Symbolic Computation of Equivalence Transformations and Parameter Reduction for Nonlinear Physical Models

0.3 MB·

by  Alexei F. Cheviakov

Symbolic Computation of Equivalence Transformations and
Parameter Reduction for Nonlinear Physical Models
Alexei F. Cheviakov†,
Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, S7N 5E6 Canada
7 January 20, 2017
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0
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n Abstract
a
J An efficient systematic procedure is provided for symbolic computation of Lie groups of
equivalence transformations and generalized equivalence transformations of systems of differ-
8
1 ential equations that contain arbitrary elements (arbitrary functions and/or arbitrary constant
parameters), using the software package GeM for Maple. Application of equivalence transfor-

] mations to the reduction of the number of arbitrary elements in a given system of equations
h
p is discussed, and several examples are considered. First computational example of a general-
- izedequivalencetransformationwherethetransformationofthedependentvariableinvolvesthe
h
arbitrary constitutive function is presented.
t
a As a detailed physical example, a three-parameter family of nonlinear wave equations de-
m
scribing finite anti-plane shear displacements of an incompressible hyperelasic fiber-reinforced
[ medium is considered. Equivalence transformations are computed and employed to radically

simplify the model for an arbitrary fiber direction, invertibly reducing the model to a simple
1
v form that corresponds to a special fiber direction, and involves no arbitrary elements.
8 The presented computation algorithm is applicable to wide classes of systems of differential
5 equations containing arbitrary elements.
2
5
0 1 Introduction
.
1
0
Themajorityofcontemporarymathematicalmodels,inparticular,physicalmodelswritteninterms
7
1 ofpartialorordinarydifferentialequations (PDEs,ODEs), involve constantorvariableparameters.
: These parameters, also called arbitrary elements, appear in the equations in the form of arbitrary
v
i constants that assume values in certain ranges, and/or arbitrary functions belonging to certain
X
classes. Equivalentformulationsofamodelathand,involvingfewerconstitutiveparametersand/or
r
a reduced forms of constitutive functions, can lead to a significant simplification of the analysis of
the model, especially when such analysis involves classifications. Such equivalent formulations can
be systematically sought using equivalence transformations.
Equivalence transformations of afamily of DE (differential equation) systems are transformations
of problem variables and arbitrary elements that map every DE system from the given family into
another system from the same family. As a basic illustration, consider a set of (1+1)-dimensional
nonlinear diffusion equations for u(x,t), given by
u =c2(u)u , (1.1)
t xx
where c(u) is an arbitrary constitutive function. (In (1.1) and throughout the paper where appro-
priate, subscripts denote derivatives.) The family (1.1) admits, for example, an obvious discrete
†Corresponding author. Electronic mail: [email protected]
1
equivalence transformation c∗(u) = c(u); in particular, any solution u(x,t) of (1.1) is mapped a
−
solution of a PDE u = (c∗(u))2u . Scalings and translations are the simplest kinds of continu-
tt xx
ous equivalence transformations. In particular, transformations that re-scale variables using their
typical values, and thus map a problem into a dimensionless one, or scaling transformations in
general, have a long history, and are commonly used to simplify and non-dimensionalize DE prob-
lems. For example, the PDE family (1.1) clearly admits scaling and translation-type equivalence
transformations
A2
x∗ = A x+A , t∗ = A t+A , u∗(x∗,t∗)= A u(x,t), c∗(u∗)= 3c(u), (1.2)
3 1 4 2 5
A
4
A ,...,A R, A > 0, so that a solution u(x,t) of a PDE (1.1) with a constitutive function c(u)
1 5 4
∈
is mapped into a solution u∗(x∗,t∗) of a PDE
u∗ = (c∗(u∗))2u∗ ,
t∗ x∗x∗
a member of the family (1.1) with a different constitutive function, as long as A2 = A .
