1. Introduction

The usual form of the equation under consideration is the following one;

(1)

Here ρ(x, y, z) is an unknown function of three independent variables. This equation arises as a reduction of the Plebansky equation[1] describing self-dual four dimensional (0 + 4); (2 + 2) gravity. In this connection it was considered in[2] and in literature is known as the Boyer-Finley equation.

Equation (1) can also be obtained as a limit of the discrete Toda chain

under appropriate rescaling (n→z). The series solutions of the symmetry equation for the Toda chain was found in[3].

Also, a solution of the two dimensional reduction of (1) (ρ = (z, y + x) was found in implicit form in[4]. Infinite series solutions of the symmetry equation corresponding to (1) were found in[5]. But the connection between series solutions of the symmetry equation with the solution of the initial system (1) has not been discovered. However general theory gives a guarantee that each solution of symmetry equation is connected with an analytical solution of the initial system in explicit or implicit form. The goal of the present paper is to fill this gap and demonstrate a way how an analytical solution of (1) is connected with the solution constructed in[3] of the symmetry equation. In[5], thePlebansky equation was represented in the form of two equations of the first order for two unknown functions, one of which satisfies the Plebansky equation by itself, the second one satisfies the corresponding symmetry equation. It is possible to find in an independent way the solution of symmetry equation in recurrence form. As was remarked the above equation under consideration in the present paper is a reduction of the Plebansky equation and so it is possible to try to solve it by the same methods[6].

2. Preliminary Manipulations. Short Excursion into[3]

Let us rewrite (1) in the form of a system of two equations of the first degree.

(2)

or as the initial equation is symmetrical with respect to exchange of the variables x, y, the following is also a possible form;

(3)

The symmetry equation arises from the initial one after differentiation by an arbitrary parameter and considering this derivative as a new unknown function. In the case under consideration this equation is

It is necessary to understand the last manipulation in such way, that if we represent the solution of the symmetry equation in the form S = T_x or S = w_y then the last equations in (2), (3) are exactly the symmetry equation by itself. Finally a linear system of equations of first order for the function u function is

(4)

In[3] we have obtained series solutions of the symmetry equation in integro-differential terms of the function u. Thus it is possible use these expressions in the system T, u and obtain two self consistent equation in-stead of only one equation for the function u. It is obvious that in this way we will not be able to obtain a general solution for the equation for u but only its partial soliton like series solutions. Solving the second equation of (2), u = θ_x, T =θ _z we rewrite (1) in the form

In[3] it was shown that the solution of the symmetry equation T, may be obtained in terms of α_n functions which satisfy the following recurrence relations

Eliminating α^n-1 from both equations we arrive at the equation for the functions αⁿ in the form

The left and right equations are the same. From these expressions it follows that there exists an obvious solution αⁿ = 1 which leads to a finite solution for T. The solution for T becomes

(5)

The second equality is obtained from the first one after the substitution

T=θ_z, u= θ_x and differentiation of the subsequent expression with respect to the argument y and integration once over z

The last equality is a series of additional conditions on the θ_x,y = θ_x θ_z,z for the function θ.

3. Generalization of R. Ward's Solution

This section explains why the analytic solutions of Ward exist at all. The simplest solution of the symmetry equation is a linear combination of derivatives of the functions u S = w_y = u_z = au_x + bu_y + cu_z. The solution of Richard Ward corresponds to choosing c = 1, a =- b. In the case c is not equal to zero we have u_z = Au_x + Bu_y and the second system under this additional condition becomes

Let us seek a solution of this system in the form

The system of equations defining derivatives of (u, w) with respect to space coordinates (x, y) is the following one;

After solving the last system,

and substitution into the previous one we arrive at a linear system of equations for θ (u, w) and σ(u, w).

The last system after eliminating (for instance) the function σ leads to an equation of the second order with separable variables

4. The Zero Order Term of Series Solution to the Symmetry Equation

In the case α⁰ = 1 from the general formula it follows that T = u or w = u and from the corresponding formulas of the previous section we obtain u_x = u_z or u_y = u_z. These are particular cases of the generalized Ward construction of the previous section. The first equations in this case lead u_y = uu_z. This is the well known Monge equation (the equation of Hamilton-Jacobi for free motion in one dimension) with general solution z + y + ux = F (u) or z + x + uy = F (u). It is not difficult to connect these solutions with the generalized Ward solution of the previous section.

