On Dummy Variables of Structure-Preserving Transformations

Abstract

A method is given for obtaining equivalence subgroups of a family of differential equations from the equivalence group of simpler equations of a similar form, but in which the arbitrary functions specifying the family element depend on fewer variables. Examples of applications to classical equations are presented, some of which show how the method can actually be used for a much easier determination of the equivalence group itself.

1. Introduction

Denote collectively by 𝒜 the set of all arbitrary functions specifying the family element in a collection ℱ of differential equations of the form Δ𝑥,𝑦_(𝑛);𝒜=0, (1.1) where 𝑥=(𝑥¹,…,𝑥^𝑝) is the set of independent variables and 𝑦_(𝑛) denotes 𝑦 and all its derivatives up to the order 𝑛. In the most general case, the function 𝒜 may depend on 𝑥,𝑦, and the derivatives of 𝑦 up to a certain order not exceeding 𝑛, but quite often 𝒜 is simply a function of 𝑥, or a constant.

Let 𝐺 be the Lie pseudo-group of point transformations of the form 𝑥=𝜑(𝑧,𝑤),𝑦=𝜓(𝑧,𝑤), (1.2) where 𝑧=(𝑧¹,…,𝑧^𝑝) is the new set of independent variables, while 𝑤=𝑤(𝑧) is the new dependent variable. The group 𝐺 is infinite because, as explained in a paper by Tresse [1, Page. 11], its elements depend in general on arbitrary functions and not on arbitrary constants, and a Lie pseudo-group is to be understood here in the sense of [2, 3], that is, as the infinite-dimensional counterpart of a local Lie group of transformations. We say that 𝐺 is the group of equivalence transformations of (1.1) if it is the largest Lie pseudo-group of point transformations that map (1.1) into an equation of the same form, that is, if in terms of the same function Δ appearing in (1.1), the transformed equation has an expression of the formΔ𝑧,𝑤_(𝑛);ℬ=0, (1.3a) where ℬ=𝑇(𝒜), (1.3b)for a certain function 𝑇, is the new set of arbitrary functions specifying the family element in the transformed equation. Equivalently, 𝐺 will map elements of ℱ into ℱ, and, when this holds, (1.2) is called an equivalence transformation of the original equation (1.1). By a result of Lie [4], (1.3b) defines another group of transformations 𝐺_𝑐 induced by 𝐺, and acting on the arbitrary functions 𝒜 of the original equation. Invariants of the group 𝐺_𝑐 are termed invariants of the differential equation (1.1). These invariants are functions which depend on the arbitrary functions 𝒜 of the original equation, and which have exactly the same expression when they are also formed for the transformed equation, and they play a crucial role in the classification of the family of equation [5, 6].

Early developments in applications of Lie groups for finding equivalence transformations of a given differential equation (DE) started in the work of Lie [7] and were later pursued by Tresse [1] and Ovsiannikov [8]. More recent developments based on Cartan equivalence methods originated in the works of Olver and collaborators [2, 9], and this has led to new methods for computing the differential invariants and Maurer-Cartan structure equations of a Lie pseudo-group.

In this paper, we show that for a given differential equation in which the arbitrary functions defining the family element are functions of independent variables alone, the equivalence group is a subgroup of a differential equation of a similar form, but in which the arbitrary functions also depend on the dependent variable and its derivatives up to a given order. An extension of the theorem to equations with arbitrary functions depending on a subset of the set consisting of all independent variables and the dependent variable and its derivatives up to a given order is provided. Examples of applications to a number of classical equations show how to obtain equivalence subgroups of a more complex family of equations from those of simpler ones.

2. Dummy Variables of the Equivalence Group

If we consider, for instance, the general linear differential equation 𝑦^(𝑛)+𝑎¹(𝑥)𝑦^(𝑛−1)+𝑎²(𝑥)𝑦^(𝑛−2)+⋯+𝑎^𝑛(𝑥)𝑦=0, (2.1) the arbitrary functions 𝒜 in (1.1) are the coefficients 𝑎^𝑖(𝑥), for 𝑖=1,…,𝑛. Thus in this case it appears that 𝒜=𝒜(𝑥) is a function of the independent variable alone. This is quite often the case with various other linear or nonlinear equations. However, more generally, 𝒜 arises as a function of the independent variables and the dependent variable 𝑦 and its derivatives up to a certain order.

