Abstract
Abstracts. A method for the group classification of differential equations is proposed. It is based on the determination of all possible cases of linear dependence of certain indeterminates appearing in the determining equations of symmetries of the equation. The method is simple and systematic and applied to a family of hyperbolic equations. Moreover, as the given family contains several known equations with important physical applications, low-order conservation laws of some relevant equations from the family are computed, and the results obtained are discussed with regard to the symmetry integrability of a particular class from the underlying family of hyperbolic equations.
1. Introduction
The group classification of differential equations, which consists in determining all symmetry classes admitted by an equation according to the values of the parameters or arbitrary functions labelling the given family of differential equations, has been carried out in the literature mostly in a more or less ad hoc manner [1–7]. This has resulted, as pointed out in [7], in a number of revised classification results published in the literature being wrong. Attempts to find a somewhat systematic method for this classification problem has, however, been made. This includes the well-known algebraic method which can be traced back to Lie’s work on symmetry algebras of ordinary differential equations (ODEs), and which has been upgraded or applied in several papers [2, 4, 8–11]. Another classification method that has emerged in recent years, often called furcate splitting, originated in [12], and its refinements and generalizations have been applied in other papers (see [13–15] and the references therein). Other methods suggested in the literature tend to be variants of the direct method, or adapted to a specific type of equations such as the one proposed for linear pdes in [6].
Some of the suggested methods still require, in general, a considerable amount of computations and analysis even for simpler cases of equations, or they are limited by the reduced scope of their applicability. For instance, the algebraic method can only be carried out when any symmetry algebra of the given family of nonlinear equations falls, in general, within the existing classification of the low-dimensional Lie algebras, in addition to other conditions. It should also be noted that the algebraic method can be complemented with other methods as done in [14, 16, 17].
In this paper, we propose a new and systematic method for the group classification of differential equations. It is based on the determination of all possible cases of linear dependence of certain indeterminates appearing in the determining equations for the Lie point symmetry algebra of the given family of equations. The method is presented through its application to the family of scalar hyperbolic equations labelled by the arbitrary function , where and are scalars. Moreover, a variable in a subscript denotes a partial derivative with respect to the variable, so that , , , and so on. This systematic method turns out to be simple, algorithmic, and yields a relatively fast classification of the family (1) of equations.
Given that the family (1) contains several well-known equations with important physical applications, including amongst others Liouville’s equation [18–20] and the sine-Gordon equation, low-order conservation laws of a number of relevant equations from this family are computed using the direct method [21–24] or some other methods [25]. Indeed, the sine-Gordon equation occurs for instance in the study of surfaces of constant negative curvature as well as in the study of crystal dislocations, and its solutions possess some soliton properties [26, 27]. Liouville’s equation on the other hand appears in the study of isothermal coordinates in classical mechanics and statistical physics [28]. More generally, as pointed out in [29], hyperbolic equations describe a large and important collection of phenomena. This includes aerodynamic, fluid, and atmospheric flows, amongst others. Moreover, their solutions tend to be more complex and more interesting than those of their parabolic or hyperbolic counterparts [29].
Some of our computations of conservation laws correspond to the case of arbitrary labelling functions of subclasses of (1). Although some of the conservation laws found are finite in number, they are infinite for other equations, and the connection between such equations and their symmetry integrability is discussed. It will naturally be assumed that is a nonconstant function.
2. Equivalence Group
In order to ease the classification problem of (1), as usual, we shall make use of its group of equivalence transformations. The complete determination of this group is summarized in the following result.
Theorem 1. The group of equivalence transformations of (1) is given as follows, where is the new dependent variable.
Case 1. . The equation reduces in this case toand the invertible point transformations are given byorfor some arbitrary constants, with. The corresponding transformed equation has the following expression:where.
Case 2. . The equation reduces in this case toand the transformations are given byfor some arbitrary constants, with, and arbitrary function. The corresponding transformed equation has the following expression:where.
Case 3. . The equivalence transformations are given byfor some arbitrary constants, with. The corresponding transformed equation has the following expression:where
Proof. The three cases outlined in the statement of the theorem clearly exhaust all possibilities for the nonconstant function To establish the stated result in Case 1, we subject (2a) to the most general invertible point transformation:
and request that the resulting transformed equation be of the form (2d) for some arbitrary function .
