Finite Rank and Complete Solutions of Euler-Poisson-Darboux Equations

1. Introduction

The main subject of this article are Euler-Poisson-Darboux (EPD) equations

$${\displaystyle w_{tt}+\frac{{\displaystyle k}}{{\displaystyle t}}{w}_t={\displaystyle \sum _{i=1}^n}\left(w_{x_i^2}+\frac{{\displaystyle {\gamma }_i}}{{\displaystyle x_i}}{w}_{x_i}\right)}$$

(1)

for a function $w(t,x_1,\dots ,x_n)$ with $n=1,2,3$; k and the ${\gamma }_i$ are constant parameters. The interest in these equations originates from its applications in electrodynamics, fluid and gas dynamics and General Relativity, among others. Special cases have been studied by Euler [6], Poisson [14] and Darboux [4]. More recent results, most of them for $n=1$, can be found in the papers of Aksenov [1], Bresters [2], Stewart [22], Shishkina [20,21], and the review by Moroşanu [13]. Explicit solutions may also be found in the collections by Polyanin [15] and Zwillinger [23].

For linear partial differential equations (PDEs) of second order in the plane, Laplace determined special transformations whose repeated application may lead to an equation that has a solution in closed form, details may be found in Goursat’s book [8]. Reversing these transformations the solution of the originally given equation may be obtained. Usually it contains an undetermined function and its derivatives up to a certain order which is called its rank. An example is Equation (1) for $n=1$, ${\gamma }_1=0$ and $k=-2$, i. e.

$$w_{tt}-w_{xx}-{\displaystyle \frac{2}{t}}{w}_t=0.$$

(2)

Its solution

$$w_{Gen}(t,x)=f\left(\alpha \right)+g\left(\beta \right)-t\left(f^{\prime }\left(\alpha \right)+g^{\prime }\left(\beta \right)\right)$$

(3)

is the sum of two rank 1 solutions corresponding to $\alpha =t+x$ and $\beta =t-x$, respectively; $f\left(\alpha \right)$ and $g\left(\beta \right)$ are undetermined functions of its argument. Laplace’s method does not generalize to higher-order equations, or to equations in more than two independent variables. Furthermore, in general these solutions may not be obtained by decomposition of the given equation either. Therefore the solution procedure proposed here searches directly for finite solutions of a fixed rank.

For every differential equation, the question arises as to how the comprehensiveness of a given solution can be characterized; here solutions with so-called partial quadratures are excluded. It is defined by the indeterminate elements it contains. These may be constants or undetermined functions depending on one or more independent variables. The following notation was introduced by Kolchin [10]. The undetermined function with the largest number of arguments determines the differential type. The number of functions with this property determines its typical differential dimension. The pair of these two quantities is called the differential dimension. A solution with the differential dimension of the equation in question is called a general solution. For Eq. (2) these values are 1 and 2, respectively, together they are denoted by the pair $\mathcal{D}\left(w_{Gen}\right)=(1,2)$, consequently (3) is a general solution of (2).

In addition to finite-rank solutions, so-called complete solutions of (1) will be determined. By definition, a solution is complete if it contains a sufficient number of constants whose elimination from this solution and its derivatives up to the order of the equation yields this equation uniquely [7], page 8. For Eq. (1) this number is $N\equiv {\displaystyle \frac{1}{2}}\left({(n+2)}^2+n\right)$, consequently the differential dimension of a complete solution is $\mathcal{D}\left(w_{Com}\right)=(0,N)$. Although they comprise only a small part of a general solution, they may have interesting applications, for example in fluid dynamics [5], page 51. It turns out that many of them may not be obtained by specialization of finite rank solutions, i.e. they are genuinly new solutions of EPD equations.

A significant part of the following calculations uses so-called Janet bases [16], which are based on the work of Maurice Janet. A good introduction to Janet’s theory can be found at Iohara and Malbos [9]. Janet bases are the differential counterpart of the Groebner bases introduced by Buchberger [3]. Both have the common property that very large expressions can occur in intermediate calculations that are practically impossible to perform by hand. Computer algebra tools in Maple [11], Mathematica [12] and Reduce [18] are available for this purpose.

The proceeding for determining finite rank solutions of Eq.(1) is described in Section 2. In Section 3 several examples in space dimension 1, 2 and 3 are elaborated in detail, and corresponding complete solutions are obtained. In the final Section 4 possible extensions of this work are discussed. Some preliminary results on this topic were reported in [19].

2. The Main Result

The proposed solution scheme for generating finite-rank solutions of Eq.(1) is based on the following theorem.

Theorem 1. 