Theorem 2.1. Denote by (𝐴) (1.1) with 𝒜=𝒜(𝑥) and by (𝐵) the same equation, but in which 𝒜 depends on 𝑥, 𝑦, and the derivatives of 𝑦 up to a certain order 𝑠, that is, in which 𝒜=𝒜(𝑥,𝑦_(𝑠)). Similarly, denote by 𝐺^𝐴 and 𝐺^𝐵 the equivalence groups for (𝐴) and (𝐵), respectively. Then 𝐺^𝐴 is a subgroup of 𝐺^𝐵.

Proof. To fix ideas, suppose that the equation has 𝑚 arbitrary functions 𝐴₁,…,𝐴_𝑚 specifying the family element in ℱ (and collectively denoted by 𝒜) and that in (B) we have 𝐴_𝑗=𝐴_𝑗(𝑥,𝑦_(𝑠)) for all 𝑗=1,…,𝑚. On the other hand, for each multi-index 𝐽=(𝑗₁,…,𝑗_𝑘), where 1≤𝑗_𝑟≤𝑝for 𝑟=1,…,𝑘, we use the notation 𝐷_𝐽𝜕𝑦≡^𝑘𝑦(𝑥)𝜕𝑥^𝑗1,𝜕𝑥^𝑗2,…,𝜕𝑥^𝑗𝑘. (2.2) Suppose now that (1.2) is an equivalence transformation of (A). To perform the transformation of (𝐵) by (1.2), we may first ignore the arguments of the function 𝒜, in which sense these arguments can be considered as dummy. Then, for every multi-index 𝐽 with 0≤#𝐽=𝑘≤𝑛, each partial derivative 𝐷_𝐽𝑦 present in the equation is replaced by a function 𝑇_𝐽(𝑧,𝑤_(𝑘)). When all such replacements are made and all remaining occurrences of independent variables are expressed in terms of 𝑧 and 𝑤 by another application of (1.2), the resulting equation is the transformed equation (3.12), in which ℬ collectively denotes the new functions 𝐵₁,…,𝐵_𝑚. According to (3.15), each transformed function 𝐵_𝑗 has an expression of the form 𝐵_𝑗=𝐹_𝑗𝑧,𝐴₁,…,𝐴_𝑚,𝑗=1,…,𝑚,𝑟_𝑗≤𝑛, (2.3) in terms of the original functions 𝐴₁,…,𝐴_𝑚, and for a certain function 𝐹_𝑗, where 𝐹_𝑗 does not depend explicitly on the dependent variable 𝑤 and its derivatives, on the assumption that (1.2) is an equivalence transformation of (𝐴). To complete the transformation of (𝐵), we may now apply again (1.2), to transform the arguments of the functions 𝐴_𝑗, and this will transform 𝐴_𝑗(𝑥,𝑦_(𝑠)) into a composition of 𝐴_𝑗 and a function of 𝑧 and 𝑤_(𝑠), and the latter transformation will thus have no effect on the form of the transformed equation. Consequently, the transformed version of (𝐵) has the same form as (𝐵), and this completes the proof of the theorem.

The proof of Theorem 2.1 suggests the following immediate extension to the case where the function 𝒜 in (A) may depend only on a subset of the set of all variables and the derivatives of 𝑦 up to a given order.

Theorem 2.2. Denote by (𝐴) (1.1) with 𝒜=𝒜(𝑋), where 𝑋 is a subset of the set consisting of all independent variables, of the dependent variable 𝑦 and its derivatives up to a certain order 𝑟. Denote also by (𝐵) the same equation, but in which 𝒜=𝒜(𝑥,𝑦_(𝑠)), with 𝑟≤𝑠. Similarly, denote by 𝐺^𝐴 and 𝐺^𝐵 the equivalence groups for (𝐴) and (𝐵), respectively. Then 𝐺^𝐴 is a subgroup of 𝐺^𝐵.