In an expression that depends polynomially on a dependent variable and its derivatives, let us assign to each monomial a weight equal to the total order of derivatives in the monomial. In this way, and for instance have a weight of two and three, respectively. It turns out that the transformed version of (2a) under (5) will conserve its maximal weight as it should if and only if Therefore, by requesting that the transformed version of (2a) be of the same form, that is an equation of the form (2d), the resulting constraints on the functions , , and reduce the admissible transformations of , , and precisely to (2b) or (2c), and the transformed equation to (2d) with the specified expression for The proof is similar for Case 2 and Case 3. This completes the proof.
Due to the properties of invertible point transformations, it is clear that in Theorem 1, the functions in the original equation and in its transformed version are related by an equivalence relation. Such an equivalence relation will be denoted . In particular, in Case 3, where , we may assume that in (1).
3. Lie Group Classification
The group classification of (1) will be performed under point transformations. We denote by , a generic symmetry vector of (1), where , , and are functions of , , and By generic symmetry vector of (1), we mean a linear combination of all symmetry generators of (1).
The determining equations for the symmetries of (1) whose expression is omitted here show that the conditions should hold. Consequently, for some functions and The remaining part of the determining equations reduces to the classifying equation: where here and in the sequel, the notation denotes the derivative of the function of a single variable with respect to its argument.
Our general procedure for the group classification problem goes as follows. In each term of (7), we consider as indeterminate in a polynomial expression those factors involving only functions of and its derivatives. In this way the vanishing coefficients of the polynomial expression can be unambiguously determined if we know exactly which of the indeterminates are nontrivially linearly dependent and which ones are not. Here, a nontrivial linear dependence is one for which all coefficients in the corresponding vanishing linear combination are nonzero.
When the expression of the determining equations can take the form of a polynomial equation in which the indeterminate involves the labelling parameters (of the family of equations) having as arguments variables which are by themselves indeterminate for the same determining equations, extensions of the principal algebra will occur if and only if one or more indeterminates are linearly dependent. The linear dependence of a single indeterminate will indeed be reduced simply to its vanishing.
Therefore, once the principal algebra has been determined in the usual way by treating all labelling parameters as arbitrary, to find any of its extension, we look for all possible nontrivially linearly dependent indeterminates. Each linear dependence of an indeterminate will generally reduce the degree of freedom of the labelling parameter, and its new value thus obtained will be inserted in the determining equations and either yield a complete determination of the corresponding symmetry algebra, or a recursive application of the process which will always end up giving a completely determined value of the parameters. In this way the complete list of nonequivalent symmetry classes of the equation is found.
To determine all possible sets of linearly dependent indeterminates, we consider all possible subsets of indeterminates, where , assuming that there is a total of indeterminates in the (single scalar) classifying equations. For each such subset of indeterminates, the linear dependence will be determined either by the direct vanishing of the linear combination of the indeterminate or equivalently by the vanishing of their Wronskian, as the labelling parameter functions are assumed to be smooth. In other words, for each value of , there will be at most valid determining equations for the labelling parameters. As for the possible vanishing of a single indeterminate, this will usually be checked by mere inspection, and corresponds indeed to having .
Given the conditions on the function in (1), we will treat separately three cases.
3.1. Case 1:
This case amounts to having in (1). Updating (7) with this expression for shows that must hold. Hence, must be a constant. With this value of , (7) reduces to
As a consequence of Theorem 1, in the determination of the group classification of (2a), one can always replace, in particular, the function by any of its nonzero scalar multiples, although admissible transformations of are quite more general than this.
For the study of equation (8), we first consider the case where the labelling function is arbitrary. The vanishing of the coefficient of then shows that , while the vanishing of the coefficient of yields Consequently, for arbitrary, the symmetry algebra of (2a) is generated by where here and in the sequel, (for ) represent arbitrary constants.
Assuming now that is not arbitrary in (8), the set of the indeterminate consists precisely of elements, none of which may vanish due to the assumption on
3.1.1. Two Indeterminates Are Linearly Dependent
The constraints on determined by the linear dependence of all possible subsets of elements from are given by
Since by assumption the function may not assume a constant value, it follows by inspection that (10) yield invalid solutions. Other values of yielding only the symmetry generator (9) are also to be excluded from the updated list of solutions of (10). The relevant values of determined by (10) and the corresponding generic symmetry generator are therefore given as follows. Here and in the sequel, (for ) are arbitrary constants. Hence, for , we have and for , we have where is a solution of the original equation (2a), while and are arbitrary functions of their arguments.For , , we have
3.1.2. Three Indeterminates Are Linearly Dependent
We now move on to consider the case where three elements in are linearly dependent. The conditions on imposed by all possible such sets of three linearly dependent indeterminates are given as follows:
The change of variables transforms (12) into and the latter has the general solution , where here and in the sequel, (for) denote arbitrary constants. Consequently, the general solution of (12) is given by and thus, this case coincides with the solved case (11c). The other equations in (12) are solved in a way more or less similar to that for equation (12). However, they yield solutions such as which are often more general solutions than those found for but which in any case do not yield symmetries other than those given by (22).
