The general structure of integrable evolution equations

This paper presents some new results in connection with the structure of integrable evolution equations. It is found that the most general integrable evolution equations in one spatial dimension which is solvable using the inverse scattering transform (i.s.t.) associated with the nth order eigenvalue problem $V_{x}=(\zeta R_{0}+P(x,t))V$ has the simple and elegant form $G(D_{\text{R}},t)P_{t}-F(D_{\text{R}},t)x[R_{0},P]=\Omega (D_{\text{R}},t)[C,P]$, where $G,F$ and $\Omega $ are entire functions of an integro-differential operatos $D_{\text{R}}$ and the bracket refers to the commutator. The list provided by this form is not exhaustive but contains most of the known integrable equations and many new ones of both mathematical and physical significance. The simple structure allows the identification in a straightforward manner of the equation in this class which is closest to a given equation of interest. The $x$ dependent coefficients enable the inclusion of the effects of field gradients. Furthermore when the partial derivative with respect to t is zero, the remaining equation class contains many nonlinear ordinary differential equation of importance, such as the Painleve equations of the second and third kind. The properties of the scattering matrix $A(\zeta,t)$ corresponding to the potential $P(x,t)$ are investigated and in particular the time evolution of $A(\zeta,t)$ is found to be $G(\zeta,t)A_{t}+F(\zeta,t)A_{\zeta}=\Omega (\zeta,t)[C,A]$. The role of the diagonal entries and the principal corner minors in providing the Hamiltonian structure and constants of the motion is discussed. The central role that certain quadratic products of the eigenfunctions play in the theory is briefly described and the necessary groundwork from a singular perturbation theory is given when $n=2or3$.