Solutions of Partial Differential Equations with a Movable

S.9.2. Solutions of Partial Differential Equations with a Movable

Pole. Description of the Method

By analogy with ordinary differential equations, solutions of partial differential equations may be sought in the form of power series expansions with movable pole singularities. The position of the pole is given by an arbitrary function.

For simplicity of exposition, we consider equations of mathematical physics in two independent variables x, t and a dependent variable w, assuming that the equations do not explicitly depend on x or t.

1 ◦ . Simplest scheme. A solution is sought near a singular manifold x−x 0 ( t) = 0 as the following series (Jimbo, Kruskal, and Miwa, 1982):

w(x, t) =

w n ( t)ε n ,

ε=x−x 0 ( t).

ε α n=0

Here, the exponent α is a positive integer (this ensures that the movable singularity is of the pole

type), and the function x 0 ( t) is assumed arbitrary.

The expression (1) is substituted into the equation under consideration. First, by equating the leading singular terms, one finds the exponent α and the leading term u 0 ( t) of the series. Then, the terms with the same powers of ε are collected. Equating the resulting coefficients of the same powers of ε to zero, one obtains a system of ordinary differential equations for the functions w n ( t).

The thus obtained solutions are general, provided that series (1) contains arbitrary functions whose number is equal to the order of the equation under consideration.

2 ◦ . General scheme. The Painlev´e test. A solution of a partial differential equation is sought in a neighborhood of the singular manifold ε(x, t) = 0 in the form of a generalized series symmetric with respect to the independent variables (Weiss, Tabor, and Carnevalle, 1983):

w(x, t) = α

w n ( x, t)ε n ,

ε = ε(x, t),

ε n=0

where ε t ε x ≠ 0. Here and in what follows, the subscripts x and t denote the corresponding partial derivatives.

Series (1) is a special case of the expansion (2), provided the equation of the singular manifold, ε(x, t) = 0, is solvable for the variable x. The requirement that there are no movable critical points implies that α is a positive integer. The solution will be general if the total number of arbitrary functions among the w n ( x, t) and ε(x, t) coincides with the order of the equation.

Substituting (2) into the equation, collecting terms with the same powers of ε, and equating them to zero, we obtain the following recurrence relations for the expansion coefficients:

P N ( n)w n = f n ( w 0 , w 1 , ...,w n−1 , ε t , ε x , . . . ).

Here, the P N ( n) is a polynomial of degree N of the integer argument n,

P N ( n) = (n + 1)(n − j 1 )( n−j 2 ) . . . (n − j N −1 ),

and N is the order of the equation under consideration. If the roots of the polynomial j 1 , j 2 , ...,j N −1 (called resonances) are nonnegative integers and the compatibility conditions

f n=j k =0

( k = 1, 2, . . . , N − 1)

hold, then one says that the conditions of the Painlev ´e test hold for the equation under consideration. Equations satisfying these conditions are often regarded as integrable equations (this is confirmed by the fact that in many known cases, such equations can be reduced to linear equations).

use a simplified scheme based on the expansion (1). The relations ( w n ) x = 0 and ε x = 1 ensure some important simplifications of technical character, as compared with the expansion (2).

The more general expansion (2) entails more cumbersome but more informative computations. It can be effectively used at the second step of the investigation, after the conditions of the Painlev ´e test have been verified. This helps to clarify many important properties of the equations and their solutions and find the form of the B¨acklund transformation that linearizes the original equation. ▲✂▼

References for Subsection S.9.2: M. Jimbo, M. D. Kruskal, and T. Miwa (1982), J. Weiss, M. Tabor, and G. Carnevalle (1983), J. Weiss (1983, 1984, 1985), W.-H. Steeb and N. Euler (1988), R. Conte (1989, 1999), R. Conte and M. Musette (1989, 1993), M. Tabor (1989), M. Musette (1998).