3.1.3. Four Indeterminates Are Linearly Dependent
We now consider the case where there are four linearly dependent elements in There is only one such possibility consisting of the whole set , and the corresponding constraint on is given as follows:
Setting and then reduces (16) to so that , , for some arbitrary constants , where . Consequently, for , . We note that is linear in this case if and only if , in which case corresponds to the already solved case (11b). On the other hand, for and corresponds to the solved case (14), while for , and , yields the symmetry (9) corresponding to arbitrary functions.
Similarly, for in the expression of , one gets , which as already seen, also yields a symmetry generator given by (9).
Theorem 2. Denote by the symmetry algebra of the hyperbolic equation and by the generic symmetry vector in . (a)For , is infinite dimensional andwhere is a solution of the original equation (2a) (b)For , is infinite dimensional and(c)For , with , has dimension 4 and(d)For any other function , has dimension 3 andThe four symmetry classes thus obtained are pairwise nonequivalent and make up all possible symmetry classes of (2a).
Proof. Statement (a), statement (b), statement (c), and statement (d) as well as the fact that the listed symmetry classes exhaust all possible symmetry classes of (2a) are just a summary of the results established immediately before the theorem, and it only remains to show that the stated symmetry classes are nonequivalent. In view of the different dimensions of the symmetry classes found, the only problem is to prove the nonequivalence in the two infinite dimensional cases. However, this follows from the fact that the corresponding functions, namely, and , are clearly nonequivalent under (2b).
3.2. Case 2:
This case amounts to having in (1), and the resulting expression of the determining equations (7) is reduced to the single equation: which will be used to classify (6a) and (6b).
When the function in (22) is arbitrary and correspondingly , an analysis similar to the one done for the preceding case shows that the generic symmetry is given by , where and is an arbitrary function. It is a remarkable fact that the class (3a) of equations has an infinite dimensional principal symmetry algebra. It might therefore be possible that this class of equations which by the way are generally easily integrable, is linearizable by certain types of transformations, although it clearly follows from Theorem 1 that (3a) is not linearizable by point transformations.
For the rest of the classification of (3a), we continue with the application of our method based on the determination of all possible cases of linearly dependent subsets of indeterminates in (22), with where in this case is the cardinality of the set of indeterminates. In the set of equations representing the constraints on for a particular value of , only those equations corresponding to new and nonredundant solutions will usually be represented.
3.2.1. Two Indeterminates Are Linearly Dependent
For , the constraints on are given by
To simplify notation, we will make use of the following vector fields, where here and in the sequel, unless otherwise stated, and are arbitrary functions of , while and are arbitrary functions of Also, is a particular solution of the linear equation , and is an arbitrary constant.
More precisely, for each value of , we denote by the th generic generator, as they consecutively occur, of the symmetry class associated with a solution of an ode representing the condition of linear dependence of indeterminates. Therefore, let us set
The solutions of (24) represented by their canonical forms under (3b) and the corresponding generic symmetry vector are given as follows:
3.2.2. Three Indeterminates Are Linearly Dependent
In order to write down more concisely the set of conditions on corresponding to , we set . The equations are then given by
In view of the equivalence transformations (2b) and the linear case already treated, one may consider that the solution of (27) satisfies . It turns out that when , and if otherwise. Since these two cases already appear in (26), it then follows that (27) does not yield any new result.
The general solution of (27) satisfies , and for , the corresponding symmetry is given by (26), while for , and the corresponding symmetry is given by , where
The next equation to consider from (27) is (27) which is up to a renaming of the unknown function’s argument the same as (12) already solved and whose solution satisfies . The symmetries corresponding to the solution are given as follows for , the case being already solved in (26). where and are the symmetries already found in (26). Hence (27) does not yield any new result either.