Proof. Using the same notation as in the proof of Theorem 2.1, and reasoning in a similar way, if we apply 𝐺^𝐴 to transform (𝐵) while first ignoring the arbitrary functions’ arguments, the form of the equation is preserved, and the new arbitrary functions 𝐵_𝑗 now take the form 𝐵_𝑗=𝐹_𝑗𝑧,𝑤_(𝑟),𝐴₁,…,𝐴_𝑚,𝑗=1,…,𝑚. (2.4) We can then transform the arguments (𝑥,𝑦_(𝑠)) in each of the functions 𝐴_𝑗 appearing in (2.4), to obtain expressions of the form 𝐴_𝑗=𝐴_𝑗(𝑧,𝑤_(𝑠)) in (2.4), and the latter transformation will not affect the form of the equation, as already seen.

It should however be noted that, as stated, Theorems 2.1 and 2.2 holds only under the assumption that, in (B), the function 𝒜 depends explicitly on 𝑥=(𝑥¹,…,𝑥^𝑝),𝑦, and all derivatives of 𝑦 up to the stated order 𝑠. This is first because when a derivative of 𝑦 of a given order 𝑠 is transformed under a given group, the resulting expression depends in general on all derivatives of the new dependent variable 𝑤 up to the order 𝑠. On the other hand, the transformed function 𝐵_𝑗 will in general depend on the full set of the new independent variables 𝑧=(𝑧¹,…,𝑧^𝑝) and not only on a subset of this set. Thus, if in (B) the function 𝒜 does not explicitly depend on all independent variables or on all derivatives of 𝑦 up to the stated order 𝑠, this restriction should be imposed on 𝐺^𝐴, and this will yield in general a smaller equivalence subgroup for (𝐵).

On the other hand, the converse is not true in Theorem 2.1, and this is simply because if (𝐴) is transformed by an equivalence transformation of (𝐵), there is no guarantee that in the transformed equation the functions 𝐵_𝑗 will not depend on 𝑤 or its derivatives. Similarly, the converse to Theorem 2.2 is not true.

3. Application to Classical Equations

3.1. The General Linear Homogeneous ODE

In the sequel, we will use the notation 𝑓_𝑎=𝜕𝑓/𝜕𝑎, for every function 𝑓 with argument 𝑎, hence subscripts in such functions will denote differentiation. We shall also use the notation 𝜕_𝑎=𝜕/𝜕𝑎 for every variable 𝑎, and a transformed equation will be assumed to be expanded as a polynomial in the dependent variable and its derivatives.