3 6 4
The notion of equivalence transformations is closely related to that of symmetries of a system of
differential equations. A symmetry is a transformation that leaves invariant the solution manifold
of a given system, i.e., maps any solution of the system into a solution of the same system. Due to
the general topological nature of this definition, a systematic way to calculate all symmetries of a
given DE system, and similarly, all equivalence transformations of a given family of DE systems,
is not available. However, equivalence transformations of particular kinds, such as Lie groups of
local and nonlocal equivalence transformations, can indeed be computed systematically.
Work on equivalence transformations was initiated in [1]. Various examples, extensions, and
applications of the notion of equivalence transformations appear in the literature, such as [2–8] and
references therein. An excellent review and work on equivalence transformations, including both
theoretical and computational aspects, is contained in the thesis [3] (see also [9]).
The current manuscript presents an algorithmic procedure to calculate Lie groups of local equiv-
alence transformations of a given DE system using a symbolic symmetry and conservation law
computation package GeM for Maple [10–12] developed by the author. A brief review of the notion
of equivalence transformations, relations between Lie groups of point equivalence transformations
and Lie point symmetries, and extended classes of equivalence transformations, is given in Section
2. The Section also contains several illustrative examples, and ends with an outline of a systematic
procedure of computation of Lie groups of generalized equivalence transformations, closely related
to the algorithm of point symmetry computation. Section 3 contains a typical sequence of steps
for complete symbolic computation of Lie groups of generalized equivalence transformations using
the GeM Maple package.
Section 4 is devoted to three examples of symbolic computation of equivalence transformations
of PDE models. Complete detailed Maple code is included for each example. In the first example,
equivalence transformations are computed for the well-known three-parameter family of dimen-
sional Korteweg-de Vries (KdV) equations, and are used to invertibly reduce the family to a single
canonical KdV equation involving no arbitrary parameters. The second example is devoted to the
calculation of nonlocal equivalence transformations of a family of nonlinear wave equations. In
particular, nonlocal equivalence transformations are computed as point transformations of a po-
tential system, and do not correspond to local equivalence transformations of the model. The final
example in Section 4 is a Lie group of generalized equivalence transformations of a PDE system,
where the transformation component corresponding to the dependent variable involves an explicit
dependance on an arbitrary element that is an arbitrary function. This is the first computational
example of this situation in the literature (see Remark 2.10).
Another example considered in Section 5 presents a recent model of finite anti-plane shear dis-
placementsinnonlinearincompressiblehyperelasicfiber-reinforcedhyperelasticmaterials[13]. Such
2
models arise in the description of multiple types of nonlinear media, including biological mem-
branes [14]. The systematically computed local equivalence transformations are used to drastically
simplify the model, by invertible transformations, through the complete elimination of arbitrary
elements. Together with the classical example of the Korteweg-de Vries model in Section 4, the
example of Section 5 addresses the questions of primary importance for the analysis of nonlinear
equations arising in physics and applied science – the identification of essential parameters of the
model, and the possibility of invertible parameter reduction.
Section 6 concludes the paper with a discussion of the presented algorithms, their efficiency, re-
sults,relatedclassificationproblems,andanoverviewofopenquestionsandpossibleworkdirections
in the area.
Throughout the paper, summation in repeated indices is assumed where appropriate.
2 Equivalence Transformations, their Computation, and Exten-
sions
2.1 Lie Groups of Equivalence Transformations
Consider a family of DE systems R x;u;K :
K
F { }
Rσ(x,u,∂u,...,∂ku,K)= 0, σ = 1,...,N, (2.1)
involvingnindependentvariablesx = (x1,...,xn),mdependentvariablesu(x) = (u1(x),...,um(x)),
and L arbitrary elements (constitutive functions and/or parameters) K =(K1,..., KL). In (2.1),
partial derivatives are denoted by uµ = ∂uµ(x)/∂xi. The symbol
i
∂pu = uµ µ = 1,...,m; i ,...,i = 1,...,n
i1...ip | 1 p
n o
∂puµ(x)
= µ = 1,...,m;i ,...,i = 1,...,n
(cid:26)∂xi1...∂xip | 1 p (cid:27)
is used to denote the set of all partial derivatives of order p, p = 1,2,.... The arbitrary elements
K in (2.1) may be either arbitrary constants or arbitrary functions within some class. Constitutive
functionsareassumedtobesufficientlysmoothfunctionsofdependentand/orindependentvariables
and/or derivatives of the dependent variables, or combinations thereof.