Let an EPD-equation (1) be given. The coefficients $a_k(t,x_1,x_2,\dots ,x_n)$ of a solution of finite rank K of the form

$${\displaystyle w(t,x_1,x_2,\dots ,x_n)={\displaystyle \sum _{k=0}^K}{a}_k(t,x_1,x_2,\dots ,x_n)f^{\left(k\right)}\left(\alpha \right)}$$

(4)

are determined by the following system of linear PDEs.

$$\begin{array}{c}{L}_1\left(a_K\right)+{\displaystyle \frac{1}{2}}{L}_2\left(\alpha \right)a_K=0,\\ L_1\left(a_{k-1}\right)+{\displaystyle \frac{1}{2}}{L}_2\left(\alpha \right)a_{k-1}+{\displaystyle \frac{1}{2}}{L}_2\left(a_k\right)=0\mathrm{for}k=1,\dots ,K,\\ L_2\left(a_0\right)=0.\end{array}$$

(5)

The operators $L_1$ and $L_2$ are defined by

$$\begin{array}{c}{\displaystyle L_2\equiv {\partial }_{tt}-{\displaystyle \sum _{i=1}^n}{\partial }_{x_i{x}_i}-\frac{{\displaystyle k}}{{\displaystyle t}}{\partial }_t-{\displaystyle \sum _{i=1}^n}\frac{{\displaystyle {\gamma }_i}}{{\displaystyle x_i}}{\partial }_{x_i},}\\ {\displaystyle L_1\equiv {\alpha }_t{\partial }_t-{\displaystyle \sum _{i=1}^n}{\alpha }_{x_i}{\partial }_{x_i}.}\end{array}$$

(6)

In addition there is a first-order PDE determining the argument $\alpha (t,x_1,\dots ,x_n)$ of f.

$${\displaystyle {\alpha }_t^2-{\displaystyle \sum _{i=1}^n}{\alpha }_{x_i}^2=0.}$$

(7)

  Proof Substituting (4) in (1) and rearranging its terms properly the following expression is obtained.

$$\begin{array}{c}{\displaystyle {\displaystyle \sum _{k=0}^K}{L}_2\left(a_k\right)f^{\left(k\right)}\left(\alpha \right)+{\displaystyle \sum _{k=0}^K}(2L_1\left(a_k\right)+L_2\left(\alpha \right)a_k)f^{(k+1)}\left(\alpha \right)}\\ {\displaystyle +({\alpha }_t^2-{\alpha }_x^2-{\alpha }_y^2-{\alpha }_z^2){\displaystyle \sum _{k=0}^K}{a}_k{f}^{(k+2)}\left(\alpha \right)=0.}\end{array}$$

(8)

The coefficients $a_k$ and the argument $\alpha$ must be determined such that (4) satisfies Eq. (1) identically. Because f is considered as an undetermined function this is only possible if the coefficients of the derivatives $f^{\left(k\right)}$ in (8) vanish individually. If $\alpha$ is a solution of (7) the last sum in (8) vanishes. The zeroth derivative, i. e. the function f itself, occurs only once in the first sum of (8), it yields the last equation of (5). The derivative $f^{(K+1)}$ in the second sum of (8) yields the first equation of (5). The derivatives $f^{\left(1\right)},\dots ,f^{\left(K\right)}$ in the first and the second sum of (4) combined yield the equations in the second line of (5). □

The equations (5) are a diagonal system of linear first-order PDEs that may be solved for $a_K,a_{K-1},\dots ,a_1,a_0$ by successive integrations. It turns out that it may be transformed in a Janet basis [16] that is more suitable for the solution procedure as is shown next.

Corollary 1. 

Let a system (5) considered in the above theorem originating from an EPD with $n\le 3$ and $\left|k\right|,|{\gamma }_i|\le 8$ be given. A corresponding Janet basis in lex term order with $a_0>a_1>\dots >a_K$ has the form

$$\begin{array}{c}{L}_1\left(a_K\right)+{\displaystyle \frac{1}{2}}{L}_2\left(\alpha \right)a_K=0,\\ l_K\left(a_K\right)=0,\\ a_k=l_k\left(a_K\right)\mathrm{for}k=0,1,\dots ,K-1.\end{array}$$

(9)

$L_1$ and $L_2$ are the operators (6); $l_K\left(a_K\right)$ is a linear PDE for $a_K$ and its derivatives w.r.t. the space variables $x_1,\dots ,x_n$, it does not necessarily occur. The equations in the last line express $a_0,a_1,\dots ,a_{K-1}$ explicitly in terms of $a_K$.