If we consider, for instance, (2.1), its equivalence transformations are given by 𝑥=𝑆(𝑧),𝑦=𝐿(𝑧)𝑤, (3.1) where 𝑆 and 𝐿 are arbitrary functions. It follows from Theorem 2.1 that, if we assume in (2.1) that the coefficients 𝑎^𝑖 depend also on 𝑦 and its derivatives up to a certain order 𝑠, then (3.1) remains an equivalence transformation of the resulting equation. Equations of the latter form frequently appear with 𝑠=0, so that 𝑎^𝑖=𝑎^𝑖(𝑥,𝑦), which gives rise to a nonlinear equation of the form 𝑦^(𝑛)+𝑎¹𝑥,𝑦𝑦^(𝑛−1)+𝑎²𝑥,𝑦𝑦^(𝑛−2)+⋯+𝑎^𝑛𝑥,𝑦𝑦=0. (3.2) After some calculations, the equivalence group of (3.2) is found for 𝑛=3 to be of the form 𝑥=𝑆(𝑧),𝑦=𝐿(𝑧)𝑤+𝐽(𝑧), (3.3) where 𝑆,𝐿, and 𝐽 are arbitrary functions. It can be shown by induction that (3.3) also holds for all 𝑛≥3 in (3.2). Theorem 2.2 states that for any other variant of (3.2) in which the coefficients 𝑎^𝑖 are of the form 𝑎^𝑖=𝑎^𝑖(𝑋), where 𝑋⊂{𝑥,𝑦}, the equivalence group must be of the predefined form (3.3), where the functions 𝑆,𝐿, and 𝐽 are to be specified. Finding the equivalence group of an equation in a more specific form naturally greatly simplifies calculations. If, for instance, we suppose that 𝑎^𝑖(𝑋)=𝑎^𝑖(𝑦), for all 𝑖, so that (3.2) reduces to 𝑦^(𝑛)+𝑎¹𝑦𝑦^(𝑛−1)+𝑎²𝑦𝑦^(𝑛−2)+⋯+𝑎^𝑛𝑦𝑦=0, (3.4) then since the 𝑎^𝑖 depend on 𝑦 alone, 𝐿 and 𝐽 must be constant functions, so that the transformation of 𝑦 reduces to 𝑦=𝑘₃𝑤+𝑘₄, for some constants 𝑘₃ and 𝑘₄. For 𝑛=3, under the corresponding transformations of 𝑥 and 𝑦, the term not involving the dependent variable 𝑤 and its derivatives as a factor in the transformation of (3.4) is (𝑘₄𝑎³𝑆³_𝑧)/𝑘₃. Since this term may however depend on 𝑤 but not on 𝑧, we infer that 𝑆_𝑧 must be a constant function, so that the transformation of the independent variable 𝑥 must also be a linear function of the form 𝑥=𝑘₁𝑧+𝑘₂. It is readily seen that the set of transformations 𝑥=𝑘₁𝑧+𝑘₂,𝑦=𝑘₃𝑤+𝑘₄ (3.5) preserves the form of (3.4) for 𝑛=3, and therefore defines its equivalence group. It is also readily found by induction on 𝑛 that this remains true for all 𝑛≥3. Using (3.3) as the predefined form of the equivalence group leads in a similar manner to a much easier determination of the equivalence group for other possible values of 𝑋. In addition, 𝑋 needs not be the same for all of the coefficients 𝑎^𝑖, and thus to apply Theorem 2.2 here we only need to have 𝑎^𝑖=𝑎^𝑖(𝑋^𝑖), with 𝑋^𝑖⊂{𝑥,𝑦}. In particular, some of the 𝑎^𝑖 might be constants.

