Follytons and the Removal of Eigenvalues for Fourth Order Differential Operators

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 $A^{\ast }A+E_0$, 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.

11 pages