Related papers: On isospectral deformations of an inhomogeneous st…
We derive isospectral flows of the mass density in the string boundary value problem corresponding to general boundary conditions. In particular, we show that certain class of rational flows produces in a suitable limit all flows generated…
A new approach to the Euler-Bernoulli beam based on an inhomogeneous matrix string problem is presented. Three ramifications of the approach are developed: (1) motivated by an analogy with the Camassa-Holm equation a class of isospectral…
In this paper we review the recent progress in the (indefinite) string density problem and its applications to the Camassa--Holm equation.
Motivated by the study of certain nonlinear wave equations (in particular, the Camassa-Holm equation), we introduce a new class of generalized indefinite strings associated with differential equations of the form…
The inverse problem which arises in the Camassa--Holm equation is revisited for the class of discrete densities. The method of solution relies on the use of orthogonal polynomials. The explicit formulas are obtained directly from the…
Given a measure $m$ on the real line or a finite interval, the "cubic string" is the third order ODE $-\phi'''=zm\phi$ where $z$ is a spectral parameter. If equipped with Dirichlet-like boundary conditions this is a nonselfadjoint boundary…
We extend the Euler-Bernoulli beam problem, formulated as a matrix string equation with a matrix-valued density, to a setting where the density takes values in a Clifford algebra, and we analyze its isospectral deformations. For discrete…
We use an inverse scattering approach to study multi-peakon solutions of the Degasperis-Procesi (DP) equation, an integrable PDE similar to the Camassa-Holm shallow water equation. The spectral problem associated to the DP equation is…
We discuss direct and inverse spectral theory for the isospectral problem of the dispersionless Camassa--Holm equation, where the weight is allowed to be a finite signed measure. In particular, we prove that this weight is uniquely…
A spectral and the inverse spectral problem are studied for the two-component modified Camassa-Holm type for measures associated to interlacing peaks. It is shown that the spectral problem is equivalent to an inhomogenous string problem…
The spectral data of a vibrating string are encoded in its so-called characteristic function. We consider the problem of recovering the distribution of mass along the string from its characteristic function. It is well-known that Stieltjes'…
This paper solves the integral equation which describes the oscillating inhomogeneous string, by using a spectral expansion method in terms of Chebyshev polynomials. The result is compared with the solution of the corresponding differential…
In [V. Barcilon Explicit solution of the inverse problem for a vibrating string. J. Math. Anal. Appl. {\bf 93} (1983) 222-234] two boundary value problems were considered generated by the differential equation of a string $$…
We consider a dynamic inverse problem for a dynamical system which describes the propagation of waves in a Krein string. The problem is reduced to an integral equation and an important special case is considered when the string density is…
We analyze in detail three classes of isomondromy deformation problems associated with integrable systems. The first two are related to the scaling invariance of the $n\times n$ AKNS hierarchies and the Gel'fand-Dikii hierarchies. The third…
We introduce a generalized isospectral problem for global conservative multi-peakon solutions of the Camassa-Holm equation. Utilizing the solution of the indefinite moment problem given by M. G. Krein and H. Langer, we show that the…
We extend the inverse spectral transform for the conservative Camassa-Holm flow on the line to a class of initial data that requires strong decay at one endpoint but only mild boundedness-type conditions at the other endpoint. The latter…
We study a class of 1+1 quadratically nonlinear water wave equations that combines the linear dispersion of the Korteweg-deVries (KdV) equation with the nonlinear/nonlocal dispersion of the Camassa-Holm (CH) equation, yet still preserves…
We study a deformation of the nonlinear Schr\"odinger equation recently derived in the context of deformation of hierarchies of integrable systems. This systematic method also led to known integrable equations such as the Camassa-Holm…
A non-local evolution equation of the Camassa-Holm type with dissipation is considered. The local well-posedness of the solutions of the Cauchy problem involving the equation is established via Kato's approach and the wave breaking scenario…