3.2. The Linear Hyperbolic Equation

Consider the PDE in two independent variables 𝑡 and 𝑥 of the form 𝑢_𝑡𝑥+𝑎¹(𝑡,𝑥)𝑢_𝑡+𝑎²(𝑡,𝑥)𝑢_𝑥+𝑎³(𝑡,𝑥)𝑢=0, (3.6) generally referred to as the linear hyperbolic second-order equation. It is well known (see e.g., [10, 11]) that the group 𝐺 of equivalence transformations of this equation is given in terms of new independent variables 𝑦 and 𝑧 and dependent variables 𝑤, by invertible transformations of the form 𝑡=𝑅𝑦,𝑥=𝑆(𝑧),𝑢=𝐿𝑦,𝑧𝑤, (3.7) where 𝑅,𝑆, and 𝐿 are arbitrary functions satisfying the nonvanishing Jacobian condition 𝑅_𝑦𝑆_𝑧𝐿≠0, (3.8) where 𝑅_𝑦=𝜕_𝑦𝑅 and 𝑆_𝑧=𝜕_𝑧𝑆. Consider now a nonlinear extension of (3.6) of the form 𝑢_𝑡𝑥+𝑎¹(𝑡,𝑥,𝑢)𝑢_𝑡+𝑎²(𝑡,𝑥,𝑢)𝑢_𝑥+𝑎³(𝑡,𝑥,𝑢)𝑢=0, (3.9) in which the dependent variable 𝑢 also appears as argument in the arbitrary coefficients. We undertake the somewhat lengthy calculation of the equivalence group of (3.9) which is not available in the literature, to illustrate how the knowledge of such a group can be utilized for a much easier determination of the equivalence group of equations of a similar form. The equivalence group 𝐻 of (3.9) must be sought in the form 𝑡=𝑅𝑦,𝑧,𝑤,𝑥=𝑆𝑦,𝑧,𝑤,𝑢=𝑇𝑦,𝑧,𝑤, (3.10) for some functions 𝑅,𝑆, and 𝑇 to be specified and which must satisfy the non-vanishing Jacobian condition −𝑅_𝑧𝑆_𝑦+𝑅_𝑦𝑆_𝑧𝑇_𝑤+𝑅_𝑧𝑆_𝑤−𝑅_𝑤𝑆_𝑧𝑇_𝑦+−𝑅_𝑦𝑆_𝑤+𝑅_𝑤𝑆_𝑦𝑇_𝑧≠0. (3.11) However, we notice that under (3.10), 𝑢_𝑡 is transformed into 𝑢_𝑡=−𝑆_𝑧𝑇_𝑦+𝑆_𝑦𝑇_𝑧+𝑆_𝑤𝑇_𝑧−𝑆_𝑧𝑇_𝑤𝑤_𝑦+𝑆_𝑦𝑇_𝑤−𝑆_𝑤𝑇_𝑦𝑤_𝑧𝑅_𝑧𝑆_𝑦−𝑅_𝑦𝑆_𝑧+𝑅_𝑧𝑆_𝑤−𝑅_𝑤𝑆_𝑧𝑤_𝑦+𝑅_𝑤𝑆_𝑦−𝑅_𝑦𝑆_𝑤𝑤_𝑧. (3.12) The occurrence of derivatives of 𝑤 in the denominator of this transformed expression for 𝑢_𝑡 will give rise in the expression of 𝑢_𝑡𝑥 to undesired terms in 𝑤_𝑧𝑧 and 𝑤_𝑦𝑦 as well as nonlinear terms of the form 𝑤^𝑖_𝑧𝑤^𝑗_𝑦, where 𝑖 and 𝑗 are some natural numbers, and they should therefore disappear. This disappearance of