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The squared eigenfunctions of the spectral problem associated with the Camassa-Holm equation represent a complete basis of functions, which helps to describe the inverse scattering transform for the Camassa-Holm hierarchy as a generalized Fourier transform. The main result of this work is the derivation of the completeness relation for the squared solutions of the Camassa-Holm spectral problem. We show that all the fundamental properties of the Camassa-Holm equation such as the integrals of motion, the description of the equations of the whole hierarchy and their Hamiltonian structures can be naturally expressed making use of the completeness relation and the recursion operator, whose eigenfunctions are the squared solutions. [ABSTRACT FROM AUTHOR]

In this paper, we will study the increasing stability in the inverse source problem for the Helmholtz equation in the plane when the source term is assumed to be compactly supported in a bounded domain Ω with a sufficiently smooth boundary. Using the Fourier transform in the frequency domain, bounds for the Hankel functions and for scattering solutions in the complex plane, improving bounds for the analytic continuation, and the exact observability for the wave equation led us to our goals which are a sharp uniqueness and increasing stability estimate when the wave number interval is growing. [ABSTRACT FROM AUTHOR]

We consider the problem of locating an obstacle in a waveguide from time domain measurements of causal waves. More precisely, we assume that we are given the scattered field due to point sources placed on a surface located inside the waveguide away from the obstacle, where the scattered field is measured on the same surface. From this multi-static scattering data we wish to determine the position and shape of an obstacle in the waveguide. To deal with this inverse problem, we adapt and analyze the time domain linear sampling method. This involves proving new time domain estimates for the forward problem, as well as analyzing several time domain operators arising in the inversion scheme. We also implement the inversion algorithm and provide numerical results in two-dimensions using synthetic data. [ABSTRACT FROM AUTHOR]

We consider a near-field inverse scattering problem for the wave equation: find the shape of a Dirichlet scattering object from time domain measurements of scattered waves. For this time-domain inverse problem, we propose a linear sampling method, a well-known technique for corresponding frequency domain inverse scattering problems. The problem setting and the algorithm incorporate two basic features. First, the data for the method consist of measurements of causal waves, that is, of waves that vanish before some moment in time. Second, the inversion algorithm directly works on the time-domain data without using a Fourier transformation. The first point is related to the applications we have in mind, which include for instance ground-penetrating radar imaging. The second feature allows us to naturally incorporate multiple (in fact, a continuum of) frequencies in the inversion algorithm. Consequently, it offers the potential of improving the quality of the reconstruction compared to frequency domain methods working with a single frequency. We demonstrate this potential by several numerical examples. [ABSTRACT FROM AUTHOR]