Proof  The form of the Janet basis (9) has been established by explicit calculation for the given parameter ranges. □ 

The special structure of system (9) suggests the following solution procedure. The linear first-order PDE for $a_K$ is solved first. Then the constraints due to the second equation for $a_K$ are imposed on this solution. The Janet basis property guarantees that this is always possible and the existence of a solution of rank K is established. It is not necessary to continue with the general solution of $a_K$, rather a non-trivial special solution suffices. Finally the remaining coefficients $a_0,a_1,\dots ,a_{K-1}$ are determined using the explicit expressions in the last line of Eq. (9).

In applications it often occurs that an equation contains undetermined parameters. In such cases it is part of the solution problem to identify those particular values of the parameters for which a nontrivial solution of the desired form may exist. To do this, the equation with undetermined parameter values is considered first. The result is a list of those special parameter values that may allow for a non-trivial solution. For small values of n and K the answer is given next.

Corollary 2. 

For $n=1,2,3$ and $0\le K\le 3$ the following parameter values in Eq. (1) lead to a Janet basis with a nontrivial solution.

$$\begin{array}{ccccc}K& 0& 1& 2& 3\\ k& 0\le k\le 2& -2\le k\le 4& -4\le k\le 6& -6\le k\le 8\\ {\gamma }_i& -2\le {\gamma }_i\le 0& -4\le {\gamma }_i\le 2& -6\le {\gamma }_i\le 4& -8\le {\gamma }_i\le 6\end{array}$$

The general rule appears to be $-2K\le k\le 2(K+1)$ and $-(2K+1)\le {\gamma }_i\le 2K$ are compatible values, it has been verified up to $K=10$.

Proof For the given range of n and K this has been shown by explicit calculation. □ 

Once a finite rank solution has been found, it is of interest to explicitly describe the equations, in the original space-time variables, whose general solution is this finite rank solution. This provides a decision procedure to decide whether any other solution can be represented by a particular finite rank solution. In particular this applies to the complete solutions to be determined later in this article. To this end, the process described in Corollary 1 is reversed as described next.

Corollary 3. 

Let a finite rank solution containing undetermined functions $f_k\left({\alpha }_k\right)$, $k=1,\dots ,K$ be known. In order that a function $\phi (t,x_1,\dots ,x_n)$ may be obtained by specialization of this finite rank solution it must obey the consistency conditions for the system expressing φ in terms of the finite rank solution, supplemented by the conditions ${\displaystyle \frac{{\displaystyle \partial f_j}}{{\displaystyle \partial {\alpha }_k}}}=0$, $j\ne k$, expressed in terms of the space-time variables $t,x_1,\dots ,x_n$.

Proof The above system is a linear homogeneous system for the undetermined functions $f_k$, the only inhomogeneity being $\phi$. Furthermore, it is known that there exist solutions for special values of $\phi$. Transforming it in a Janet basis with a priory undetermined $\phi$, the resulting consistency conditions determine the general case for their existence. □

The special structure of EPD equations entails that polynomials of a certain order d are completely transformed to order $d-2$ when inserted. This means that the existence of polynomial solutions can be considered for each degree separately. As a result, the linear algebra problem of determining the coefficients of a complete solution, or a polynomial solution in general, is much less complex.

The proceedings described above will now be applied to several concrete examples for space dimensions one, two or three.

3. Solving EPD Equations in Low Space Dimensions

From now on, the names x, y and z are chosen for the space variables, and for the parameters ${\gamma }_i$ the variables a, b, and c.

Example 1. 

The following equation in one space dimension containing parameters $k=a=-2$

$$w_{tt}-w_{xx}-{\displaystyle \frac{{\displaystyle 2}}{{\displaystyle t}}}{w}_t-{\displaystyle \frac{{\displaystyle 2}}{{\displaystyle x}}}{w}_x=0$$

(10)

with $\mathcal{D}\left(w\right)=(1,2)$ is considered first. Equation (7) is ${\alpha }_t^2-{\alpha }_x^2=0$, it yields the solutions $t\pm x$. With $\alpha =t+x$ the operators

$$L_2={\partial }_{tt}-{\partial }_{xx}-{\displaystyle \frac{{\displaystyle 2}}{{\displaystyle t}}}{\partial }_t-{\displaystyle \frac{{\displaystyle 2}}{{\displaystyle x}}}{\partial }_x\mathrm{and}{L}_1={\partial }_t+{\partial }_x+{\displaystyle \frac{{\displaystyle t-x}}{{\displaystyle tx}}}$$

are obtained, they generate the system (5). The corresponding Janet basis is

$$a_{1,t}+a_{1,x}+{\displaystyle \frac{t-x}{tx}}{a}_1=0,a_0+a_{1,x}+{\displaystyle \frac{{\displaystyle t+x}}{{\displaystyle tx}}}{a}_1=0.$$

