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The article gives the motivations for the measurement of the hyperfine splitting (hfs) in the ground state of muonic hydrogen to explore the properties of the proton at low momentum transfer. It summarizes these proposed measurement methods and finally describes the FAMU experiment in more detail.

We propose a method to numerically compute fractional derivatives (or the fractional Laplacian) on the whole real line via Riesz fractional integrals. The compactified real line is divided into a number of intervals, thus amounting to a multi-domain approach; after transformations in accordance with the underlying $Z_{q}$ curve ensuring analyticity of the respective integrands, the integrals over the different domains are computed with a Clenshaw-Curtis algorithm. As an example, we consider solitary waves for fractional Korteweg-de Vries equations and compare these to results obtained with a discrete Fourier transform.

The FAMU experiment at RIKEN-RAL is a muonic atom experiment with the aim to determine the Zemach radius of the proton by measuring the 1s hyperfine splitting in muonic hydrogen. The activity of the FAMU Collaboration in the years 2015-2023 enabled the final optimisation of the detector-target setup as well as the gas working condition in terms of temperature, pressure and gas mixture composition. The experiment has started its data taking in July 2023. The status of the detector setup for the 2023 experimental runs, for the beam characterisation and muonic X-ray detection in the 100-200 keV energy range, is presented and discussed., Comment: Submitted to JINST

We report the first experimental determination of the collision-energy dependence of the muon transfer rate from the ground state of muonic hydrogen to oxygen at near-thermal energies. A sharp increase by nearly an order of magnitude in the energy range 0 - 70 meV was found that is not observed in other gases. The results set a reliable reference for quantum-mechanical calculations of low-energy processes with exotic atoms, and provide firm ground for the measurement of the hyperfine splitting in muonic hydrogen and the determination of the Zemach radius of the proton by the FAMU collaboration., Comment: 30 pages, 10 figures

We present a detailed numerical study of the stability under periodic perturbations of line solitons of two-dimensional, generalized Zakharov-Kuznetsov equations with various power nonlinearities. In the $L^{2}$-subcritical case, in accordance with a theorem due to Yamazaki we find a critical speed, below which the line soliton is stable. For higher velocities, the numerical results indicate an instability against the formation of lumps, solitons localized in both spatial directions. In the $L^2$-critical and supercritical cases but subcritical for the 1D generalized Korteweg-de Vries equation), the line solitons are shown to be numerically stable for small velocities, and strongly unstable for large velocities, with a blow-up observed in finite time., Comment: New figures added

The study of complex geometric optics solutions to a system of d-bar equations appearing in the context of electrical impedance tomography and the scattering theory of the integrable Davey-Stewartson II equations for large values of the spectral parameter $k$ in \cite{KlSjSt20} is extended to the reflection coefficient. For the case of potentials $q$ with compact support on some domain $\Omega$ with smooth strictly convex boundary, improved asymptotic relations are provided.

We present a numerical approach for generalised Korteweg-de Vries (KdV) equations on the real line. In the spatial dimension we compactify the real line and apply a Chebyshev collocation method. The time integration is performed with an implicit Runge-Kutta method of fourth order. Several examples are discussed: initial data bounded but not vanishing at infinity as well as data not satisfying the Faddeev condition, i.e. with a slow decay towards infinity.

An efficient high precision hybrid numerical approach for integrable Davey-Stewartson (DS) I equations for trivial boundary conditions at infinity is presented for Schwartz class initial data. The code is used for a detailed numerical study of DS I solutions in this class. Localized stationary solutions are constructed and shown to be unstable against dispersion and blow-up. A finite-time blow-up of initial data in the Schwartz class of smooth rapidly decreasing functions is discussed.

The first measurement of the temperature dependence of the muon transfer rate from muonic hydrogen to oxygen was performed by the FAMU collaboration in 2016. The results provide evidence that the transfer rate rises with the temperature in the range 104-300 K. This paper presents the results of the experiment done in 2018 to extend the measurements towards lower (70 K) and higher (336 K) temperatures. The 2018 results confirm the temperature dependence of the muon transfer rate observed in 2016 and sets firm ground for comparison with the theoretical predictions.

A multi-domain spectral method is presented to compute the Hilbert transform on the whole compactified real line, with a special focus on piece-wise analytic functions and functions with algebraic decay towards infinity. Several examples of these and other types of functions are discussed. As an application solitons to generalized Benjamin-Ono equations are constructed.