derivatives is translated into the conditions 𝑅_𝑧𝑆_𝑤−𝑅_𝑤𝑆_𝑧=0,−𝑅_𝑦𝑆_𝑤+𝑅_𝑤𝑆_𝑦=0, (3.13) under which the Jacobian condition (3.11) is reduced to −𝑅_𝑧𝑆_𝑦+𝑅_𝑦𝑆_𝑧𝑇_𝑤≠0. (3.14) If we suppose that 𝑅_𝑤=0 and 𝑆_𝑤≠0, then it follows from (3.13) that 𝑅_𝑦=𝑅_𝑧=0, which contradicts (3.14). Thus we have 𝑅_𝑤=0 if and only if 𝑆_𝑤=0. On the other hand, if we assume that 𝑅_𝑤≠0, then using (3.13) we may write 𝑆_𝑦=𝑅_𝑦𝑆_𝑤𝑅_𝑤,𝑅_𝑧=𝑆_𝑧𝑅_𝑤𝑆_𝑤, (3.15) which also contradicts (3.14), so that 𝑅_𝑤=𝑆_𝑤=0, and the resulting expressions in (3.10) take the form 𝑡=𝑅𝑦,𝑧,𝑥=𝑆𝑦,𝑧,𝑢=𝑇𝑦,𝑧,𝑤. (3.16) Under the new change of variables, the terms in 𝑤_𝑦𝑦 and 𝑤_𝑧𝑧 in the transformed equation are given by 𝑅_𝑧𝑆_𝑧−𝑅_𝑧𝑆_𝑦+𝑅_𝑦𝑆_𝑧𝑇_𝑤𝑤_𝑦𝑦,𝑅_𝑦𝑆_𝑦−𝑅_𝑧𝑆_𝑦+𝑅_𝑦𝑆_𝑧𝑇_𝑤𝑤_𝑧𝑧, (3.17) respectively. On account of (3.14), the vanishing of these terms is reduced to the condition 𝑅_𝑧𝑆_𝑧=𝑅_𝑦𝑆_𝑦=0, (3.18) which on account of the Jacobian condition (3.8) readily yields 𝑅_𝑧=𝑆_𝑦=0. (3.19) Consequently, (3.10) reduces to 𝑡=𝑅𝑦,𝑥=𝑆(𝑧),𝑢=𝑇𝑦,𝑧,𝑤. (3.20) Under this new set of transformations, (3.9) takes the form 𝑤_𝑦𝑤_𝑧𝑇_𝑤,𝑤𝑇_𝑤+𝑤_𝑧𝑎²𝑅_𝑦+𝑇_𝑦,𝑤𝑇_𝑤+𝑤_𝑦𝑎¹𝑆_𝑧+𝑇_𝑧,𝑤𝑇_𝑤+𝑎³𝑇𝑅_𝑦𝑆_𝑧+𝑎¹𝑆_𝑧𝑇_𝑦+𝑎²𝑅_𝑦𝑇_𝑧+𝑇_𝑦,𝑧𝑇_𝑤+𝑤_𝑦,𝑧=0, (3.21) which shows that 𝑇 must be linear in 𝑤. Consequently, the required change of variables must be of the form 𝑡=𝑅𝑦,𝑥=𝑆(𝑧),𝑢=𝐿𝑦,𝑧𝑤+𝐽𝑦,𝑧, (3.22) and (3.22) transforms (3.9) into 𝐽𝑎³𝑅_𝑦𝑆_𝑧𝐿+𝐿_𝑧𝐿+𝑎¹𝑆_𝑧𝑤_𝑦+𝐿_𝑦𝐿+𝑎²𝑅_𝑦𝑤_𝑧+𝑤𝑎²𝐿_𝑧𝑅_𝑦+𝑎¹𝐿_𝑦𝑆_𝑧+𝐿𝑎³𝑅_𝑦𝑆_𝑧+𝐿_𝑦,𝑧𝐿+𝑤_𝑦,𝑧=0, (3.23) which is of the required form since we may in this case write the constant term 𝐽𝑎³𝑅_𝑦𝑆_𝑧/𝐿 in the transformed equation in the form 𝑤(𝐽𝑎³𝑅_𝑦𝑆_𝑧/(𝐿𝑤)). Therefore, (3.22) defines the equivalence group 𝐻 of (3.9). It differs from the equivalence group 𝐺 of (3.6) only by the additional term 𝐽(𝑦,𝑧) in the transformation of 𝑢, which has the effect of adding a sort of nonhomogeneous term to the transformed equation.

