Follytons and the Removal of Eigenvalues for Fourth Order Differential Operators

TL;DR

This paper investigates how a nonlinear functional governs the eigenvalue changes of fourth order differential operators, providing explicit examples and conditions for isospectral operators, with implications for solving related PDEs.

Contribution

It introduces a nonlinear functional framework for eigenvalue manipulation of fourth order operators and constructs explicit isospectral examples with potential applications to PDE solutions.

Findings

Explicit examples of eigenvalue loss and gain

Conditions for isospectral operators with a negative eigenvalue

Exact solutions to associated time-dependent PDEs

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.