5. The First Term of the Symmetry Equation Series Solution

In this case α¹ = 1. It is possible represent the solution of the symmetry equation in different variables and in connection of this it will possible to obtain two different solutions (at least known to us) of the continuous Toda chain.

5.1. The First Possibility

In connection with the recurrence procedure we obtain and the solution for T takes the form

or after passing to the function θ, u=θ_x, T= θ_z, (obeying the equation θ_y,x= θ_xθ_z,z) it appears as

In the transformation from the left to the right hand sides we have used the equation for θ and the once integrated result on z. From the second form of the equation above it follows that θ is a solution of the Monge-Ampere equation of the third order and it is possible to use its known general solution. We will go by more direct way (which is obviously equivalent to the previous one). Now let us introduce the notations

After differentiation on the right hand equation by x and z respectively and taking into account the above definition of λ we arrive at the system of equations

The last system is well known and its general solution in implicit form is the following:[7]

where F ¹, F ² are arbitrary functions of their two arguments. But in our case we have additional conditions (λ₁)_z = (λ₂)_x and this fact will limit the functions F ¹, F ². After differentiation of the two equations by x, z, y respectively and solving a linear system of algebraic equations we obtain all derivatives of the functions (λ₁, λ₂) with respect to these three arguments. The result is the following:

(6)

where

(7)

is the determinant of linear system under consideration. From the additional condition above and calculated values of corresponding derivatives of the function λ we obtained in the last equation we have F ¹ = G_λ1 ; F ² = G _λ2. In the same way we obtain for derivatives of the functions with respect the argument y

The last calculations show that the implicitly defined functions from the equations above satisfy the necessary system of equations and allow us to obtain a further constraint on the function G which arises from the fact that the equation for the function θ must be satisfied.. Namely

After substitution in the last equation of all expressions obtained above we pass to

After the introduction of new variables s=λ₁ λ₂, the last equations lead to a two dimensional equation with separable variables. Indeed

Substituting these expressions into equation for G we seek its solution in the form and pass to

Thus for the function G we obtain finally

where c₁, c₂ are arbitrary functions of argument k and g₁, g₂ are two fundamental solutions of the linear equation of second order above. Thus for the solution of the Toda chain u =θ_x =λ₁ we obtain an implicit solution for it

where G is the general solution of the linear equation above containing two arbitrary functions.

5.1.1. An example

By direct calculations it is not difficult to check that G = is a partial solution of equation of the subsection above. Thus in connection of the main result

and λ₁ = u is a solution of the continuous Toda chain. Eliminating λ₂ from the equations above we obtain for u

Calculation of necessary derivatives

5.2. The Second Possibility

In the case α₁ =1 in connection with the recurrence procedure we obtain and the solution for T takes the form

and the equations which are necessary to solve are the following;

The first and the last equations above lead to a relation between the derivatives (after integration once with respect to the argument z) in the form ln ,. As in the previous subsection it means that U is a solution of the Monge-Ampere equation of the third order. It is possible to substitute its general solution into the last equation and find some restrictions to determine the arbitrary functions in the solution. We will go more directly.

Let us seek a solution of these equations using the following parameterization

from which expressions for second order derivatives follow immediately;

X = W_β (β, γ, y); Z = W_γ (β, γ, y) and the equation transform to a linear equation of second order with separable variables

It is possible determine the dependence of the function W upon its argument after solution of two equations which arise after differentiation of the previous equations by the argument y

Remembering that after trivial manipulations we rewrite the last equations in a form

we obtain for W and explicit expressions for x,z

where W ^L is solution of the linear equation obtained above, which do not depend on argument y. The solution of the Toda chain of the beginning of this paper is given by connection and dependence β, γ functions on independent arguments x, z, y is defined in explicit form by formulas above.

5.3 Second Example

This example may help the reader to understand the difficulties in trying to obtain solutions in explicit form. It is easy to check that W ^L =γ e^-β is an explicit solution of the linear equation and thus. The implicit form of the solution is given by

From these expressions the equation

follows immediately. It is equivalent to our linear system above. With the help of this equation it is not difficult to check that

satisfy the equations

We rewrite equations which define an implicit solution in a form u =

in equivalent form

After eliminating terms without y on the right hand side we arrive at a quadratic equation to determinate the variable γ

Substituting the solution of this equation into the first or second equations we reach an equation determining in implicit form the function u. It is obvious that to obtain this equation is not a very simple problem.

