Coefficient characterization of linear differential equations with maximal symmetries

By a result of Lie [1], a linear ordinary differential equation (ode) of a general order $n$ is known to have a symmetry algebra of maximal dimension $d_{n}$ if it is reducible by a point transformation to the equation $y^{(n)}=0$, which will henceforth be referred to as the canonical form of the linear equation. In a much recent paper Krause and Michel [2] proved the converse of this result and also showed that a linear equation is iterative if and only if its symmetry algebra has the maximal dimension $d_{n}.$ (By the cited result of Lie [1], $d_{n}=n+4$ for $n\geq 3$). Characterizing linear equations having a symmetry algebra of maximal dimension is therefore the same as characterizing linear equations that are reducible by a point transformation to the canonical form. The latter characterization for the third-order equation $y^{(3)}+c_{2}\,y^{\prime\prime}+c_{1}\,y^{\prime}+c_{0}\,y=0$ is due to Lie [3] and Laguerre [4] who showed independently that this equation is reducible to the canonical form if and only if its coefficients satisfy the equation

This characterization also clearly applies to all nonlinear odes which are linearizable by point transformations [5, 6], as the latter transformations do not alter the dimension of the symmetry algebra.

In this paper, we extend this characterization to equations of higher orders. It turns out that for each equation of order $n$ there will be $n-2$ characterizing equations, and the limitation of our presentation of the characterizing equations only up to the order five is simpy due to their very large size. However, we give a description of the method for deriving this characterization for equations of any order. The derivation of these characterizing equations is also based on the canonical normal form of linear equations admitting a maximal symmetry algebra that was obtained in [5] from a symmetry approach, and in [7] from an iterative approach. These characterizing equations therefore also represent a generalization of the results of [5] and [7]. We then deduce that the most general form of a linear equation admitting a maximal symmetry algebra can be expressed in standard form in terms of only two arbitrary functions. We also deduce that the Laguerre-Forsyth form of a linear equation reduces to the canonical form if and only if the equation has maximal symmetries.

Although we do not give the characterizing equations for each linear equation of order $n$, we note however that among the $n-2$ characterizing equations exactly one of them represents a semi-invariant of the equation, that is a function of the coefficients of the equation whose expression does not change when the dependent variable is transformed. We obtain an expression for these semi-invariants for equations of all orders and describe some of their properties.

Finally, using some simplifying assumptions and the method of [8], we give expressions for both the symmetry generator $X_{n}$ of $G_{\!S}$ and $X_{n}^{0}$ of the induced pseudo group of transformations $G_{c},$ and for all three most general canonical forms of linear equations of a general order $n.$ Here, $G_{\!S}$ denotes the symmetry group of the general linear equation in which the arbitrary functions are considered as additional dependent variables.

A method based on a symmetry approach has been proposed in [5] for characterizing the coefficients of linear ordinary differential equations (odes) that admit a maximal symmetry algebra, but only for equations in reduced normal form (in which the term of second highest order vanishes). In a more recent paper [7] a similar characterization based on an iterative approach was proposed, in which according to a result of Krause and Michel [2] a linear equation admitting a maximal symmetry is simply viewed as an iterative equation. By iterative equation, we mean an equation of the form

and where $r=r(x)$ and $s=s(x)$ are the parameters of the source equation $\Psi^{1}[y]=0$. This characterization shows that in its reduced normal form, a general linear equation depends solely on one arbitrary function $a=a(x)$. For equations of orders three to five, the corresponding equations are given as follows:

However, as a linear equation need not occur in its reduced normal form, but rather in the most general standard form, it is thus useful to obtain the corresponding characterization for equations in standard form. We let the general linear equation be given in standard form as

where $C=(c_{0},\dots,c_{n-1}).$ Suppose that such an equation has a symmetry algebra of maximal dimension and let its corresponding reduced normal form be given by

where the $B_{j}$ for $j=0,\dots,n-2$ are its coefficients and depend as already noted above on a single arbitrary function $a=B_{n-2}$ and its derivatives. Let

be the corresponding standard form of (5), which may be obtained by a transformation of the form

