We report the experimental measurement of the density of states (DOS) associated with the soliton gas emerging during the development of the noise-induced modulation instability (MI) in optical fibres. By employing a time-lens-based heterodyne detection technique (SEAHORSE), we reconstruct the complex optical field and compute its nonlinear discrete spectrum within the framework of the inverse scattering transform. Our results show that, at early stages of the MI, the DOS matches the Weyl distribution predicted for an ‘ideal’ critically dense soliton gas, thereby confirming the relevance of the SG description for this nonlinear random wave regime. At larger effective propagation distances, we observe a progressive deformation of the DOS in the complex plane. We compare these observations with numerical simulations of a generalised nonlinear Schrödinger equation that includes losses, third-order dispersion and stimulated Raman scattering (SRS). Our simulations reproduce the main experimental trends and demonstrate that SRS is the dominant mechanism responsible for the spectral deformation. These findings highlight the need to extend the kinetic theory of soliton gas beyond purely integrable evolutions. In particular, our results call for a generalised kinetic equation (or generalised hydrodynamics description) that accounts for weak non-integrable perturbations such as SRS.

We show that there is nonuniqueness for the Calderón problem with partial data for Riemannian metrics with Hölder continuous coefficients in dimension greater than or equal to three. We provide simple counterexamples in the case of cylindrical Riemannian manifolds with boundary having two ends. The coefficients of these metrics are smooth in the interior of the manifold and are only Hölder continuous of order $\unicode[STIX]{x1D70C}<1$ at the end where the measurements are made. More precisely, we construct a toroidal ring $(M,g)$ and we show that there exist in the conformal class of $g$ an infinite number of Riemannian metrics $\tilde{g}=c^{4}g$ such that their corresponding partial Dirichlet-to-Neumann maps at one end coincide. The corresponding smooth conformal factors are harmonic with respect to the metric $g$ and do not satisfy the unique continuation principle.

We extend the full wavefield modeling with forward scattering theory and Volterra Renormalization to a vertically varying two-parameter (velocity and density) acoustic medium. The forward scattering series, derived by applying Born-Neumann iterative procedure to the Lippmann-Schwinger equation (LSE), is a well known tool for modeling and imaging. However, it has limited convergence properties depending on the strength of contrast between the actual and reference medium or the angle of incidence of a plane wave component. Here, we introduce the Volterra renormalization technique to the LSE. The renormalized LSE and related Neumann series are absolutely convergent for any strength of perturbation and any incidence angle. The renormalized LSE can further be separated into two sub-Volterra type integral equations, which are then solved noniteratively. We apply the approach to velocity-only, density-only, and both velocity and density perturbations. We demonstrate that this Volterra Renormalization modeling is a promising and efficient method. In addition, it can also provide insight for developing a scattering theory-based direct inversion method.

The inverse problem considered in this paper is to determine the shape and the impedance of crack from a knowledge of the time-harmonic incident field and the corresponding far field pattern of the scattered waves in two-dimension. The combined single- and double-layer potential is used to approach the scattered waves. As an important feature, this method does not require the solution of u and ∂u/∂v at each iteration. An approximate method is presented and the convergence of this method is proven. Numerical examples are given to show that this method is both accurate and simple to use.

We study the inverse problem of recovering the scatterer shape from the far-field pattern(FFP) in the presence of noise. Furthermore, only a discrete partial aperture is usually known. This problem is ill-posed and is frequently addressed using regularization. Instead, we propose to use a direct approach denoising the FFP using a filtering technique. The effectiveness of the technique is studied on a scatterer with the shape of the ellipse with a tower. The forward scattering problem is solved using the finite element method (FEM). The numerical FFP is additionally corrupted by Gaussian noise. The shape parameters are found based on a least-square error estimator. If ũ∞ is a perturbation of the FFP then we attempt to find Γ, the scatterer shape, which minimizes ∣∣ũ∞ − ũ∞∣∣ using the conjugate gradient method for the denoised FFP