In general, equivalence transformations for a DE family (2.1) can be defined as follows.
K
F
Definition 2.1. An equivalence transformation for a DE family is a change of variables and
K
F
arbitrary elements (x,u,K) (x∗,u∗,K∗) which maps every DE system R x;u;K into a
K
→ { } ∈ F
DE system R x∗;u∗;K∗ .
K
{ } ∈ F
Definition 2.1 is rather general, and there is no systematic way to compute explicit forms of
transformations described therein. For the practical purposes, the following definition describing a
special subclass of equivalence transformations is often used [1,3].
Definition 2.2. A one-parameter Lie group of point equivalence transformations of a family of
K
F
DE systems (2.1) is a one-parameter Lie group of transformations acting in the space of (x,u,K),
given by
(x∗)i = fi(x,u;ε) = xi +εξi(x,u)+O(ε2), i = 1,...,n,
(u∗)µ = gµ(x,u;ε) = uµ+εηµ(x,u)+O(ε2), µ = 1,...,m, (2.2)
(K∗)ℓ = Gℓ(Qℓ;ε) = Kℓ+εκℓ(Q)+O(ε2), ℓ = 1,...,L,
which maps a PDE system R x;u;K into another PDE system R x∗;u∗;K∗ in the same
K
{ } ∈ F { }
family.
3
Informulas(2.2),theargumentsQℓofthecorrespondingtransformationGℓ andthetangentvector
field component κℓ dependon the nature of the arbitrary element Kℓ, ℓ = 1,...,L. In particular, if
Kℓ is a constant parameter, Qℓ may be the set of all constant parameters of the DE system family
(2.1). If Kℓ is a arbitrary (constitutive) function, Qℓ may also involve independent/dependent
variables Kℓ depends on, other constitutive functions with compatible dependencies, etc. Within
Definition 2.2, transformations (2.2) form a group on the (x,u) space; the transformations for x,u
do not involve arbitrary elements.
Remark 2.3. The notion of a Lie group of equivalence transformations (2.2) of a DE system is
intimately related to that of a Lie group of point symmetries; the appropriate notation, including
that for infinitesimal generators, and further details, can be found in the following Section 2.2.
Multiple examples of equivalence transformations are known in the literature that go beyond the
Definition 2.2. Some of such examples are considered in Section 2.3 below.
Example 2.4. The transformations (1.2) involve five one-parameter groups of equivalence trans-
formations. Indeed, they can be written in the global group form as
x∗ = eε3x+ε , t∗ = eε4t+ε , u∗(x∗,t∗) = eε5u(x,t), c∗(u∗) = eε3−ε4/2c(u), (2.3)
1 2
with group parameters ε , i = 1,...,5. The corresponding infinitesimal generators are given by
i
∂ ∂ ∂ ∂ ∂ 1 ∂ ∂
X = , X = , X = x +c , X = t c , X = u . (2.4)
1 2 3 4 5
∂x ∂t ∂x ∂c ∂t − 2 ∂c ∂u
Example 2.5 (Equivalence transformations and parameter reduction). As a second example, con-
sider a classical family of “dimensional Korteweg-de Vries” equations for the unknown function
u(x,t), given by
u +au +buu +qu = 0, (2.5)
t x x xxx
involving three constant parameters a,b,q R, b,q = 0. The model (2.5) arises in multiple
∈ 6
physical contexts [15], for example, it describes the evolution of the fluid surface elevation u(x,t)
in a limit considering small-amplitude long waves in a shallow channel. We now recollect the well-
known equivalence transformations of the PDE family (2.5). (An example of symbolic software
computations of these equivalence transformations is given below in Section 4.1.)