Up to this point, the constraints on resulting from the condition of linear dependence of indeterminates have been expressed typically as th order odes, and we have luckily been able to solve all such odes. However, some of them, such as (27), are hard to solve and we have to resort to the more direct method of expressing the condition of linear dependence as the vanishing of a linear combination of the indeterminate with some arbitrary constant coefficient. In that way all such constraints turn out in this instance to be mere linear first order odes. The drawback with this method lies in the rigidity of arbitrary constants in the resulting general solutions which albeit straightforward to find, are somewhat much harder to reduce to a simpler and suitable form. This is due to the fact that the coefficients in the linear combination are in fact also arbitrary parameters of the required solution.
For equation (27), the corresponding vanishing of the linear combination of indeterminate takes the form with solution
where the constant of integration together with the coefficient , , and are to be considered as arbitrary parameters in the solution (30b). With a bit of manipulation, the latter solution can be put into the form where the parameters , , and are different from those in (30b). In fact, the renaming of parameters in a transformed expression to those appearing in the original expression will often be assumed in the sequel. Applying now the equivalence relation (3c) to the latter expression for reduces it to the form for some new arbitrary constant Finally inserting the latter expression for into the determining equations (22) and solving shows that the only existing symmetries for equation (27) are those for arbitrary and given by (23).
The symmetries corresponding to the third order ODE (27) are found with the same procedure as above for (27). The linear dependence constraints on takes the form with the solution
Thanks to the equivalence relation (3c), one has , and substituting the latter expression into (22) gives rise to two relevant cases to consider, namely, the case for which correspond to a case already solved, and the more general case given as follows together with its corresponding generic symmetry. where
3.2.3. Four Indeterminates Are Linearly Dependent
Assuming now that of the indeterminate in (22) are linearly dependent gives rise to a maximum of four possible conditions on the function , only two of which yield meaningful solutions and are given as follows:
Equation (35) is just equation (16) whose general solution in terms of satisfies . In view of the preceding cases, we have to assume , and hence, the resulting relevant cases are given by (arbitrary), and which all yield no new symmetries apart from the one given by (23). The other relevant case is which is also an already solved case. In other words, equation (35) yields no new symmetries.
On the other hand, Equation (35) has general solution: and by (3c), one has
In view of the preceding results, the latter value for , and hence Equation (35), yields no new symmetries.
3.2.4. Five Indeterminates Are Linearly Dependent
For the last case in (22) where there are five linearly dependent indeterminates in , the corresponding single equation is given by
Setting , and then reduces (37) to the simpler equation , whose solution is However, none of the corresponding solutions of the original equation (37) yields a new symmetry.
Theorem 3. For the hyperbolic equation , the list of all possible nonequivalent values of yielding, under point transformations, nonequivalent symmetry algebras with generic vectors are given by Tables 1 and 2, and consists of a total of eight nonequivalent symmetry classes.
Proof. The fact that Tables 1 and 2 list all possible nonequivalent or distinct values of and the corresponding symmetry algebras follows from the discussion carried out in Section 3.2. The simplification of the canonical form of each function listed in the table naturally follows from (3c). However, it remains to be ascertained if all functions in these lists are pairwise nonequivalent. This also follows by a straightforward application (3c). This completes the proof.
It should be noted that by using a simple symmetry argument based on the structure of the resulting equation with respect to the variables and , the results from this section implicitly include those corresponding to the case .
3.3. Case 3:
As already noted, in view of (1), this case corresponds to having in (1). The corresponding determining equation (7) then reduces to
When is arbitrary in (38), it readily follows that the functions , , and from (6a) and (6b) are given by , , and , for some arbitrary constants , , and . In other words, the principal algebra has a generic vector:
Given that in (38) is a function of while (38) itself depends explicitly on and , in order to apply our classification procedure for the extension of the principal algebra to (38), it is more appropriate to first transform the whole of (1) under the new dependent variable: which turns out to be invertible, within verse and rather apply the classification procedure for the determination of all admissible functions to the resulting transformed equation. Indeed, since (40a) and (40b) is invertible, there is a one-to-one correspondence between any function associated with (1) and a function associated with the transformed version of (1) under (40a) and (40b) whose explicit expression takes the form
Denoting by the generic symmetry generator in the symmetry algebra of (41), the associated determining equations show that for some arbitrary functions and Moreover, the remaining determining equations take the form
As far as the determination of an indeterminate in (43) is concerned, only equation (43) may be considered, as it contains all the indeterminate in (43). Hence, the list of indeterminates for (41) is given by
The admissible functions corresponding to this list and extending the principal algebra of (41) are by construction the same as those extending the principal algebra of the original equation
Finding these admissible functions the usual way as in Subsections 3.1 and 3.2 yield the following complete classification of (45).