(11)

The equation $l_1\left(a_1\right)$ in (9) does not occur in this example. The system (11) has the solutions $a_0={\displaystyle \frac{{\displaystyle 1}}{{\displaystyle x}}}$ and $a_1=-{\displaystyle \frac{{\displaystyle t}}{{\displaystyle x}}}$, they yield

$$w_f={\displaystyle \frac{{\displaystyle 1}}{{\displaystyle x}}}f\left(\alpha \right)-{\displaystyle \frac{{\displaystyle t}}{{\displaystyle x}}}{f}^{\prime }\left(\alpha \right)$$

(12)

with an undetermined function $f\left(\alpha \right)$. The other solution $\beta =t-x$ of Eq.(7) leads to a similar expression containing an undetermined function $g\left(\beta \right)$. The two solutions combined yield the solution

$$w_{Gen}(t,x)={\displaystyle \frac{{\displaystyle 1}}{{\displaystyle x}}}\left(f\left(\alpha \right)+g\left(\beta \right)\right)-{\displaystyle \frac{{\displaystyle t}}{{\displaystyle x}}}\left(f^{\prime }\left(\alpha \right)+g^{\prime }\left(\beta \right)\right)$$

(13)

of Eq. (10) with $\mathcal{D}\left(w_{Gen}\right)=(1,2)$.

The system described in Corollary 3 is

$$\begin{array}{c}{\displaystyle \frac{{\displaystyle 1}}{{\displaystyle x}}}\left(f\left(\alpha \right)+g\left(\beta \right)\right)-{\displaystyle \frac{{\displaystyle t}}{{\displaystyle x}}}\left({\displaystyle \frac{{\displaystyle \partial f}}{{\displaystyle \partial \alpha }}}\right)+{\displaystyle \frac{{\displaystyle \partial g}}{{\displaystyle \partial \beta }}})=\phi (t,x),{\displaystyle \frac{{\displaystyle \partial f}}{{\displaystyle \partial \beta }}}=0,{\displaystyle \frac{{\displaystyle \partial g}}{{\displaystyle \partial \alpha }}}=0.\end{array}$$

(14)

Replacing $\alpha$ and $\beta$ by t and x and transforming it in a Janet basis yields the only consistency condition ${\phi }_{tt}-{\phi }_{xx}-{\displaystyle \frac{{\displaystyle 2}}{{\displaystyle t}}}{\phi }_t-{\displaystyle \frac{{\displaystyle 2}}{{\displaystyle x}}}{\phi }_x=0$; this follows from the fact that (13) is a general solution of Eq. (10).

A complete solution of equation (10) containing five undetermined constants $C_1,\dots ,C_5$ is

$$w_{Com}(t,x)=C_1+C_2(3t^2-x^2)+C_3{t}^3+C_4(15t^4+10t^2{x}^2-x^4)+C_5(3t^5+5t^3{x}^2).$$

(15)

Due to the generality of $w_{Gen}$ each of its components may be obtained by specialization from it. It is conjectured that there are d independent polynomial solutions for degree d. □

The next examples differ significantly from the preceding one due to the fact that the finite rank solutions obtained are special, i.e. they may be obtained by specialization of a general solution, as the following example for $n=2$ shows.

Example 2. 

The following equation in two space dimensions with $k=a=b=-2$

$$w_{tt}-w_{xx}-w_{yy}-{\displaystyle \frac{{\displaystyle 2}}{{\displaystyle t}}}{w}_t-{\displaystyle \frac{{\displaystyle 2}}{{\displaystyle x}}}{w}_x-{\displaystyle \frac{{\displaystyle 2}}{{\displaystyle y}}}{w}_y=0$$

(16)

with $\mathcal{D}\left(w\right)=(2,2)$ is considered. Eq. (7) yields the solutions $\sqrt{2}t\pm x\pm y$. At first $\alpha \equiv \sqrt{2}t+x+y$ is chosen. The operators $L_1$ and $L_2$ defined by (6) yield the system (5). Transforming it in a Janet basis the following system for the coefficient $a_1$ is obtained.

$$\begin{array}{c}{a}_{1,xxxxx}-4a_{1,xxxy}+6a_{1,xxyy}-4_{a_1,xyyy}+a_{1.yyyy}\hfill \\ -{\displaystyle \frac{{\displaystyle 4(x-y)}}{{\displaystyle xy}}}(a_{1,xxx}-3a_{1,xxy}+3a_{1,xyy}-a_{1,yyy})\hfill \\ -{\displaystyle \frac{{\displaystyle 12}}{{\displaystyle xy}}}(a_{1,xx}-2a_{1,xy}+a_{1,yy})=0,\hfill \end{array}$$

