Series solution of the 1+2 continuous Toda Chain

A way to obtain the series solutions of the 1 + 2 dimensional continuous Toda chain is presented.


Introduction
The usual form of the equation under consideration is the following one; ρ y,x = (e ρ ) z,z , (ln u) y,x = u z,z , u ≡ e ρ (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 selfdual four dimensional (0 + 4), (2 + 2) gravity. In this connection it was considered in [2]. Equation (1) can also be obtained as a limit of the discrete Toda chain ρ y,x = e ρ n+1 − 2e ρn + e ρ n−1 under appropriate rescaling (n → z). This is the reason for the title of the present paper. A solution of the two dimensional reduction of (1) (1) (ρ = ρ(z, y + x) was found in implicit form in [3]. Infinite series solutions of the symmetry equation corresponding to (1) were found in [4]. But the connection between series solutions of the symmetry equation with the solution of the initial system (1) has not been found. But general theory gives a guarantee that with 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 fill this gap and demonstrate a way how an analytical solution of (1) is connected with the solution constructed in [4] of the symmetry equation. In [5], the Plebansky 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 by an independent way the solution of symmetry equation a in recurrence form. As was remarked the above equation under consideration in the present paper is a reduction of Plebansky equation and so it is possible to try to solve it by the same methods.

Preliminary manipulations. Short excursion into [4]
Let us rewrite (1) in the form of a system of two equations of the first degree or as the initial equation is symmetrical with respect to exchange of the variables x, y, the following is also a possible form; The symmetry equation arises from the initial one after differentiation by some arbitrary parameter and considering this derivative as a new unknown function. In the case under consideration this equation is; Thus we see that the linear system of equations of the first few lines of this section is connected with the symmetry equation of the initial system. In [4] we have obtained series solutions of the symmetry equation in integrodifferential terms of the function u. And thus it is possible use these expressions in the system T, u and obtain two self consistent equation instead 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. Resolving the second equation of (2) u = θ x , T = θ z we rewrite (1) in the form In [4] it was shown that solution of symmetry equation T may be obtained in terms of r α n functions which satisfy the the following recurrence relations Eliminating α n−1 from both equations we arrive at the equation for α n function in the form The left and right equations are the same. From these expressions it follows that there exists an obvious solution α n = 1 which leads to a finite solution for T . The solution for T becomes The second equality is obtained from the first one after the substitution T = θ z , u = θ x and differentiation of the subsequent θz θx with respect to the argument y and integration once over z.

Generalization of R.Ward's solution.
This section explains why 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 The case considered by Ward corresponds to the limiting case A → ∞, B → ∞, B A → −1.

The zero order term of series solution to the symmetry equation
In the case α 0 = 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 cases lead u y = uu z . This well the 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.

The first term of the symmetry equation series solution
In the case α 1 = 1 in connection with the recurrence procedure we obtain α 0 = dyu z and solution for T takes the form and the equations which are necessary to resolve are the following; Taking into account the last equation the first one leads to a relation between the derivatives (after integration once with respect to the argument z) in the form ln U y = 1 2 U 2 z + U x , U y = e 1 2 U 2 z +Ux . Let us seek a solution of these equations using the following parametrisation where indices β, γ denote respectively the first and second arguments of the functions X, Z. X = W β , Z = W γ and equation transorm to a linear equation of second order with separable variables −2W β,γ + W γ,γ = W 1,1 , W = e kβ U(γ), −2kU γ + U γ,γ = k 2 U It is possible obtain the dependence of the function W after solution of two equations which arise after differentiation of the previous equations by the argument y X β u x,y + X γ u z,y + X y = 0, Z β u x,y + Z γ u z,y + Z y = 0, remembering that U y = e 1 2 U 2 z +Ux . After trivial manipulations we obtain for W where W L is solution of the linear equation obtained above. Solution of Toda chain of the begining of this paper is given by connection u = U y = e

One simple example
It is easy to check that W L = 2e −β is an explicit solution of the linear equation and thus W = ye 1 2 U 2 z +Ux + 2e −β . The implicit form of the solution is given by From these expressions immediately follows the equation U z,z + 2U z U z,x − U x,x = 0, which is equivalent to our linear system above. With the help of this equation it is not difficult to check that T = U y U z = 2e After excluding terms containing y we arrive at a quadratic equation for determining the variable γ Substituting the solution of this equation into the first or second equations we come to equation determining in implicit form the function u. It is obvious that to obtain this equation is not a very simple problem. In Appendix we present an alternative method of solution of the problem of this section.

Second step
In the case where α 2 = 1 in connection with the recurrence procedure we obtain α 1 = dyu z = U z ,α 0 = dy u 2 α 1 2u = dy(U y,z U z + 1 2 U y U z,z ) = 1 2 (U 2 z + U x ) and the solution for T takes the form The equations which are necessary to resolve are the following ones; Let us introduce notations Equations above together with introduced notations lead to the following system of equations As in the cases above we will seek solution of these equation by implicit substitution After differentiation all these equalities with respect to independent arguments of problem and introduction matrix V = Substituting these expressions in linear system equations of the first order we obtain The last equations allow to reconstruct explicit form matrix V LV The first two columns are direct consequence of equations above. The last column arises from the fact T race(V LV −1 ) n = T raceL n . Now we come to linear system of equations for determining X, Z, Y functions e c Y we will denote by Y . System of 9 equations are the following one 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 = 1 2 R b,c . Element M 2,2 allow to conclude function f depend only from one argument b. Elements M 3,1 and M 1,3 are the sane and lead to equation (R b +αR c ) a = 1 2 R c,c . And at last elements M 3,2 and M 2,3 pass to a third equation in the form Thus we have three equations which it is necessary to resolve For further calculations it will be more suitable Variables b, α = 1 2 (a + b), c. In these variables the system equations above looks as First equation give R = Q α , W = Q c Substituting both others equation we pass to system of two equations Q α,α = 2Q b,c , Q α,b + bQ α,α + αQ α,c = Q c,c Let us seek solution of this linear system above in Laplace-Furier forma Q = dkdpe kα+pb+ k 2 2p c f (p, k) 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 under the integral function f (k, p), which for the function F ≡ k 3 f is with the obvious solution where φ is an arbitray function of the argument k 2 p .