A basic set of equivalence transformations of the PDE family (2.5) is given by infinitesimal
generators
∂ ∂ ∂ ∂ ∂ ∂
Y = , Y = , Y = b , Y = t + ,
1 2 3 4
∂x ∂t ∂u − ∂a ∂x ∂a
∂ ∂ ∂ ∂ ∂ ∂ ∂
Y = x +t +2q , Y = x +3t 2u 2a , (2.6)
5 6
∂x ∂t ∂q ∂x ∂t − ∂u − ∂a
∂ ∂ ∂ ∂
Y = x +3t 2a 2b .
7
∂x ∂t − ∂a − ∂b
The corresponding point transformations can be written, for example, as
A A
x∗ = 5 (x A t)+A , t∗ = 5 t+A , u∗(x∗,t∗) = A2(u(x,t)+A ),
A A − 4 1 A3A3 2 6 3
6 7 6 7 (2.7)
a∗ = A2A2(a A b A ), b∗ = A2b, q∗ = A2q.
6 7 − 3 − 4 7 5
with seven arbitrary constants A ,...,A R.
1 7
∈
4
The equivalence transformations (2.7) can be used to invertibly map the PDE (2.5) into one of
the standard forms of the KdV equation, for example,
u +uu +u = 0, (2.8)
t x xxx
thus reducing the number of constant parameters from three to zero. To do this, one removes the
term au (a = 0), and re-scales the variables. In common KdV derivations, one of two common
x
6
waysisusedtoremovethetermau . Thiscanbeachieved bypassingtoatravelingwavecoordinate
x
x∗ = x at, t∗ = t, u∗(x∗,t∗) = u(x,t), that is, using the transformation (2.7) with
−
A = A = A = 0, A = a, A = A = A = 1.
1 2 3 4 5 6 7
Another way consists in a pure shift of the dependent variable, i.e., the transformation (2.7) with
A = A = A = 0, A = a/b, A = A = A = 1, (2.9)
1 2 4 3 5 6 7
leading to the choice of variables x∗ = x, t∗ = t, u∗(x∗,t∗) = u(x,t) + A . Both of the above
3
choices, as well as any other combined transformation leading to a∗ a A b A = 0 in (2.7),
3 4
∝ − −
yield an equivalent PDE from the family (2.5), given by (after the omission of the asterisks)
u +buu +qu = 0. (2.10)
t x xxx
Due to the freedom in changing the signs of x and u, i.e., in the choice A , A , A = 1 in
5 6 7
±
(2.7), one can assume without loss of generality that in (2.10), b,q > 0. Finally, using again the
transformations (2.7) with
A = A = A = A = 0, A = q−1/2, A = 1, A = b−1/2,
1 2 3 4 5 6 7
one transforms the PDE (2.10) into the standard form (2.8).
2.2 Equivalence Transformations and Point Symmetries
Liegroupsofequivalencetransformations(2.2)areclosely related toLiegroupsofpointsymmetries
of differential equations (e.g., [1,2]). Let R x;u given by (2.1) be a specific DE system
K
{ } ∈ F
within the indicated family, for a fixed set of constitutive quantities K.
Consider a one-parameter Lie group of point transformations
(x∗)i = fi(x,u;ε) = xi +εξi(x,u)+O(ε2), i =1,...,n,
(2.11)
(u∗)µ = gµ(x,u;ε) = uµ+εηµ(x,u)+O(ε2), µ = 1,...,m,
with the infinitesimal generator
∂ ∂
X= ξi(x,u) +ηµ(x,u) , (2.12)
∂xi ∂uµ
and the k-th prolongation of (2.12) is given by
∂ ∂ ∂
X(k) = ξi(x,u) +ηµ(x,u) +η(1)µ(x,u,∂u)
∂xi ∂uµ i ∂uµ
i (2.13)
∂
+...+η(k)µ (x,u,∂u,...,∂ku) .