(17)

$$\begin{array}{c}{a}_{1,t}-{\displaystyle \frac{{\displaystyle 1}}{{\displaystyle \sqrt{2}}}}{a}_{1,x}-{\displaystyle \frac{{\displaystyle 1}}{{\displaystyle \sqrt{2}}}}{a}_{1,y}+{\displaystyle \frac{{\displaystyle \sqrt{2}xy+tx+ty}}{{\displaystyle \sqrt{2}txy}}}{a}_1=0.\end{array}$$

(18)

A complete solution of (17) involving 5 parameters $C_i$ is

$$a_1=(C_1+C_{2}x)xy(x-y)+(C_3+C_{4}x+C_{5}y)xy(x^2-y^2);$$

the $C_i$ are constants w.r.t. the space variables x and y, but may depend on t. This latter dependence has to be determined such that (18) is also satisfied. It turns out that a valid choice is $C_i=0$ for $i\ne 1$ and $C_1$ a solution of $C_1^{\prime }+{\displaystyle \frac{{\displaystyle 1}}{{\displaystyle t}}}{C}_1=0$, i. e. $C_1={\displaystyle \frac{{\displaystyle 1}}{{\displaystyle t}}}$. There follows $a_0={\displaystyle \frac{{\displaystyle 1}}{{\displaystyle xy}}}$ and $a_1=-{\displaystyle \frac{{\displaystyle \sqrt{2}t}}{{\displaystyle xy}}}$. A corresponding rank 1 solution involving an undetermined function f of $\alpha =\sqrt{2}t+x+y$ is

$$w_f(t,x,y)={\displaystyle \frac{1}{xy}}f\left(\alpha \right)-{\displaystyle \frac{\sqrt{2}t}{xy}}{f}^{\prime }\left(\alpha \right).$$

By a similar analysis rank 1 solutions corresponding to $\beta \equiv \sqrt{2}t+x-y$ and $\gamma \equiv \sqrt{2}t-x-y$ may be obtained. The linear combination

$$w(t,x,y)={\displaystyle \frac{{\displaystyle 1}}{{\displaystyle xy}}}\left(f\left(\alpha \right)+g\left(\beta \right)+h\left(\gamma \right)\right)-{\displaystyle \frac{{\displaystyle \sqrt{2}t}}{{\displaystyle xy}}}\left(f^{\prime }\left(\alpha \right)+g^{\prime }\left(\beta \right)+h^{\prime }\left(\gamma \right)\right)$$

(19)

with undetermined functions f, g and h is a solution of (16) with differential dimension $\mathcal{D}\left(w\right)=(1,3)$.

The system defined in Corollary 3 is

$$\begin{array}{c}{\displaystyle \frac{{\displaystyle 1}}{{\displaystyle xy}}}\left(f\left(\alpha \right)+g\left(\beta \right)+h\left(\gamma \right)\right)-{\displaystyle \frac{{\displaystyle \sqrt{2}t}}{{\displaystyle xy}}}\left({\displaystyle \frac{{\displaystyle \partial f}}{{\displaystyle \partial \alpha }}}+{\displaystyle \frac{{\displaystyle \partial g}}{{\displaystyle \partial \beta }}}+{\displaystyle \frac{{\displaystyle \partial h}}{{\displaystyle \partial \gamma }}}\right)=\phi (t,x,y),\\ {\displaystyle \frac{{\displaystyle \partial f}}{{\displaystyle \partial \beta }}}=0,{\displaystyle \frac{{\displaystyle \partial f}}{{\displaystyle \partial \gamma }}}=0,{\displaystyle \frac{{\displaystyle \partial g}}{{\displaystyle \partial \alpha }}}=0,{\displaystyle \frac{{\displaystyle \partial g}}{{\displaystyle \partial \gamma }}}=0,{\displaystyle \frac{{\displaystyle \partial h}}{{\displaystyle \partial \alpha }}}=0,{\displaystyle \frac{{\displaystyle \partial h}}{{\displaystyle \partial \beta }}}=0.\end{array}$$

(20)

Replacing $\alpha$, $\beta$ and $\gamma$ by t, x and y, and transforming it in a Janet basis the following consistency conditions for $\phi (t,x,y)$ are obtained.

