Follytons and the Removal of Eigenvalues for Fourth Order Differential Operators

Details Follytons and the Removal of Eigenvalues for Fourth Order Differential Operators Follytons and the Removal of Eigenvalues for Fourth Order Differential Operators

2003-11-09

arxiv.org/abs/math-ph/0311011v1

Physics

Mathematical Physics

11 pages

Scientific paper

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.

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