i1...ik ∂uµ
i1...ik
(1)µ (k)µ
Here the extended infinitesimals η , ..., η are defined through the formulas
i i1...ik
η(1)µ = D ηµ (D ξj)uµ, η(k)µ = D η(k−1)µ (D ξj)uµ , (2.14)
i i − i j i1...ik ik i1...ik−1 − ik i1...ik−1j
for µ = 1,...,m, and i,i = 1,...,n for p = 1,...,k (e.g., [2,16,17]).
p
5
Definition 2.6. A one-parameter Lie group of point transformations (2.11) acting on the inde-
pendent and dependent variables of the DE system R x;u (2.1) is a one-parameter Lie group of
{ }
point symmetries of this system if and only if its kth extension (2.13) leaves invariant the solution
manifold of R x;u in (x,u,∂u,...,∂ku)-space.
{ }
The notion of a point symmetry naturally extends to include higher-order local symmetries, non-
local symmetries, approximate symmetries, and other related objects (see, e.g., [2,10,12,18–20]).
Thesymmetry structureof aDE system contains essential information abouttheproperties of that
system, and thus can vary greatly. For example, differential equations exist that have no point
symmetries, a several-parameter group of point symmetries, or an infinite number of point symme-
tries. Infinite-dimensional groups of point symmetries arise when the symmetry components ξi,ηµ
in (2.12) involve arbitrary functions. Such symmetries are particularly useful for the construction
of families of new exact solutions from known ones, as well as other applications. Moreover, for
a nonlinear DE system with an infinite point (or nonlocal) symmetry group, one can determine
whether or not this system can be mapped into a linear system by an invertible (or, respectively,
a non-invertible) transformation, and find an explicit form of such a transformation ( [2,18] and
references therein).
Remark 2.7. For families of DE systems, classifications of their analytical properties with re-
spect to constitutive elements are of high theoretical and practical importance. Examples of such
problems include local and nonlocal symmetry and conservation law classifications. Classification
tables are usually presented modulo known equivalence transformations, i.e., only for forms of
constitutive functions and/or parameters that are not related by an equivalence transformation.
Such classifications are available for multiple families of equations of mathematical physics; see, for
instance, [2,19].
Equivalence transformations can be used directly to simplify symmetry determining equations
during symbolic computations. A symbolic software package SymmetryClassification for Maple
developed by Huang and Lisle includes a detEqsForEquiv routine which computes rather a re-
strictedclassofpointequivalencetransformations,namely,oneswherecomponentsforeachvariable
orarbitraryelementdependonlyonthatspecificvariableorarbitraryelement. Areducedsimplified
version of the rifsimp-generated symmetry classification tree is consequently obtained [9,21]. The
work contains detailed examples of symmetryclassification forafamily ofnonlinear heatequations,
and a family of nonlinear convection-diffusion (Richards) equations.
Remark 2.8. The following obvious relationships hold between the group of equivalence transfor-
mations of a DE family (2.1) and the groups of point symmetries of its members:
A point symmetry (2.11) of a DE system R x;u is an equivalence transformation of
K
• { } ∈ F
the family (2.1) if (2.11) is point symmetry for all systems in .
K K
F F
An equivalence transformation (2.2) of the DE family (2.1) is a point symmetry of its
K
• F
every member if and only if it does not involve components corresponding to the arbitrary
elements of the family.
In practice, one-parameter Lie group of point symmetries are found from the following invariance
condition (e.g., [1,2,16,17]).
Theorem 2.9. Consider a one-parameter Lie group of point transformations (2.11), with the
corresponding infinitesimal generator (2.12) and its kth extension (2.13). Then the transformation
(2.11) is a point symmetry of a nondegenerate DE system R x;u (2.1) if and only if for each
{ }
α = 1,...,N,
X(k)Rα(x,u,∂u,...,∂ku) = 0, (2.15)
6
whenever
Rσ(x,u,∂u,...,∂ku)= 0, σ = 1,...,N. (2.16)
The computation of point symmetries from the determining equations (2.15), (2.16) proceeds
through the Lie’s algorithm:
1. Obtain the determining equations by evaluating (2.15) on the solution manifold of the given
PDE system, i.e., through the substitution of the DEs (2.16) and their differential conse-
quences into the invariance condition (2.15).