$$\begin{array}{c}{\phi }_{tt}-2{\phi }_{yy}-{\displaystyle \frac{{\displaystyle 2}}{{\displaystyle t}}}{\phi }_t-{\displaystyle \frac{{\displaystyle 2}}{{\displaystyle y}}}{\phi }_y=0,{\phi }_{xx}-{\phi }_{yy}-{\displaystyle \frac{{\displaystyle 2}}{{\displaystyle x}}}{\phi }_x-{\displaystyle \frac{{\displaystyle 2}}{{\displaystyle y}}}{\phi }_y=0,\\ {\phi }_{txy}-{\phi }_{tyy}+\sqrt{2}{\phi }_{xyy}-\sqrt{2}{\phi }_{yyy}+{\displaystyle \frac{{\displaystyle 2t+\sqrt{2}y}}{{\displaystyle 2ty}}}{\phi }_{tx}-{\displaystyle \frac{{\displaystyle 4tx-2ty+\sqrt{2}xy}}{{\displaystyle 2txy}}}{\phi }_{ty}\\ +{\displaystyle \frac{{\displaystyle 2\sqrt{2}}}{{\displaystyle y}}}{\phi }_{xy}-{\displaystyle \frac{{\displaystyle \sqrt{2}x-\sqrt{2}y}}{{\displaystyle xy}}}{\phi }_{yy}+{\displaystyle \frac{{\displaystyle 2t-\sqrt{2}x+\sqrt{2}y}}{{\displaystyle 2txy}}}{\phi }_t+{\displaystyle \frac{{\displaystyle 2\sqrt{2}}}{{\displaystyle xy}}}{\phi }_y=0.\end{array}$$

(21)

These equations determine the constraints for any function to be represented as a finite rank solution (19).

A complete solution of Eq. (16) involving nine undetermined constants is

$$\begin{array}{c}w(t,x,y)=C_1+C_2(3t^2-x^2)+C_3(3t^2-y^2)+C_4{t}^3+C_5(15t^4+10t^2{x}^2-x^4)\hfill \\ +C_6(15t^4+10t^2{y}^2-y^4)+C_7(9t^4+3t^2{x}^2+3t^2{y}^2-x^2{y}^2)\hfill \\ +C_8(3t^5+5t^3{x}^2)+C_9(3t^5+t^3{y}^2).\hfill \end{array}$$

(22)

Substitution in (21) shows that all non-constant solutions of (22) do not satisfy system (21), i.e. they cannot be obtained as special finite-rank solutions (19). It is conjectured that there are ${\displaystyle \frac{1}{2}}(d-1)$ independent polynomial solutions for odd d and ${\displaystyle \frac{1}{2}}(d+2)$ for even d. □

Example 3. 

In addition to the time variable t this example contains three space variables x, y and z; as usual $k=a=b=c=-2$.

$$w_{tt}-w_{xx}-w_{yy}-w_{zz}-{\displaystyle \frac{{\displaystyle 2}}{{\displaystyle t}}}{w}_t-{\displaystyle \frac{{\displaystyle 2}}{{\displaystyle x}}}{w}_x-{\displaystyle \frac{{\displaystyle 2}}{{\displaystyle y}}}{w}_y-{\displaystyle \frac{{\displaystyle 2}}{{\displaystyle z}}}{w}_z=0.$$

(23)

Eq. (7) leads to the solutions $\sqrt{3}t\pm x\pm y\pm z$. With $\alpha \equiv \sqrt{3}t+x+y+z$ the operators $L_1$ and $L_2$ defined by (6) follow, they yield the system (5). In the next step, this system is transformed in a Janet basis as described in Corollary 1. Because of its size, it cannot be fully specified here. It leads to a rank 1 solution $w_f$ containing an undetermined function f of $\alpha$.

$$w_f={\displaystyle \frac{{\displaystyle 1}}{{\displaystyle xyz}}}f\left(\alpha \right)-{\displaystyle \frac{{\displaystyle \sqrt{3}t}}{{\displaystyle xyz}}}{f}^{\prime }\left(\alpha \right).$$

(24)

Three more solutions of Eq. (7) are

$$\begin{array}{c}\beta \equiv \sqrt{3}t+x+y-z,\gamma \equiv \sqrt{3}t+x-y-z,\delta \equiv \sqrt{3}t-x-y-z.\end{array}$$

They yield three additional solutions $w_g$, $w_h$ and $w_k$ of (23) with undetermined functions $g,h$ and k, respectively. The linear combination

$$w(t,x,y,z)={\displaystyle \frac{{\displaystyle 1}}{{\displaystyle xyz}}}\left(f\left(\alpha \right)+g\left(\beta \right)+h\left(\gamma \right)+k\left(\delta \right)\right)-{\displaystyle \frac{{\displaystyle \sqrt{3}t}}{{\displaystyle xyz}}}\left(f^{\prime }\left(\alpha \right)+g^{\prime }\left(\beta \right)+h^{\prime }\left(\gamma \right)+k^{\prime }\left(\delta \right)\right).$$

(25)

is a solution of (23) with $\mathcal{D}\left(w\right)=(1,4)$.

