The Schr?dinger equation with convection term
o
Consider the Schr?dinger equation with convection term
o
?? ??
(17)
i + ??? = Vk ,
?t ?xk
where ? = ?(t, x) and Vk = Vk (t, x) (k = 1, n) are complex functions. For exten-
sion of symmetry, we regard the functions Vk as dependent variables. Note that the
requirement that the functions Vk are complex is essential for the symmetry of (17).
Let us investigate the symmetry of (17) in the class of ?rst-order di?erential
operators
X = ? µ ?xµ + ??? + ? ? ??? + ?k ?Vk + ??k ?Vk , (18)
?



where ? µ , ?, ? ? , ?k , ??k are functions of t, x, ?, ? ? , V , V ? .
Theorem 4. Equation (17) is invariant under the in?nite-dimensional Lie algebra
with the in?nitesimal operators
QA = 2A?t + Axr ?xr ? iAxr (?Vr ? ?Vr? ) ? A(Vr ?Vr + Vr? ?Vr? ),
? ? ?
Qkl = Bkl (xl ?xk ? xk ?xl + Vl ?Vk ? Vk ?Vl + Vl? ?Vk ? Vk ?Vl? ) ?
?
?

?
? iBkl (xl ?V ? xk ?V ? xl ?V ? + xk ?V ? ), (19)
k l k l
?
Qa = U a ?xa ? iU a (?Va ? ?Va ),
?


Z1 = ??? , Z2 = ? ? ??? , Z3 = ?? , Z4 = ??? ,

where A, B kl (k < l, k, l = 1, n) , U a (a = 1, n) are arbitrary smooth functions of t,
B kl = ?B lk , we mean summation over the index r and no summation over indices
a, k, and l.
176 W.I. Fushchych, Z.I. Symenoh

This theorem is proved by the standard Lie algorithm in the class of operators (18).
Note that algebra (19) includes as a particular case the Galilei operator of the
form:

Ga = t?xa ? i?Va + i?Va . (20)
?



This operator generates the following ?nite transformations:
xb > xb = xb + ?b t?ab ,
t > t = t,
? > ? = ?, ? ? > ? ? = ? ? ,
Vb > Vb = Vb ? i?b ?ab , Vb? > Vb? = Vb? + i?b ?ab ,
where ?b is an arbitrary real parameter and we mean summation from 1 to n over the
repeated index b. Note that the wave function ? is not changed for these transforma-
tions. Operator (20) is essentially di?erent from the standard Galilei operator
i
xa (??? ? ? ? ??? ). (21)
Ga = t?xa +
2?
of the free Schr?dinger equation (Vk = 0). Note that we cannot derive operator (21)
o
from algebra (19). Thus, we have two essentially di?erent representations of the Galilei
operator: (20) for the Schr?dinger equation with convection term and (21) for the free
o
Schr?dinger equation.
o
Remark 2. If we assume that the functions Vk are real in equation (17) and study
symmetry in the class of operators

X = ? µ ?xµ + ??? + ? ? ??? + ?a ?Va , (22)

where the unknown functions ? µ , ?, ? ? , ?a depend on t, x, ?, ? ? , V , then the maxi-
mal invariance algebra of equation (17) is su?ciently restricted. Namely, in the class
of operators (22), equation (17) is invariant under the Lie algebra with the basis
operators
P0 , Pa , Jab = xa ?xb ? xb ?xa + Va ?Vb ? Vb ?Va ,
D = 2t?t + xr ?xr ? Vr ?Vr , Z1 = ??? , Z2 = ? ? ??? , Z3 = ?? , Z4 = ??? .

Thus, in the case of real functions Vk , equation (17) is not invariant under the Galilei
transformations.
Consider now the system of equation (17) with the additional condition for the
potentials Vk , namely, the complex Euler equations:
?? ??
i + ??? = Vk ,
?t ?xk
(23)
?Vk ?Vk ??
? Vl
i = F (|?|) .
?t ?xl ?xk
Here, ? and Vk are complex dependent variables of t and x, F is a smooth function
of |?|. The coe?cients of the second equation of (23) provide the broad symmetry of
this system.
Symmetry of equations with convection terms 177

Let us investigate symmetry classi?cation of system (23). Consider the following
?ve cases.
Case 1. F is an arbitrary smooth function. The maximal invariance algebra is
P0 , Pa , Jab , Ga , where

Jab = xa ?xb ? xb ?xa + Va ?Vb ? Vb ?Va + Va ?Vb? ? Vb? ?Va ,
? ?


Ga = t?xa ? i?Va + i?Va .
?



Case 2. F = C|?|k (C is an arbitrary complex constant, C = 0, k is an arbitrary
real number, k = 0 and k = ?1). The maximal invariance algebra is P0 , Pa , Jab , Ga ,
D(1) , where
2
D(1) = 2t?t + xr ?xr ? Vr ?Vr ? Vr? ?Vr? ? (??? + ? ? ??? ).
1+k
C
Case 3. F = (C is an arbitrary complex constant, C = 0). The maximal
|?|
invariance algebra is P0 , Pa , Jab , Ga , Z = Z1 + Z2 , where

Z = ??? + ? ? ??? , Z2 = ? ? ??? .
Z1 = ??? ,

Case 4. F = C = 0 (C is an arbitrary complex constant). The maximal invariance
algebra is P0 , Pa , Jab , Ga , D(1) , Z3 , Z4 , where

Z3 = ?? , Z4 = ??? .

Case 5. F = 0. The maximal invariance algebra is P0 , Pa , Jab , Ga , D, A, Z1 , Z2 , Z3 ,
Z4 , where

D = 2t?t + xr ?xr ? Vr ?Vr ? Vr? ?Vr? ,
A = t2 ?t + txr ?xr ? (ixr + tVr )?Vr + (ixr ? tVr? )?Vr? .

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. 39
( 70 .)



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