We present a detailed numerical study of solutions to the Zakharov-Kuznetsov equation in three spatial dimensions. The equation is a three-dimensional generalization of the Korteweg-de Vries equation, though, not completely integrable. This equation is $L^2$-subcritical, and thus, solutions exist globally, for example, in the $H^1$ energy space. We first study stability of solitons with various perturbations in sizes and symmetry, and show asymptotic stability and formation of radiation, confirming the asymptotic stability result in \cite{FHRY2020} for a larger class of initial data. We then investigate the solution behavior for different localizations and rates of decay including exponential and algebraic decays, and give positive confirmation toward the soliton resolution conjecture in this equation. Finally, we investigate soliton interactions in various settings and show that there is both a quasi-elastic interaction and a strong interaction when two solitons merge into one, in all cases always emitting radiation in the conic-type region of the negative $x$-direction.

In a previous work on the large $|k|$ behavior of complex geometric optics solutions to a system of d-bar equations, we treated in detail the situation when a certain potential is the characteristic function of a strictly convex set with real-analytic boundary. We here extend the results to the case of sets with smooth boundary, by using almost holomorphic functions.

We present a multidomain spectral approach with an exterior compactified domain for the Maxwell equations for monochromatic fields. The Sommerfeld radiation condition is imposed exactly at infinity being a finite point on the numerical grid. As an example, axisymmetric situations in spherical and prolate spheroidal coordinates are discussed., Comment: Examples added

Complex geometric optics solutions to a system of d-bar equations appearing in the context of electrical impedance tomography and the scattering theory of the integrable Davey-Stewartson II equations are studied for large values of the spectral parameter $k$. For potentials \( q\in \langle \cdot \rangle^{-2} H^{s}(\mathbb{C}) \) for some $s \in]1,2]$, it is shown that the solution converges as the geometric series in $1/|k|^{s-1}$. For potentials $q$ being the characteristic function of a strictly convex open set with smooth boundary, this still holds with $s=3/2$ i.e., with $1/\sqrt{|k|}$ instead of $1/|k|^{s-1}$. The leading order controbutions are computed explicitly. Numerical simulations show the applicability of the asymptotic formulae for the example of the characteristic function of the disk., Comment: references updated

We propose a way to retrieve the secure key generated by the coherent one way protocol without reading the information transmitted on the quantum channel., Comment: 6 pages, LaTeX, 1 figure. References added

We present a detailed numerical study of solutions to the (generalized) Zakharov-Kuznetsov equation in two spatial dimensions with various power nonlinearities. In the $L^{2}$-subcritical case, numerical evidence is presented for the stability of solitons and the soliton resolution for generic initial data. In the $L^2$-critical and supercritical cases, solitons appear to be unstable against both dispersion and blow-up. It is conjectured that blow-up happens in finite time and that blow-up solutions have some resemblance of being self-similar, i.e., the blow-up core forms a rightward moving self-similar type rescaled profile with the blow-up happening at infinity in the critical case and at a finite location in the supercritical case. In the $L^{2}$-critical case, the blow-up appears to be similar to the one in the $L^{2}$-critical generalized Korteweg-de Vries equation with the profile being a dynamically rescaled soliton.

We present an efficient high-precision numerical approach for the Davey-Stewartson (DS) II equation, treating initial data from the Schwartz class of smooth, rapidly decreasing functions. As with previous approaches, the presented code uses discrete Fourier transforms for the spatial dependence and Driscoll's composite Runge-Kutta method for the time dependence. Since the DS equation is a nonlocal, nonlinear Schr\"odinger equation with a singular symbol for the nonlocality, standard Fourier methods in practice only reach accuracies of the order of $10^{-6}$ or less for typical examples. This was previously demonstrated for the defocusing case by comparison with a numerical approach for DS via inverse scattering. By applying a regularization to the singular symbol, originally developed for D-bar problems, the presented code is shown to reach machine precision. The code can treat integrable and non-integrable DS II equations. Moreover, it has the same %complexity numerical complexity as existing codes for DS II. Several examples for the integrable defocusing DS II equation are discussed as test cases. In an appendix by C.~Kalla, a doubly periodic solution to the defocusing DS II equation is presented, providing a test for direct DS codes based on Fourier methods.

We generalise a previously published approach based on a multi-domain spectral method on the whole real line in two ways: firstly, a fully explicit 4th order method for the time integration, based on a splitting scheme and an implicit Runge--Kutta method for the linear part, is presented. Secondly, the 1D code is combined with a Fourier spectral method in the transverse variable both for elliptic and hyperbolic NLS equations. As an example we study the transverse stability of the Peregrine solution, an exact solution to the one dimensional nonlinear Schr\"odinger (NLS) equation and thus a $y$-independent solution to the 2D NLS. It is shown that the Peregrine solution is unstable against all standard perturbations, and that some perturbations can even lead to a blow-up for the elliptic NLS equation., Comment: Hyperbolic case added

