Interior Elastic Scattering by a Non-Penetrable Partially Coated Obstacle and Its Shape Recovering

In this paper, the interior elastic direct and inverse scattering problem of time-harmonic waves for a non-penetrable partially coated obstacle placed in a homogeneous and isotropic medium is studied. The scattering problem is formulated via the Navier equation, considering incident circular waves due to point-source fields, where the corresponding scattered data are measured on a closed curve inside the obstacle. Our model, from the mathematical point of view, is described by a mixed boundary value problem in which the scattered field satisfies mixed Dirichlet-Robin boundary conditions on the Lipschitz boundary of the obstacle. Using a variational equation method in an appropriate Sobolev space setting, uniqueness and existence results as well as stability ones are established. The corresponding inverse problem is also studied, and using some specific auxiliary integral operators an appropriate modified factorisation method is given. In addition, an inversion algorithm for shape recovering of the partially coated obstacle is presented and proved. Last but not least, useful remarks and conclusions concerning the direct scattering problem and its linchpin with the corresponding inverse one are given.

Is a Miracle-less WIMP Ruled out?

We examine a real electroweak triplet scalar field as dark matter, abandoning the requirement that its relic abundance is determined through freeze out in a standard cosmological history (a situation which we refer to as `miracle-less WIMP’). We extract the bounds on such a particle from collider searches, searches for direct scattering with terrestrial targets, and searches for the indirect products of annihilation. Each type of search provides complementary information, and each is most effective in a different region of parameter space. LHC searches tend to be highly dependent on the mass of the SU(2) charged partner state, and are effective for very large or very tiny mass splitting between it and the neutral dark matter component. Direct searches are very effective at bounding the Higgs portal coupling, but ineffective once it falls below \lambda_{\text{eff}} \lesssim 10^{-3}λeff≲10−3. Indirect searches suffer from large astrophysical uncertainties due to the backgrounds and JJ-factors, but do provide key information for \sim∼ 100 GeV to TeV masses. Synthesizing the allowed parameter space, this example of WIMP dark matter remains viable, but only in miracle-less regimes.

A Hybrid Adjoint Network Model for Thermoacoustic Optimization

A method is proposed that allows the computation of the continuous adjoint of a thermoacoustic network model based on the discretized direct equations. This hybrid approach exploits the self-adjoint character of the duct element, which allows all jump conditions to be derived from the direct scattering matrix. In this way, the need to derive the adjoint equations for every element of the network model is eliminated. This methodology combines the advantages of the discrete and continuous adjoint, as the accuracy of the continuous adjoint is achieved whilst maintaining the flexibility of the discrete adjoint. It is demonstrated how the obtained adjoint system may be utilized to optimize a thermoacoustic configuration by determining the optimal damper setting for an annular combustor.

A Hybrid Adjoint Network Model for Thermoacoustic Optimization

A method is proposed that allows the computation of the continuous adjoint of a thermoacoustic network model based on the discretized direct equations. This hybrid approach exploits the self-adjoint character of the duct element, which allows all jump conditions to be derived from the direct scattering matrix. In this way, the need to derive the adjoint equations for every element of the network model is eliminated. This methodology combines the advantages of the discrete and continuous adjoint, as the accuracy of the continuous adjoint is achieved whilst maintaining the flexibility of the discrete adjoint. It is demonstrated how the obtained adjoint system may be utilized to optimize a thermoacoustic configuration by determining the optimal damper setting for an annular combustor.

Riemann–Hilbert problem for the generalized modified Korteweg–de Vries equation with N distinct arbitrary-order poles

In this work, the generalized modified Korteweg–de Vries (gmKdV) equation is constructed by the first time and is solved by the Riemann–Hilbert method with the zero boundary condition. In the direct scattering transform, the analytical and asymptotic properties related to the Jost functions and the scattering matrix are given. On the basis of the above results, the appropriate Riemann–Hilbert problem (RHP) is constructed. By solving the RHP, we obtain the exact solution of the gmKdV equation in the case of no reflection potential when the scattering data [Formula: see text] has simple poles and higher-order poles. Furthermore, the three special solutions under different zero points are given and the phenomenon of their spread is described, respectively.

High-orders methods and high-precision arithmetics make direct scattering transform for the Korteweg-De Vries equation robust.

<p>Similar to the theory of direct scattering transform for nonlinear wave fields containing solitons within the focusing one-dimensional nonlinear Schrödinger equation [1], we revisit the theory associated with the Korteweg–De Vries equation. We study a crucial fundamental property of the scattering problem for multisoliton potentials demonstrating that in many cases position parameters of solitons cannot be identified with standard machine precision arithmetics making solitons in some sense “uncatchable”. Using the dressing method we find the landscape of soliton scattering coefficients in the plane of the complex spectral parameter for multisoliton wave fields truncated within a finite domain, allowing us to capture the nature of such anomalous numerical errors. They depend on the size of the computational domain L leading to a counterintuitive exponential divergence when increasing L in the presence of a small uncertainty in soliton eigenvalues. Then we demonstrate how one of the scattering coefficients loses its analytical properties due to the lack of the wave-field compact support in case of L→∞. Finally, we show that despite this inherent direct scattering transform feature, the wave fields of arbitrary complexity can be reliably analyzed using high-precision arithmetics and high-order algorithms based on the Magnus expansion [2, 3] providing accurate information about soliton amplitudes, velocities<span>, positions</span> and intensity of the radiation. This procedure is robust even in the presence of noise opening broad perspectives in analyzing experimental data on propagation of surface waves on shallow water.</p><p>The work is partially funded by Russian Science Foundation grant No 19-79-30075.</p><p>[1] Gelash A., Mullyadzhanov R. Anomalous errors of direct scattering transform // Physical Review E 101 (5), 052206, 2020.</p><p>[2] Mullyadzhanov R., Gelash A. Direct scattering transform of large wave packets // Optics Letters 44 (21), 5298-5301, 2019.</p><p>[3] Gudko A., Gelash A., Mullyadzhanov R. High-order numerical method for scattering data of the Korteweg—De Vries equation // Journal of Physics: Conference Series 1677 (1), 012011, 2020.</p><p> </p><p> </p>