Theorem 4. Denote by the symmetry algebra of the hyperbolic equation (45), which is equivalent under (4a) to with , and denote by the generic symmetry vector in . (a)For , is infinite dimensional andwhere is any solution of the linear equation (b)For , is infinite dimensional and(c)For , with , has dimension 4 and(d)For any other function , has dimension 3 and
The four symmetry classes thus obtained are pairwise nonequivalent and make up all possible symmetry classes of (45).
Proof. The proof is similar to that given for Theorem 2 and Theorem 3, and the details are omitted.
4. Conservation Laws
As discussed in Section 1, the many physical applications associated with several equations of the form (1) motivates the study in this section of conservation laws of this class of equations. Let us recall that a conservation law of (1) is a divergence expression: that vanishes on the solution space of (1), where and are the usual total differential operators acting on the jet space of the underlying space of independent variables and dependent variable . Moreover, the notation denotes a differential function of , that is, a function depending on , , and and the derivatives of up to an arbitrary but fixed order. The vector is then called a conserved current, and may also be called a flux vector, as we shall do.
Conservation laws are also determined up to an equivalence class, whereby two conservation laws are termed equivalent if they differ by a trivial conservation law, that is, by one which vanishes for all smooth functions and not only on the solution space of (1). Such trivial conservation laws are said to be of the second kind as opposed to trivial conservation laws of the first kind in which the function itself, in (50), vanishes on all solutions of the equation [21]. Trivial conservation laws of the first kind will be eliminated by the choice of derivatives appearing as arguments in
To each conservation law (50), there corresponds a multiplier given by and two multipliers are equivalent if they differ by a trivial multiplier, that is, by one which vanishes itself on the solution space of the equation. For normal and totally nondegenerate equations of the form (1), there is a one-to-one correspondence between equivalent classes of multipliers and equivalent classes of conservation laws.
For Lagrangian equations, every characteristic of a variational symmetry is a multiplier and can be uniquely identified with a conservation law. However, Lagrangian equations constitute a very restricted type of equation and for more general equations for which the concept of variational symmetry in particular does not apply, multipliers are found as generalized integrating factors. That is, as factors for which the product is a total divergence expression, and hence, a null Lagrangian.
Our search for conservation laws will be focussed on those of low order, as physically relevant properties such as energy, mass-energy, and momentum conservation are always associated with these low-order conservation laws. In mathematical terms, these are conservation laws for which every derivative of a dependent variable in the equation can be obtained by differentiating with respect to some independent variable a similar derivative in the expression of the multiplier.
Conservation laws of (1) with are given for arbitrary values of in [30], although [30] does not make any reference to the corresponding multiplier. It is indeed possible to find a specific flux vector directly, albeit generally more intricate, by solving (50) on the solution surface of the equation. We complement the result of [30] by finding the multiplier which is to be sought in the form or . In view of obvious symmetry considerations associated with the structure of (1), it is enough to search for . The determining equations for the function are as follows:
with solution , for a certain constant parameter , which we may assume to be equal to thanks to the linearity of the Euler Operator. Indeed, is determined by the condition , where is the Euler Operator with respect to The corresponding determining equations for the flux vector takes the form
The latter determining equation has the following solution: where , , and are arbitrary functions. It turns out that this flux vector contains as a term a trivial flux, that is, one corresponding to a trivial conservation law, namely, the term
Using the symmetry argument related to the structure of (1) together with the above results show that the multipliers and corresponding conservation laws of (1) are given for arbitrary by
We shall make use of the same procedure used thus far in this section to find the conservation laws of other relevant equations of the form (1). In this way, the next equation from the class (1) with which we consider is the Liouville equation:
Although the conservation laws of (57), which is a well-studied equation [19, 20, 28] are likely to have been computed, we find it interesting to gather all such results in a single short paper. After calculations, the multipliers and corresponding conservations laws for (57) are given as follows: where and are arbitrary functions. This shows in particular that (57) has infinitely many conservation laws of low order. It has in fact been proved, that (58) is symmetry integrable, that is, it has an infinite series of generalized symmetries of arbitrary orders. Moreover [31], all symmetry integrable equations of the form (2a) are given up to scalings and shifts by functions satisfying
Similar integrability properties might also hold for equation (1) with , given that not only such an equation is similar by its type to (2a), but it also satisfies by Theorem 4, in the same way as (2a) does, the remarkable property of having a symmetry algebra of infinite dimension.
