The inverse spectral problem for the discrete cubic string

Abstract

Given a measure $m$ on the real line or a finite interval, the "cubic string" is the third order ODE $-{\varphi }^{\prime \prime \prime }=zm\varphi$ where $z$ is a spectral parameter. If equipped with Dirichlet-like boundary conditions this is a nonselfadjoint boundary value problem which has recently been shown to have a connection to the Degasperis-Procesi nonlinear water wave equation. In this paper we study the spectral and inverse spectral problem for the case of Neumann-like boundary conditions which appear in a high-frequency limit of the Degasperis--Procesi equation. We solve the spectral and inverse spectral problem for the case of $m$ being a finite positive discrete measure. In particular, explicit determinantal formulas for the measure $m$ are given. These formulas generalize Stieltjes' formulas used by Krein in his study of the corresponding second order ODE $-{\varphi }^{\prime \prime }=zm\varphi$.

24 pages. LaTeX + iopart, xypic, amsthm. To appear in Inverse Problems (http://www.iop.org/EJ/journal/IP)

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Cited by in corpus (5)

• Dynamics of interlacing peakons (and shockpeakons) in the Geng-Xue equation
• A view of the peakon world through the lens of approximation theory
• Inverse problems associated with integrable equations of Camassa-Holm type; explicit formulas on the real axis, I
• Continuous and discontinuous piecewise linear solutions of the linearly forced inviscid Burgers equation
• Inverse spectral problem for a third-order differential operator with non-local potential