Follytons and the Removal of Eigenvalues for Fourth Order Differential Operators
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چکیده
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 = ∂ + ∂ 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 eigenfunctions of L, we find u and v such that L is isospectral to the free operator L0 = ∂ 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. 1. FACTORIZATION OF THE OPERATOR L = ∂ + ∂ u ∂ + v. Let us assume that u and v are real-valued functions and u, v ∈ S (R), where S (R) denotes the Schwarz class of rapidly decaying functions. Let L be a linear fourth order selfadjoint operator in L (R) (1.1) L := ∂ + ∂ u ∂ + v defined on functions from the Sobolev class H(R). This operator is bounded from below and we assume that its lowest eigenvalue E0 < 0 is of double multiplicity and therefore there exist two orthogonal in L(R) eigenfunctions ψ+ and ψ− satisfying the equation (1.2) Lψ = E0 ψ. As shown in the appendix, the Wronskian (1.3) W (x) := ψ+(x)ψ ′ −(x)− ψ−(x)ψ +(x) is necessarily non-vanishing, W (x) 6= 0, x ∈ R. Let us try to factorize L− E0 as (1.4) A∗A = ( −∂ − f∂ + g − f ′ ) ( −∂ + f∂ + g )
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تاریخ انتشار 2008