文档介绍：Supplement B
Methods of Generalized and Functional
Separation of Variables in Nonlinear
Equations of Mathematical Physics  
. Introduction
. Preliminary Remarks
Separation of variables is the mon approach to solve linear equations of mathematical
physics. This approach involves searching for exact solutions in the form of the product of functions
depending on different arguments (see Section ).
¢ £
As far as nonlinear equations with two independent variables ¡ , and a dependent variable
are concerned, some of these equations also have solutions with the form
¡ ¢ ¤ ¡ ¥ ¢ £ ¡ ¢ ¤ ¡ ¥ ¢
£ ( , ) = ( ) ( ) or ( , ) = ( ) + ( )
that are called multiplicatively and additively separable,respectively. We call such solutions ordinary
separable solutions. In particular, integrating a few classes of first-order nonlinear partial differential
equations is based on searching for additively separable solutions [., see Appell (1953), Kamke
(1965), Markeev (1990), Zwillinger (1998), Polyanin, Zaitsev, and Moussiaux (2001)].
Over the last decade, more sophisticated, generalized and functional separable solutions have
been obtained for a number of second-order nonlinear equations of mathematical physics. For
example, Galaktionov and Posashkov (1989) and Galaktionov, Posashkov, and Svirshchevskii
¡ ¦ ¤ ¦ ¥ ¡ § ¦
(1995) obtained generalized separable solutions with the forms £ ( , ) = ( ) ( ) + ( ) and
¡ ¦ ¤ ¦ ¥ ¡ § ¡
£ ( , ) = ( ) ( ) + ( ) for some classes of parabolic and hyperbolic equations with quadrat-
ic nonlinearities. In Galaktionov and Posashkov (1994), Galaktionov (1995), and Svirshchevskii
(1995), plicated generalized separable solutions are presented. The results of Galaktionov
and Posashkov (1994) and Galaktionov (1995) are based on finding finite-dimensional subspaces
that are invariant under appropriate nonlinear differential operators (in practice, the authors had
to find a system of coordinate functions in one of the varia