Follytons and the Removal of Eigenvalues for Fourth Order Differential Operators

A non-linear functional $Q[u,v]$ is given that governs the loss, respectively gain, of (doubly degenerate) eigenvalues of fourth order differential operators $L = \partial^4 + \partial u \partial + v$ on the line. Apart from factorizing $L$ as $A^{*}A + E_{0}$, providing several explicit examples, and deriving various relations between $u$, $v$ and eigenfunctions of $L$, we find $u$ and $v$ such that $L$ is isospectral to the free operator $L_{0} = \partial^{4}$ up to one (multiplicity 2) eigenvalue $E_{0} < 0$. Not unexpectedly, this choice of $u$, $v$ leads to exact solutions of the corresponding time-dependent PDE's.