2. Obtain the split system of determining equations, using the fact that the symmetry compo-
nents ξi,ηµ do not depend on the derivatives of u: in the determining equations, set to zero
all coefficients at all independent combinations of derivatives of dependent variables.
The application of the Lie’s algorithm and its extensions to nontrivial DE systems often requires
extensive algebraic manipulation. Indeed, split symmetry determining equations are linear overde-
termined PDE systems for the unknown symmetry components ξi,ηµ; in practice, such systems
may involve from a dozen to hundreds or thousands of equations. To treat such huge systems,
symbolic computer software is commonly used (see, e.g., [10,12]). In particular, methods based
on differential Gro¨bner bases and the characteristic set method are successfully used in computer
algebra packages [3,22–24]. A review of relevant work and symbolic software can be found in [2]
(Section 5.1).
In practical computations, groups of equivalence transformations (2.2) of DE families are system-
atically sought usingthe above Lie’s algorithm, similarly to findingLie groups of point symmetries,
with arbitrary elements treated as dependent variables, and the dependence of the transformation
components restricted accordingly. Details are given in Section 2.4 below.
2.3 Extensions of the Notion of Equivalence Transformations
Definition 2.2 can be extended in several different ways. Details and further examples of such
extensions can be found in [3,4,7,25,26] and references therein, and are out of scope of this
paper. We briefly review several generalizations of equivalence transformations that are relevant
to systematic symbolic computations.
Ageneralapproachtotheextensionofthenotionofone-parameterLiegroupsofpointequivalence
transformations (2.2) can be formulated in terms of seeking transformations of the form
(x∗)i = fi[x,u,K], i = 1,...,n,
(u∗)µ = gµ[x,u,K], µ = 1,...,m, (2.17)
(K∗)ℓ = Gℓ[x,u,K], ℓ = 1,...,L,
that map PDE systems in the family into PDE systems in the same family. Generally, trans-
K
F
formations (2.17) may be neither point, nor local transformations; the bracket notation in (2.17)
denotes possible local or nonlocal dependencies of the transformation components on the indepen-
dent and dependent variables and arbitrary elements, as well as, possibly, on derivatives of the
dependent variables.
The following specific instance of general transformations (2.17) have been considered in the
literature.
(A) Generalized equivalence transformations. An extension of one-parameter Lie groups of
point equivalence transformations (2.2), in particular, Lie groups with the corresponding infinites-
imal generators
∂ ∂ ∂
X= ξi(x,u,K) +ηµ(x,u,K) +ζν(x,u,K) , (2.18)
∂xi ∂uµ ∂Kν
7
were defined in [4] (cf. [3]), and are referred to as groups of “generalized equivalence transforma-
tions”. The essential difference here is that the transformation components for the independent
and/or dependent variables ξi(x,u,K), ηµ(x,u,K) may additionally depend on arbitrary elements
K. Examples of such transformations are known in the literature. For example, they were com-
puted in [27] for a class of nonlinear (1+1)-dimensional Schro¨dinger equations with a potential
V(x,t) and a power nonlinearity γ = const given by
iψ +ψ + ψ γψ+V(x,t)ψ = 0,
t xx
| |
For an infinite-parameter group of generalized equivalence transformations derived in that work,
transformations for the dependent variable ψ(x,t) depend on the arbitrary constant parameter γ.