The system described in Corrollary 3 is

$$\begin{array}{c}{\displaystyle \frac{{\displaystyle 1}}{{\displaystyle xyz}}}\left(f\left(\alpha \right)+g\left(\beta \right)+h\left(\gamma \right)+k\left(\delta \right)\right)-{\displaystyle \frac{{\displaystyle \sqrt{3}t}}{{\displaystyle xyz}}}\left({\displaystyle \frac{{\displaystyle \partial f}}{{\displaystyle \partial \alpha }}}+{\displaystyle \frac{{\displaystyle \partial g}}{{\displaystyle \partial \beta }}}+{\displaystyle \frac{{\displaystyle \partial h}}{{\displaystyle \partial \gamma }}}+{\displaystyle \frac{{\displaystyle \partial k}}{{\displaystyle \partial \delta }}}\right)=\phi (t,x,y,z),\\ {\displaystyle \frac{{\displaystyle \partial f}}{{\displaystyle \partial \beta }}}=0,{\displaystyle \frac{{\displaystyle \partial f}}{{\displaystyle \partial \gamma }}}=0,{\displaystyle \frac{{\displaystyle \partial f}}{{\displaystyle \partial \delta }}}=0,{\displaystyle \frac{{\displaystyle \partial g}}{{\displaystyle \partial \alpha }}}=0,{\displaystyle \frac{{\displaystyle \partial g}}{{\displaystyle \partial \gamma }}}=0,{\displaystyle \frac{{\displaystyle \partial g}}{{\displaystyle \partial \delta }}}=0,{\displaystyle \frac{{\displaystyle \partial h}}{{\displaystyle \partial \alpha }}}=0,{\displaystyle \frac{{\displaystyle \partial k}}{{\displaystyle \partial \alpha }}}=0,{\displaystyle \frac{{\displaystyle \partial k}}{{\displaystyle \partial \beta }}}=0,{\displaystyle \frac{{\displaystyle \partial k}}{{\displaystyle \partial \gamma }}}=0,\end{array}$$

(26)

Replacing $\alpha$, $\beta$, $\gamma$ and $\delta$ by t, x, y and z and transformating it in a Janet basis yields the consistency conditions

$$\begin{array}{c}{\phi }_{tt}-3{\phi }_{zz}-{\displaystyle \frac{{\displaystyle 2}}{{\displaystyle t}}}{\phi }_t-{\displaystyle \frac{{\displaystyle 6}}{{\displaystyle z}}}{\phi }_z=0,{\phi }_{xx}-{\phi }_{zz}+{\displaystyle \frac{{\displaystyle 2}}{{\displaystyle x}}}{\phi }_x-{\displaystyle \frac{{\displaystyle 2}}{{\displaystyle z}}}{\phi }_z=0,\\ {\phi }_{yy}-{\phi }_{zz}+{\displaystyle \frac{{\displaystyle 2}}{{\displaystyle y}}}{\phi }_y-{\displaystyle \frac{{\displaystyle 2}}{{\displaystyle z}}}{\phi }_z=0,\\ {\phi }_{xy}-{\phi }_{xz}+{\phi }_{yz}-{\phi }_{zz}-{\displaystyle \frac{{\displaystyle y-z}}{{\displaystyle yz}}}{\phi }_x+{\displaystyle \frac{{\displaystyle x+z}}{{\displaystyle xz}}}{\phi }_y-{\displaystyle \frac{{\displaystyle 2xy-xz+yz}}{{\displaystyle xyz}}}{\phi }_z+{\displaystyle \frac{{\displaystyle x-y+z}}{{\displaystyle xyz}}}+\phi =0,\\ {\phi }_{txz}-{\phi }_{tzz}+\sqrt{3}{\phi }_{xzz}-\sqrt{3}{\phi }_{zzz}+{\displaystyle \frac{{\displaystyle 3t+\sqrt{3}z}}{{\displaystyle 3tz}}}{\phi }_{tx}-{\displaystyle \frac{{\displaystyle 6tx-3tz+\sqrt{3}xz}}{{\displaystyle 3txz}}}{\phi }_{tz}\\ +{\displaystyle \frac{{\displaystyle 2\sqrt{3}}}{{\displaystyle z}}}{\phi }_{xz}-{\displaystyle \frac{{\displaystyle \sqrt{3}x-z}}{{\displaystyle xz}}}{\phi }_{zz}+{\displaystyle \frac{{\displaystyle 3t-\sqrt{3}y+\sqrt{3}z}}{{\displaystyle 3tyz}}}{\phi }_t+{\displaystyle \frac{{\displaystyle 2\sqrt{3}}}{{\displaystyle xz}}}{\phi }_z=0.\end{array}$$

(27)

These are the sufficient conditions for any function $\phi (t,x,y,z)$ to be a special case of (25).