Now that the equivalence group (3.22) of (3.9) is available, we can clearly demonstrate how Theorem 2.2 can be used for a much easier determination of equivalence groups for variants of (3.9), in which the arbitrary coefficients 𝑎^𝑖 have arguments of the form 𝑋^𝑖⊂{𝑡,𝑥,𝑢}. Indeed, since by Theorem 2.2 any such group is a subgroup of 𝐻, its equivalence transformations must therefore be sought in the form (3.22), which greatly simplifies calculations. We show this by considering two examples.

To begin with, consider a variant of (3.9) of the form 𝑢_𝑡𝑥+𝑎¹(𝑥)𝑢_𝑡+𝑎²(𝑡)𝑢_𝑥+𝑎³(𝑡,𝑥)𝑢=0. (3.24) Since the coefficient 𝑎³(𝑡,𝑥) of 𝑢 does not involve 𝑢 itself, we may assume that 𝐽(𝑦,𝑧)=0 in (3.22), and we thus look for equivalence transformations of (3.24) in the form 𝑡=𝑅𝑦,𝑥=𝑆(𝑧),𝑢=𝐿𝑦,𝑧𝑤, (3.25) and under which the coefficients 𝜃^𝑦 of 𝑤_𝑦 and 𝜃^𝑧 of 𝑤_𝑧 take the form 𝜃^𝑦=𝑎¹𝑆_𝑧+𝐿_𝑧𝐿,𝜃^𝑧=𝑎²𝑅_𝑦+𝐿_𝑦𝐿. (3.26) Since we have 0=𝜕𝜃^𝑦=𝜕𝑦𝜕𝜃^𝑧,𝜕𝑧𝜕𝜃^𝑦=𝜕𝑦𝜕𝜃^𝑧=𝜕𝑧𝐿𝐿_𝑦,𝑧−𝐿_𝑦𝐿_𝑧𝐿², (3.27)𝐿 must satisfy the condition 𝜕_𝑦(𝐿_𝑧/𝐿)=0, and hence 𝐿=𝑒^{𝑓(𝑦)+𝑔(𝑧)}, for some arbitrary functions 𝑓 and 𝑔. This new expression for 𝐿 reduces (3.25) to 𝑡=𝑅𝑦,𝑥=𝑆(𝑧),𝑢=𝑒^{𝑓(𝑦)+𝑔(𝑧)}𝑤, (3.28) and the latter change of variables transforms (3.24) into 𝑤𝑓_𝑦𝑔_𝑧+𝑎¹𝑆_𝑧+𝑅_𝑦𝑎²𝑔_𝑧+𝑎³𝑆_𝑧+𝑔_𝑧+𝑎¹𝑆_𝑧𝑤_𝑦+𝑓_𝑦+𝑎²𝑅_𝑦𝑤_𝑧+𝑤_𝑦,𝑧=0, (3.29) which is of the prescribed form, and this shows that (3.28) represents the equivalence transformations of (3.24).

For the second example of determination with a variant of (3.9), consider the equation 𝑢_𝑡𝑥+𝑎¹(𝑡)𝑢_𝑡+𝑎²(𝑡)𝑢_𝑥+𝑎³(𝑡,𝑥)𝑢=0, (3.30) in which the coefficient of 𝑢_𝑡 and 𝑢_𝑥 depend on 𝑡 alone. Here again, the equivalence group is to be sought in the form (3.25), and under this change of variables, the coefficients 𝜃^𝑦 of 𝑤_𝑦 and 𝜃^𝑧 of 𝑤_𝑧 take the form 𝜃^𝑦=𝑎¹𝑆_𝑧+𝐿_𝑧𝐿,𝜃^𝑧=𝑎²𝑅_𝑦+𝐿_𝑦𝐿, (3.31) where 𝑎¹=𝑎¹(𝑅(𝑦)) and 𝑎²=𝑎²(𝑅(𝑦)), and the conditions 0=𝜕_𝑧𝜃^𝑦 and 0=𝜕_𝑧𝜃^𝑧 lead to the differential equations −𝐿²_𝑧+𝐿𝐿_𝑧𝑧𝐿²+𝑎¹𝑆_𝑧𝑧=0,−𝐿_𝑦𝐿_𝑧+𝐿𝐿_𝑦,𝑧𝐿²=0. (3.32) It follows from the arbitrariness of the coefficient 𝑎¹ in the first equation of (3.32) that 𝑆=𝑘₁𝑧+𝑘₂ for some constants 𝑘₁ and 𝑘₂, while −𝐿²_𝑧+𝐿𝐿_𝑧𝑧=0. (3.33) The latter equation has solution 𝐿=𝑔(𝑦)𝑒^{𝑧𝑓(𝑦)}, where 𝑔≠0 and 𝑓 are arbitrary functions. A substitution of this expression for 𝐿 in the second equation of (3.32) leads to 𝑔𝑓_𝑦𝑒^2𝑧𝑓=0, and hence 𝑓=𝑘₃ is a constant function, so that the corresponding change of variables (3.25) now takes the form 𝑡=𝑅𝑦,𝑥=𝑘₁𝑧+𝑘₂,𝑢=𝑔𝑦𝑒^𝑘3𝑧𝑤. (3.34) The transformation of (3.30) under (3.34) yields the equation 𝑎¹𝑘₁+𝐿_𝑧𝐿𝑤_𝑦+𝑎²𝑅_𝑦+𝐿_𝑦𝐿𝑤_𝑧+𝑤_𝑦,𝑧+𝑤𝐴𝑘₁𝐿_𝑦+𝑘₁𝐿𝑎³𝑅_𝑦+𝑎²𝐿_𝑧𝑅_𝑦+𝐿_𝑦,𝑧𝐿=0, (3.35) which is of the required form, and thus (3.34) represents the group of equivalence transformations of (3.30).