Remark 2.10. It is also worth mentioning that Lie groups of generalized equivalence transfor-
mations (2.18) where the components ξi(x,u,K) and/or ηµ(x,u,K) of the independent and/or
dependent variables would depend on arbitrary functions present in the given system, rater than
arbitrary constant parameters, reliable examples of such verified examples have not appeared in
the literature been available to this day. Despite multiple re-citations, the example given in the
original work [4] does not seem to be correct, as can be verified by a direct substitution. The first
and only example of the indicated kind the author is aware of is due to Opanasenko, Bihlo and
Popovych, appearing in their yet unpublished work [28]. Consider the KdV-Burgers PDE family
for the unknown function u(x,t) given by
r
u +Cuu = A u +B, (2.19)
t x k k
Xk=0
where u ∂ku/∂xk, k = 1,...,r, r 2 is a fixed integer, and A = A (x,t), B = B(x,t), C =
k k k
≡ ≥
C(x,t) are constitutive functions (arbitrary elements). The work [28] contains several statements
concerning the structure of the generalized equivalence transformation groupoid of the PDE family
(2.19). In particular, equivalence transformations may be used to invertibly map any PDE within
(2.19) into one with A = 1 and A = 0. Moreover, in [28], it is shown that when A = 1 and
r 1 r
A = 0, the family (2.19) admits the equivalence transformations with the transformed variables
1
given by
x = F x+F , t = G, u (x ,t ) = U u+U , (2.20)
∗ 1 0 ∗ ∗ ∗ ∗ 1 0
where F = F (t), G = G(t) and U = U (t) are arbitrary functions, F = F (t) is related to G(t),
0 0 1 1 1 1
U is given by
0
F′x+F′
U = U (x,t) = 1 0U , (2.21)
0 0 1
F C
1
and the transformations for A ,B,C are given by complex formulas found in [28].
k
Importantly, the transformation of the dependent variable u(x,t) (2.21) involves the arbitrary
element C(x,t). These transformations are interpreted as the generalized equivalence group of the
PDEfamily(2.19)ifoneformallyextendsthesetofarbitraryelementstoincludethetimederivative
C as well as x-derivatives of C of orders 1,...,r. The transformations (2.20) do not arise as point
t
generalized equivalence transformations of the PDE family (2.19) without such prolongation.
The above example is readily verified using GeM symbolic software discussed in this contribution
(see Sections 3 and 4.3 below).
Remark 2.11. One may also note that it is possible for the equivalence transformations to involve
constant arbitrary elements, and therefore belong to the class (2.17), even when these transfor-
mations are point equivalence transformations (2.2). An instance of such a transformation is in
fact contained in Example 2.5 above. Indeed, in the equivalence transformations (2.7), the choice
A = a in (2.9) corresponds to a generalized equivalence transformation, since a is a constant
3
constitutive parameter.
8
(B)Generalizedextendedequivalence transformations. Onecanconsiderfurtherextensions
of the notion of equivalence transformations, for example, the ones involving nonlocal transforma-
tions of variables and/or constitutive components. For example, generalized extended equivalence
groups include transformations of the form (2.17), where the expressions for the transformed quan-
tities are not necessarily point, but become a point transformation with respect to the independent
and dependent variables after fixing the values of the arbitrary elements. Examples of transforma-
tions (2.17) involving arbitrary functions that arise as solutions of ordinary differential equations
are contained, for instance, in [7]; we present an example from that paper below.
Example 2.12. Consider the PDE family
f(x)u = (g(x)u ) +h(x)um, (2.22)
t x x
wheref(x),g(x), h(x)arearbitrarysmoothconstitutive functionssatisfyingf(x)g(x)h(x) = 0, and
6
m is a constant constitutive parameter. It is shown in [7], for example, that the transformations
x∗ = φ(x), t∗ = δ t+δ , u∗ = ψ(x)u, m∗ = m,
1 2
(2.23)
δ δ δ φ δ
f∗ = 0 1 f, g∗ = 0 x g, h∗ = 0 h
φ ψ2 ψ2 φ ψm+1
x x
map the PDE family (2.22) into itself, for any choice of smooth φ(x) = 0, constants δ , δ , δ
0 1 2
6
satisfying δ δ = 0, if ψ(x) is an arbitrary solution of an ODE
0 1
6
gψ
x
= 0.
(cid:18) ψ2 (cid:19)
x
Generalized extended equivalence transformations similar to (2.23) have been extensively used in
symmetry and conservation law classification problems in [7,25,29,30] and other works of the same
group. In fact, the following early example of generalized extended equivalence transformations
appears in [31], and is discussed in detail in [3].
