Lax pairs: a novel type of separability

An attempt is made to place into historical context the fundamental concept of Lax pairs. For economy of presentation, emphasis is placed on the effectiveness of Lax pairs for the analysis of integrable nonlinear evolution PDEs. It is argued that Lax pairs provide a deeper type of separability than the classical separation of variables. Indeed, it is shown that: (a) the solution of the Cauchy problem of evolution equations is based on the derivation of a nonlinear Fourier transform pair, and this is achieved by employing the spectral analysis of one of the two eigenvalue equations forming a Lax pair; thus, although this methodology still follows the reverent philosophy of the classical separation of variables and transform methods, it can be applied to a class of nonlinear PDEs. (b) The solution of initial-boundary-value problems of evolution equations is based on the simultaneous spectral analysis of both equations forming a Lax pair and hence, in a sense, it employs the synthesis instead of the separation of variables; this methodology does not have a direct classical analogue, however, it can be considered as the nonlinearization of a method which combines Green's function classical integral representations with an analogue of the method of images, but which are now formulated in the spectral (Fourier) instead of the physical space. In addition to presenting a general methodology for analysing initial- and initial-boundary-value problems for nonlinear integrable evolution equations in one and two spatial variables, recent progress is reviewed for the derivation and the solution of integrable nonlinear evolution PDEs formulated in higher than two spatial dimensions.