Jens Hoppe, Ari Laptev and J rgen stensson
Follytons
and the Removal of Eigenvalues for Fourth Order Differential Operators.
(29K, Latex2E)
ABSTRACT. 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.