Example 2.13. Consider a family of PDE systems of nonlinear telegraph equations given by
v = u , v = c2(u)u +b(u), (2.24)
t x x t
with dependent variables u = u(x,t), v = v(x,t) and two arbitrary functions b(u), c(u). Then for
any fixed value of the parameter ε, the change of variables
u c2(z)
x∗ = x εv, v = u , t∗ = t ε dz
t x
− − Z 1 εb(z) (2.25)
u0 −
u∗ = u, v∗ = v
maps each nonlinear telegraph system to another system from the same family, with new arbitrary
functions given by
b(u) c(u)
b∗(u∗)= , c∗(u∗) = . (2.26)
1 εb(u) 1 εb(u)
− −
The transformations defined by (2.25), (2.26) are nonlocal transformations that do not form a
Lie group of point transformations in the (x,t,u,v) space. Indeed, the transformed quantities
x∗,t∗,u∗,v∗ depend not only on the values of the independent and dependent variables x,t,u,v,
but also on an a priori specification of the arbitrary elements b(u), c(u), and hence belong to the
class of generalized extended equivalence transformations.
9
Remark 2.14. Fromthecomputationalpointofview,itisimportanttoremarkthatingeneral,the
generalized extended equivalence transformations do not form a Lie group of point transformations
even in the space of variables extended by arbitrary elements. For example, the transformations
defined by (2.25), (2.26) do not form a Lie group in the space of (x,t,u,v,b,c), and hence do not
belong to the set of generalized equivalence transformations (2.18).
(C) Potential equivalence transformations. Another example of transformations which are
outside of the class of local Lie groups of equivalence transformations (2.2) is given by potential
equivalence transformations. Suchnonlocaltransformationscan ariseasLiegroupsofpointequiva-
lence transformations of a potential system. Considera family of nonlinear wave equations in (1+1)
dimensions [2,8] given by
u = (c2(u)u ) , (2.27)
tt x x
for the unknown u = u(x,t), with an arbitrary constitutive function c(u). One can show that the
infinitesimal generators of the group of point equivalence transformations of the family (2.27) are
given by
∂ ∂ ∂ ∂
Z = , Z = , Z = , Z = t ,
1 2 3 4
∂t ∂x ∂u ∂u
(2.28)
∂ ∂ ∂ ∂ ∂
Z = u , Z = t +x , Z = t c ,
5 6 7
∂u ∂t ∂x ∂t − ∂c
and the global group has the form
x∗ = a x+a , t∗ = a a t+a , u∗ = a u+a t+a , c∗(u∗) =a−1c(u), (2.29)
6 2 6 7 1 5 4 3 7
where a ,...,a are arbitrary constants with a a a = 0. Additional equivalence transformations
1 7 5 6 7
6
arise as local equivalence transformations of a PDE system nonlocally related to (2.27) [2,8,32,
33]. In particular, for an arbitrary c(u), PDEs of the family (2.27) admit local conservation laws
corresponding to multipliers 1,t,x,xt. For the multiplier t, the conservation law is
∂ ∂
(tu u) (tc2(u)u ) = 0. (2.30)
t x
∂t − − ∂x
The corresponding family of potential systems with a nonlocal variable w(x,t), is given by
w = tu u, w = tc2(u)u ; (2.31)
x t t x
−
every PDE of this family is nonlocally related and equivalent (in a non-invertible manner) to a
PDE of the family (2.27). A computation of Lie groups of point equivalence transformations of the
family of potential systems (2.27) yields the eight-dimensional Lie algebra of generators
W = Z , i = 2,4,7,
i i
∂ ∂
W = , W = Z x ,
1 3 3
∂w − ∂w
(2.32)
∂ ∂ ∂
W = Z t x , W = Z +w ,
5 5 6 6
− ∂t − ∂x ∂w
∂ ∂ ∂ ∂
W = tu +w +u2 2uc .
8
∂t ∂x ∂u − ∂x
In particular, the time translation Z disappears, and two new generators, the obvious W and
1 1
an additional generator W , appear as local equivalence transformation generators of the potential
8
system (2.31).
10