A complete solution of Eq. (23) is

$$\begin{array}{c}{w}_{Com}=C_1+C_2(t^2-3x^2)+C_3(t^2-3y^2)+C_4(t^2-3z^2)+C_5{x}^3+C_6{y}^3+C_7{z}^3\hfill \\ +C_8(t^4-10t^2{x}^2-15x^4)+C_9(t^4-10t^2{y}^2-15y^4)+C_{10}(t^4-10t^2{z}^2-15z^4)\hfill \\ +C_{11}(5t^2{x}^3+3x^5)+C_{12}(5t^2{y}^3+3y^5)+C_{13}(5t^2{z}^3+3z^5)\hfill \\ +C_{14}(t^2{x}^3+t^2{y}^3+t^2{z}^3-3x^3{y}^2-3x^2{y}^3-3x^2{z}^3).\hfill \end{array}$$

Eliminating its 14 constants yields uniquely (23). Its non-constant components do not satisfy system (27), i.e. they are not special cases of the rank 1 solution (25). It is conjectured that there are ${\displaystyle \frac{1}{8}}(d^2-1)$ independent polynomial solutions for odd d and ${\displaystyle \frac{1}{8}}(d+2)(d+4)$ for even d.     □

4. Summary and Outlook

The main results of this article are twofold. On the one hand new closed-form solutions of various classes of EPD equations (1) have been determined, i.e. finite-rank solutions of differential dimension $\mathcal{D}\left(w\right)=(1,n)$, and complete solutions of differential dimension $\mathcal{D}\left(w\right)=(0,N)$ with $N={\displaystyle \frac{1}{2}}({(n+2)}^2+n)$. Second, the method for determining finite rank solutions described in Theorem 1 and its Corrolaries is completely algorithmic and more versatile than the Laplace transform method. Therefore, it can be applied to linear partial differential equations of any order and any number of independent variables with the aim of determining more classes of closed form solutions with higher differential dimension. This makes them more comprehensive, potentially they describe new phenomena in applications and may serve as a tool for testing numerical methods. Furthermore, for each n complete solutions were determined that are not special solutions of finite rank.

Of particular interest are, of course, general solutions, i.e. the most comprehensive ones at all. If they exist in closed form, it is sometimes possible to determine them using procedures based on Theorem 1 and its corollaries. An example is the equation

$$w_{tt}-w_{tx}-w_{ty}+w_{xy}+{\displaystyle \frac{{\displaystyle k}}{{\displaystyle x+y}}}(2w_t-w_x-w_y)=0.$$

(28)

For $k=-3$ one obtains the following rank 2 solution for an undetermined function $f({\alpha }_1,{\beta }_1)$ with ${\alpha }_1=t+x,{\beta }_1=y$.

$$w_f(t,x,y)={\displaystyle \frac{{\displaystyle 12t+18x+6y}}{{\displaystyle {(x+y)}^5}}}f+{\displaystyle \frac{{\displaystyle 6t+8x+2y}}{{\displaystyle {(x+y)}^4}}}\left(f_{{\alpha }_1}-f_{{\beta }_1}\right)+{\displaystyle \frac{{\displaystyle t+x}}{{\displaystyle {(x+y)}^3}}}\left(f_{{\alpha }_1,{\alpha }_1}-2f_{{\alpha }_1,{\beta }_1}+f_{{\beta }_1,{\beta }_1}\right).$$

(29)

A similar solution $w_g(t,x,y)$ follows for an undetermined function $g({\alpha }_2,{\beta }_2)$ with ${\alpha }_2=t+y$, ${\beta }_2=x$. The two solutions combined $w_{Gen}=w_f+w_g$ are a general solution of (28) of rank 2 and differential dimension $\mathcal{D}\left(w\right)=(2,2)$. It is conjectured that general solutions of (28) may be obtained for any integer $k\le -3$ and rank $|k+1|$. To perform the necessary calculations, computer algebra software is available in the ALLTYPES computer algebra system [18].

   References