6. Second Step

In the case where α₂ = 1 in connection with the recurrence procedure we obtain

and the solution for T takes the form

(8)

The equations which are necessary to solve are the following;

Let us introduce the definitions

Equations above together with the notation introduced lead to the following system of equations

As in the cases above we will seek solution of these equation by implicit substitution

After differentiation of these equalities with respect to the independent arguments of the problem and introduction of the matrix

we have

Substituting these expressions in linear system equations of the first order we obtain

The last equations allow us to reconstruct the explicit form matrix V LV^-1

The first two columns are a direct consequence of the equations above. The last column arises from the fact Trace(V LV^-1)ⁿ = Trace Lⁿ.

Now we arrive at a linear system of equations for determining the functions X, Z, Y.

e^cY we will denote by Y. A system of 9 equations follows;

Elements M_1,1 and M_3,3 lead to a parametrization X = R_a, Y = R_c, Z = R_b + f(a,b). Elements M_2,1 and M_1,2 both lead to equation (R_a + bR_c)_a = Element M_2,2 allows us to conclude that the function f depends only from one argument b. Elements M_3,1 and M_1,3 are the sane and lead to equation (R_b + R_c)_a = ¹₂ R_c,c. And finally elements M_3,2 and M_2,3 pass to a third equation in the form (R_b + f(b) + R_c)_b = (R_a + bR_c)_c. Thus we have three equations which it is necessary solve

For further calculations the variables b, , c will be more suitable. In these variables the system equations above appears as

These equations can be interpreted as the vanishing of the curl of some vector which means that vector by itself is gradient of some scalar function, or and that the same system of equations of the second order above can be rewritten as the system of equation of the first order for two unknown functions R, Q

Eliminating Q we come back to equations for R. Eliminating R, we pass to a system of equations for Q in the form

(9)

The last is the usual Laplace equation in three dimensions, which is invariant with respect to transformations of the five dimensional rotation group. One is operators of which is exactly The last equation is a consequence of the two first ones. From them it follows that (DQ)_α,α = 2(DQ)_b,c and operator D commutes with the Laplace operator. Thus the last equation above is consequence of two first ones. Let us seek a solution the last system of equations in the form of Laplace-Fourier transform.

After substituting this into the system of equations above under the sign of integral we obtain two equations

(10)

Eliminating terms with derivatives we arrive at a condition of self consistency

The first possibility k_c +1 = 0 we call the degenerate solution the second one non degenerate one.

6.1. Degenerate Case

Let us seek solution Q of Laplace equation under additional condition DQ = 0. In this case simultaneously Q_c + Q = f(b) or. Now

. The solution of the last equation is and finally the degenerate solution of the problem is . But in this case the third equation is no more a consequence of two first ones and it must be satisfied independently. We have

This ordinary differential equation may be solved in terms of Bessel functions. Coming back to the function R we obtain

By direct calculations it is simple to check that all equations for R above are satisfied. A solution of the continuous Toda chain u = e^-c in implicit form is determined from the equations

6.2. Non Degenerate Case

In this case The first equation of (10) is rewritten in the form

The solution the last equation of the first order is

where F is a scalar function of its argument. Finally we obtain

(11)

The first equation gives R = Q_α, W = Q_c. Substituting into both other equations we pass to a system of two equations

Let us seek a solution of this linear system above by a Laplace-Fourier trans-form

The first equation is satisfied automatically. The second one leads to a differential equation of the first order in partial derivatives for the determination of the function f(k, p) under the integral sign, which for the function f is

with the obvious solution

(12)

where is an arbitrary function of the argument

7. Conclusions

We have presented a new idea, unknown up to now to the best of our knowledge in the theory of integrable systems connected with the symmetry equation of the initial system. We have also presented some non-trivial solutions of the 2 + 1 continuous Toda chain. These solutions are often only given in implicit form. An explicit solution is however obtained as a specific example. It may be that this is the best that can be hoped for, i.e. that in the general case the solution is only obtainable in implicit form. This is the case for many non-linear systems, including that of the simple non-linear wave (Monge) equation. The method of solution presented here is far from obvious or straightforward. We hope that a more direct method of finding a solution can be found; now we know that a solution is possible