Then (4) and (6) must be identical, and in particular the nonzero coefficient $A_{n-1}$ introduced by the transformation (7) satisfies $A_{n-1}~{}=~{}c_{n-1}$, and more generally we have

Note that the coefficients $c_{j}$ in (4) are mere symbols and we wish to find a relationship among them. Given that in (5) the function $B_{n-2}$ is precisely the arbitrary function $a(x)$ labeling the equation, it can be shown by a recursive procedure, or even by induction on $n$ that

Therefore, solving the equation $c_{n-2}=A_{n-2}$ for $a$ gives

Consequently, the characterizing equations for linear equations in standard form with maximal symmetry algebra are given by the remaining $n-2$ equations

in which the function $a$ and its derivatives are substituted with the corresponding expressions given by (9).

As an immediate consequence of the corollary, linear equations such as $y^{(3)}+f(x)y=0$ or $y^{(4)}+f(x)y^{\prime}=0$ have maximal symmetry algebras if and only if the function $f(x)$ vanishes identically. We now make use of (10) and (9) to explicitly derive the characterizing equations for maximal symmetry algebras for equations of orders three to five.

For $n=3$, it is readily found that in (6) we have

while the corresponding expression for $a$ in (9) reduces to

Applying (12) into (11) and substituting the resulting expression for $A_{0}$ into (10) gives exactly the already cited equation (1) found by Lie [3] and Laguerre [4] and given by

The most general form of a linear third-order equation admitting a maximal symmetry algebra can thus be expressed in terms of only two arbitrary functions $c_{1}(x)$ and $c_{2}(x)$ in the form of

Equation (13) naturally reduces to (3a) for $c_{2}=0.$

For $n=4,$ we successively get

Substituting (14a) into (14b) and (14c) gives the two equations

which represent the characterizing equations for maximal symmetry algebra for equations of order $4$. Note that conversely any linear fourth order equation whose coefficients satisfy (15) must be iterative, which is why conditions such as (15) are termed characterizing equations. Indeed, if the coefficients of a fourth order equation of the form (4) satisfy (5), then its reduced normal form has, after the substitution of the expressions for $c_{0}$ and $c_{1}$ given by (5) in terms of $c_{2},c_{3},$ and their derivatives, the form

The coefficients $Q_{j}$ thus obtained clearly satisfy the conditions

prescribed by (3b) for iterative equations, as required.

For equations of order $n=5$, by proceeding as above for the orders three and four, we obtain the following $n-2=3$ characterizing equations

The group of equivalence transformations of the general linear equation (4) is given by invertible point transformations of the form

and they preserve the linearity and the homogeneity of the equation. Let

be the transformed version of (4) under (18). By a semi-invariant of (4) we shall mean a function $F=F(c_{0},c_{1},\dots,c_{n-1})$ of the coefficients of the equation which have the same expression for the transformed equation when the dependent variable (alone) changes. It is well known that under (18) the expression of the semi-invariant for the transformed equation is related to that for the original equation [9, 10] by the equality

where $\mu$ is an integer. In this case we say that the semi-variant $F$ has index $\mu.$ To each expression of the form $d^{k}c_{j}/dx^{k},$ let us assign the weight $(n-j)+k$, and we let this weight function be multiplicative so that the product $c_{p}c_{q}$ has weight $(n-p)+(n-q).$ It is well known that for a given semi-invariant all terms have the same weight and that this weight coincides with the index of the semi-invariant [9, 10] .

A closer look at the set of characterizing equations (10) shows that precisely one of them corresponds to a semi-invariant of the equation, namely the relation $c_{n-3}=A_{n-3}$, which gives rise to the semi-invariant $F=A_{n-3}-c_{n-3}$.