3.3. Some Applications

It is worthwhile making some comments at this point on some applications of equivalence transformations with regards to the integration of differential equations. By essence, equivalence transformations are a key tool by means of which equations can be classified and then represented in each equivalence class by a simpler and more tractable canonical form. If we consider, for instance, the variant (3.24) of the linear hyperbolic equation it follows from the form of the transformed equation (3.29) that if in the corresponding equivalence transformations (3.28) we set 𝑓=𝑘₁−𝑎²(𝑅)𝑅_𝑦𝑑𝑦,𝑔=𝑘₂−𝑎¹(𝑆)𝑆_𝑧𝑑𝑧, (3.36) for some constants 𝑘₁ and 𝑘₂ and then set 𝑅_𝑦𝑆_𝑧=1, then the transformed equation is reduced to 𝑏𝑦,𝑧𝑤+𝑤_𝑦,𝑧=0,with𝑏𝑦,𝑧=𝑎³−𝑎¹𝑎², (3.37) which gives a much simpler canonical form for all equations of the form (3.24). In particular, if in the latter equation we have 𝑎³=𝑎¹𝑎², (3.38) then (3.24) is always reducible to the linear wave equation 𝑤_𝑦,𝑧=0. (3.39) The complete classification of families of equations can be realized by means of invariant functions of these equations, which as already mentioned are defined by the equivalence transformations. For instance, using the functions 𝐻=𝑎¹_𝑡+𝑎¹𝑎²−𝑎³,𝐾=𝑎²_𝑥+𝑎¹𝑎²−𝑎³ (3.40) known as Laplace invariants [10], Ovsiannikov obtained the contact invariants 𝑃=𝐻/𝐾 and 𝑄=(ln|𝐻|)_𝑡,𝑥/𝐻 and used them to achieve the classification of a subfamily of the linear hyperbolic equation (3.24) [8, 12]. Similar classifications of differential equations based on invariant functions have also been carried out by other more recent writers on the topic [5, 6, 13, 14]. Using Cartan’s equivalence method in the form developed by Fels and Olver [9], Morozov [13] gave a complete solution to the equivalence problem for a number of families of partial differential equations, under the pseudo-group of contact transformations.

Significant simplifications of the equation, which may include a reduction of its order, are often achieved by the simple vanishing of the invariants of the equation. For instance, the condition 𝑎³=𝑎¹𝑎² obtained in (3.38) by a direct analysis of the equivalence transformations of (3.24) and which have led to the reduction of this equation to the linear wave equation 𝑤_𝑦,𝑧=0, actually corresponds exactly to the vanishing of the invariant 𝑃=𝐻/𝐾 of this equation. Similar reductions or derivations of integrability properties of differential equations based on the vanishing of invariant functions were carried out by Laguerre [15], followed by Brioschi [16] for third-and fourth-order ODEs, as well as by Liouville [17] and many others.

4. Concluding Remarks

In this paper, we considered two families (𝐴) and (𝐵) of DEs of a similar form depending on arbitrary functions, where the arguments of each of the arbitrary functions in (𝐴) are a subset of that for the corresponding arbitrary function in (𝐵). We then showed in Theorem 2.2 that the equivalence group 𝐺^𝐴 of (𝐴) must be a subgroup of the equivalence group 𝐺^𝐵 of (𝐵). We also showed through some examples of applications how the theorem can be used either for an easier determination of equivalence subgroups or just equivalence transformations of a complex family of DEs or for finding the equivalence group 𝐺^𝐴 of type (𝐴) DEs when 𝐺^𝐵 is known.

Acknowledgment

This publication was made possible in part by a grant from the Carnegie Corporation of New York.