14 - Linear Evolution Equations Associated with Isospectral Evolutions of Differential Operators

Summary

INTRODUCTION.

The spectral transform of a (matrix-valued) function of a (space) variable x is a powerful tool to solve, and investigate, several (classes of) wave equations that model both dispersion and nonlinear interaction. An introduction to this method can be found in Calogero and Degasperis (1982). Here, in order to keep the following discussion within a simple context, consider the prototype nonlinear evolution equation

that can be solved by the spectral transform technique; this is the celebrated Korteweg-de Vries equation with two arbitrary parameters, a_o and a₁.

Two necessary ingredients of the spectral transform method are the following: the linear ordinary differential equation

and the linear partial differential equation

where H(t) and M are differential operators, and k is the spectral variable. In the case of the KdV equation (1.1), these operators read

Note that the variable t enters in the ODE (1.2) only parametrically, and that the validity of both the equations (1.2) and (1.3) for all values of k implies the operator evolution equation (Lax, 1968)

indeed this equation, with (1.4) and (1.5), is equivalent to the KdV equation (1.1).

In general the nonlinear PDE (1.1) is the equation of applicative and/or mathematical interest, while the linear equations (1.2) and (1.3) merely play an auxiliary role in the solution of (1.1). Regarding these two equations most of the attention has been devoted to the direct and inverse spectral problems associated with the ODE (1.2) with (1.4a), their solution being essential to the very application of the spectral transform method. Here our attention is instead focussed on the other linear equation, i.e., the PDE (1.3) with (1.5). More precisely, the aim of this paper is to investigate a class of linear PDE's, with x and t dependent coefficients, that includes the evolution equation (1.3); however, we first confine our consideration to the linear PDE

or, more explicitly, (see (1.4a) and (1.5))

and then a larger class of linear PDE's is reported in the last section. Here, and in the following, we consider evolution equations of dispersive type; this implies that the four constants a_o a₁, b_o and b1 be real.