First of all, using the method of either [7] or [5], it can be proved that the coefficient $B_{n-3}$ in (5) satisfies $B_{n-3}=\frac{n-2}{2}a^{\prime}.$ Consequently, using the expression of the function $a$ in (9) it follows by induction on $n$ that the coefficient $A_{n-3}$ in (6) is given by

so that the corresponding invariant function $I_{n}$ has expression

The fact that the function $I_{n}=I_{n}(c_{0},c_{1},\dots,c_{n-1})$ in (22) is a semi-invariant can readily be verified. First each term in this expression has weight three, and we readily see that

which proves the assertion.

Although the invariant functions $I_{n}$ in (22) are originally defined only for $n\geq 3,$ their expression shows that they vanish identically for $n=1$ or $n=2$, by letting $c_{j}=0$ for $j<0.$ This vanishing can be interpreted by the fact that all first order and all second order linear equations are all equivalent through a point transformation to the equations $y^{\prime}=0$ and $y^{\prime\prime}=0$, respectively, and therefore they do not have nontrivial invariant functions.

On the other hand it should be noted that the other equations in the characterizing system (10) do not give rise to invariant functions except for the value $j=n-3$ in that system of equations. Indeed, denote collectively by $C$ and $Q$ the coefficients in equations (4) and (19), respectively, and for $n=4$ denote by $J(C)=1600(c_{0}-A_{0})$ the normalized function obtained from 2.9 with $j=0$. Then it can be seen that although each term in the expression of $J(C)$ has weight four, we have

clearly showing that the function $J$ is not a semi-invariant.

The equivalence group $G$ in (18) of the general linear equation (4) induces another Lie pseudo group $G_{c}$ acting on the coefficients of (4) [3]. For linear equations with maximal symmetries, their most general form depends as already noted on only two arbitrary functions, instead of $n$. For instance, the most general form of linear equations of order four admitting a maximal symmetry algebra is given on account of (15) by

and it is expressible solely in terms of the coefficients $c_{n-1}$ and $c_{n-2}$, here $c_{3}$ and $c_{2}$.

Although Eq. (23) is a very special case of the general Eq. (4), its equivalence group is the same group $G$ in (18) because equivalent equations have similar symmetry groups. Consequently the infinitesimal generators $X^{0}$ of the group $G_{c}$ for (4) will also be valid for equations with maximal symmetries. In particular to obtain the specific infinitesimal generators for equations with maximal symmetries expressed only in terms of the two arbitrary functions, it will be sufficient to substitute the characterizing equations (10) into the expression for $X^{0}$.

A method for finding the infinitesimal generator $X^{0}$ has been proposed in [8]. If we denote by

the infinitesimal generator of (4) in which the coefficients

are also considered as dependent variables, then the method of [8] consists of finding a set of minimum conditions for which the projection $V=\xi\,\partial_{x}+\eta\,\partial_{y}$ of $X$ on the $(x,y)$-space reduces to the infinitesimal generator $V^{0}=\left\{\xi^{0},\eta^{0}\right\}$ of the equivalence group $G$. This set of minimal conditions imposed to $\phi=(\phi_{0},\phi_{1},\dots,\phi_{n-1})$ yields a function $\phi^{0}=(\phi_{0}^{0},\phi_{1}^{0},\dots,\phi_{n-1}^{0})$ so that the expression for $X^{0}$ takes the form

In practice, the determination of the symmetry generator $X$ for the general linear equation (4) is computationally exhaustive, and a popular Lie symmetry software such as MathLie (See [11]) computes $X$ only for $n\leq 4$ due to computer memory problems (on an Intel Core2 Quad CPU machine) while another well-known similar Lie symmetry software such as SYM [12] does not compute symmetries such as $X$ that involve several dependent variables for a single independent variable.