We now let in (2a)–(2d), so that the resulting equation takes the form
One sees that low-order multipliers are linear combinations of and . Letting the multiplier , the corresponding flux vector has the following expression: where , , and are arbitrary functions of their arguments. The pure flux vector associated with the multiplier , that is, the one deprived of any trivial flux vector is, however, given by
In other words, the aggregate of arbitrary functions in (61) only yields a trivial flux vector. These results and the symmetrical structure of (2a)–(2d) with respect to the variables and show that all multipliers and corresponding conservation laws of (60) are given by
In particular, (60) has only two conservation laws of low order.
For the counterpart (3a)–(3c) of (2a)–(2d) for which so that the equation reduces to , no general result of a similar nature concerning the symmetry integrability of this equation seems to be available. To begin with, for arbitrary values of , (3a) has only trivial multipliers and hence trivial conservation laws. Nevertheless, it turns out that for the particular case , the corresponding equation also has an infinity of conservation laws of low order.
Indeed, using our usual procedure described above, the general expression for the multiplier of low-order conservation laws of (64) takes the form where is an arbitrary function of its arguments, and is a solution of the second order linear hyperbolic equation
Although a general solution of (65b) is not available, we can find its particular solutions, and this will be enough to verify this claim. A practical and relatively simple way to obtain these particular solutions is to use the similarity reduction method yielding group-invariant solutions [21, 32–34] (see also [35, 36] for more recent applications of this reduction technique). In addition to the solution symmetry , where is any given solution of (65b), the other symmetries of this equation are given by
Group-invariant solutions , , and associated with the above symmetries , , and of (65b) and the corresponding multipliers , , and of (64) are given by where , the and are arbitrary constants for , and the arbitrary function is given by (65a). Finally, the flux vectors corresponding to the multipliers are given by
Clearly, the nontrivial conservation law for each of the pure flux vectors above are given by , and such low-order conservation laws are infinitely many as they are dependent on the arbitrary function , with .
5. Concluding Remarks and Future Outlook
The group classification of differential equations has proved to be a very challenging exercise, even for the simplest types of differential equations that odes represent, and we have made a breakthrough in this paper by finding a simple and systematic way of solving this type of problem, based on the determination of all possible cases of linearly dependent indeterminates in the determining equations. This has allowed us to give a complete classification of the family (1) of equations in a relatively brief manner. Although the method has only been discussed in application to the particular case of Equation (1), it seems, however, quite clear that it can be extended in a straightforward way to at least any scalar differential equation, and in fact to any system of odes or pdes.
It should also be noted that the difficulty with group classification is not really dependent on the dimension of the original equation or of the classifying determining equations, but rather, and with regard in particular to our method, on the number of indeterminate variables which occur themselves as arguments of arbitrary labelling functions. In other words, the classification problem of (1) would be more complicated to solve if one instead had for instance . The full treatment of such cases would give more insight into the method and hopefully point to ways for improvements.
Our method of course caters typically for cases where the arbitrary functions together with their arguments, or the arbitrary parameters viewed as definite functions of given arguments can be treated as indeterminate in the polynomial expression of determining equations. In that case the method is generally self-sufficient and yields the complete classification result. Else it might be used in combination with other methods.
The proposed method could be used to review some of the relevant cases of group classification already appearing in the literature and often cited in this paper, such as those given in papers [13] to [17] for instance. This will entail in particular the consideration of families of equations depending on more than one arbitrary function, each with its own argument, as well as cases where the classifying equations consist of a system of several equations. Moreover, the case of arbitrary labelling functions involving the independent variables should also be considered. Such an undertaking will allow not only to confirm the validity of existing classification results, but also to gain more insight into the method itself.
Calculations done in Section 4 show in particular that Equation (60) has only a finite number of low-order conservation laws, while (57) and (64) have infinitely many of them. However, while (57) is known to be up to two other related cases listed in (59) the only equations of the form (2a) that are symmetry integrable, nothing is known about the symmetry integrability of (64) and in particular about that for the more general case (3a). The symmetry integrability property of a differential equation remains a very active and challenging domain of research with very limited results, and it would be interesting to find out if (64) is symmetry integrable, and if so, whether such property also relates to the symmetry integrability of the whole family of equations (3a).
Data Availability
No data were used in this study.
Conflicts of Interest
The author declares that there is no conflict of interest for this publication.
Acknowledgments
This work was supported by the NRF Incentive Funding for Rated Researchers grant (Grant Number 97822); and the University of Venda (Grant Number I538).