Solitons and the removal of eigenvalues for fourth-order differential operators

A nonlinear functional Q[u, v] is given that governs the loss, respectively gain, of (doubly degenerate) eigenvalues of fourth-order differential operators L = ∂⁴ + ∂u∂ + v on the line. Apart from factorizing L as A^∗A + E0, providing several explicit examples, and deriving various relations between u, v, and the eigenfunctions of L, we find u and v such that L is isospectral to the free operator L0 = ∂⁴ up to one (multiplicity 2) eigenvalue E0 < 0. Not unexpectedly, this choice of u, v leads to exact solutions of the corresponding time-dependent PDE's. Removal of eigenvalues allows us to obtain a sharp Lieb-Thirring inequality for a class of operators L whose negative eigenvalues are of multiplicity two.