We therefore need an efficient simplifying ansatz for the manual computation of $X^{0}$ at orders higher than the fourth. For this, we note that as the full symmetry group of (4) with $C$ considered also as dependent variable should leave the equation invariant, the transformation of the dependent and the independent variables should preserve the form of the equation, except for the introduction of a constant term independent of $y$ which should be offset by the subsequent transformations of the coefficient $C$. This means that in (24), we must have

A verification of (26) is possible by direct calculation for equations of order not higher than the fourth using the MathLie software, while for orders higher than four, the validity of the generators $X$ and $X^{0}$ found can be tested through the satisfaction of the infinitesimal condition of invariance applied to the general linear equation (4), and to the semi-invariants $I_{n}$ found in (22), respectively. Recall that the infinitesimal criterion of invariance for the infinitesimal generator $X$ of (4) is given by

where $X^{[n]}$ represents the $n$-th prolongation of $X.$ Regarding the verification of the infinitesimal condition of invariance for semi-invariants, we note that if for some group element $\alpha\in G_{c}$ we set $Q=\alpha\cdot C,$ then every semi-invariant of $G_{c}$ satisfies $F(\alpha\cdot C)=\mathbf{w}(\alpha)\cdot F(C)$ for some weight function $\mathbf{w},$ and $X^{0}$ is an infinitesimal generator of $G_{c}$ if and only if

for all such functions $F,$ where $\mathbf{w}(e)$ is the differential of $\mathbf{w}$ at the identity element $e$ of $G_{c}.$ In the actual case of (4) and $G_{c}$ (which is the same as G except that it acts on the space of coefficients), for $\alpha\equiv(f,g)$ specified in (18) we have $\mathbf{w}(\alpha)=f^{\prime}(z)^{3},$ and for each generator $X^{0}\equiv X^{0}(n)$ found, it is readily verified that

as required.

To our knowledge the infinitesimal generators $X^{0}$ of the induced pseudo group $G_{c}$ has been computed only for third order equations, or for the normal or the Laguerre-Forsyth forms of equations of low orders not exceeding five [13, 14, 9]. This is due in part as already mentioned to the intensive computational requirements for the calculation of these generators, but also because the more systematic method for finding them proposed in [8] is relatively recent.

We list in the next three theorems the general expressions for the infinitesimal generators $X_{n}$ of $G_{\!S}$ and $X_{n}^{0}$ of $G_{\!c}$ and for the three most general canonical forms of linear equations, where the subscript $n$ denotes the order of the equation.

The Laguerre-Forsyth form of the general linear equation is the equation of the form (4) in which the coefficients $c_{n-1}$ and $c_{n-2}$ of terms of second and third highest orders have vanished. In principle, such a transformation can be realized by means of the change of variables of the form

is the Schwarzian derivative, and $z^{\prime}=dz/dx.$ The Laguerre-Forsyth form of (4) is therefore of an implicit nature in the sense that (36) can not always be solved explicitly for $z$. Nevertheless, such a form is still of interest, in particular because linear equations often occur in this form.

Thanks to the ansatz (26) a direct computation of $X_{n}$ and $X_{n}^{0}$ for equations of low orders up to seven has been performed and confirms the validity of the expressions given in the three preceding theorems. It should also be noted that unlike the case of equations in standard or in normal forms, the generator $X_{n}^{0}$ of $G_{\!c}$ in the case of the Laguerre-Forsyth form involves only a finite number of constant parameters. This means that the invariant functions for this form of the general linear equation are much easier to compute, as already noted by Forsyth [10] who obtained an expression for them by a direct analysis.

As noted earlier, for equations with a maximal symmetry algebra which are already expressed solely in terms of the two coefficients $c_{n-1}$ and $c_{n-2},$ to obtain the corresponding infinitesimal generator $X^{0},$ it suffices to substitute in the expression for $X_{n}^{0}$ corresponding to the general linear equation (4) the corresponding characterizing equations which give an expression for the other coefficients solely in terms of $c_{n-1}$ and $c_{n-2}$ alone. For instance, for $n=4,$ the expression for $X_{n}^{0}$ corresponding to the normalized equation (23) has, on account of (15) and (29), an expression given by