We consider the cosmological application of a (variant of) relatively newly proposed model \cite{1609.06915} unifying inflation, dark energy, dark matter, and the Higgs mechanism. The model was originally defined using additional non-Riemannian measures but it can be reformulated into effective quintessential model unifying inflation, dark energy and dark matter. Here we demonstrate numerically that it is capable of describing the entire Universe evolution in a seamless way, but this requires some revision of the model setup. The main reason is that there is a strong effective friction in the model, a feature which has been neglected in the pioneer work. This improves the model potential for proper description of the Universe evolution, because the friction ensures a finite time inflation with dynamically maintained low-value slow-roll parameters in the realistic scenarios. In addition, the model predicts the existence of a constant scalar field in late Universe., Comment: 17 pages, 6 figures, accepted for publication in International Journal of Modern Physics A

We report the first measurement of the temperature dependence of muon transfer rate from $\mu$p atoms to oxygen between 100 and 300 K. Data were obtained from the X-ray spectra of delayed events in gaseous target H$_2$/O$_2$ exposed to a muon beam. Based on the data, we determined the muon transfer energy dependence up to 0.1 eV, showing an 8-fold increase in contrast with the predictions of constant rate in the low energy limit. This work set constraints on theoretical models of muon transfer, and is of fundamental importance for the measurement of the hyperfine splitting of $\mu$p by the FAMU collaboration., Comment: 11 pages, 4 figure

We present a multidomain spectral approach for Fuchsian ordinary differential equations in the particular case of the hypergeometric equation. Our hybrid approach uses Frobenius' method and Moebius transformations in the vicinity of each of the singular points of the hypergeometric equation, which leads to a natural decomposition of the real axis into domains. In each domain, solutions to the hypergeometric equation are constructed via the well-conditioned ultraspherical spectral method. The solutions are matched at the domain boundaries to lead to a solution which is analytic on the whole compactified real line $\mathbb{R}\cup {\infty}$, except for the singular points and cuts of the Riemann surface on which the solution is defined. The solution is further extended to the whole Riemann sphere by using the same approach for ellipses enclosing the singularities. The hypergeometric equation is solved on the ellipses with the boundary data from the real axis. This solution is continued as a harmonic function to the interior of the disk by solving the Laplace equation in polar coordinates with an optimal complexity Fourier--ultraspherical spectral method.

The inverse scattering approach for the defocusing Davey-Stewartson II equation is given by a system of D-bar equations. We present a numerical approach to semi-classical D-bar problems for real analytic rapidly decreasing potentials. We treat the D-bar problem as a complex linear second order integral equation which is solved with discrete Fourier transforms complemented by a regularization of the singular parts by explicit analytic computation. The resulting algebraic equation is solved either by fixed point iterations or GMRES. Several examples for small values of the semi-classical parameter in the system are discussed.

In this work we present spectral algorithms for the numerical scattering for the defocusing Davey-Stewartson (DS) II equation with initial data having compact support on a disk, i.e., for the solution of d-bar problems. Our algorithms use polar coordinates and implement a Chebychev spectral scheme for the radial dependence and a Fourier spectral method for the azimuthal dependence. The focus is placed on the construction of complex geometric optics (CGO) solutions which are needed in the scattering approach for DS. We discuss two different approaches: The first constructs a fundamental solution to the d-bar system and applies the CGO conditions on the latter. This is especially efficient for small values of the modulus of the spectral parameter $k$. The second approach uses a fixed point iteration on a reformulated d-bar system containing the spectral parameter explicitly, a price paid to have simpler asymptotics. The approaches are illustrated for the example of the characteristic function of the disk and are shown to exhibit spectral convergence, i.e., an exponential decay of the numerical error with the number of collocation points. An asymptotic formula for large $|k|$ is given for the reflection coefficient., Comment: Introduction change, various minor updates

The main goal of the FAMU experiment is the measurement of the hyperfine splitting (hfs) in the 1S state of muonic hydrogen $\Delta E_{hfs}(\mu^-p)1S$. The physical process behind this experiment is the following: $\mu p$ are formed in a mixture of hydrogen and a higher-Z gas. When absorbing a photon at resonance-energy $\Delta E_{hfs}\approx0.182$~eV, in subsequent collisions with the surrounding $H_2$ molecules, the $\mu p$ is quickly de-excited and accelerated by $\sim2/3$ of the excitation energy. The observable is the time distribution of the K-lines X-rays emitted from the $\mu Z$ formed by muon transfer $(\mu p) +Z \rightarrow (\mu Z)^*+p$, a reaction whose rate depends on the $\mu p$ kinetic energy. The maximal response, to the tuned laser wavelength, of the time distribution of X-ray from K-lines of the $(\mu Z)^*$ cascade indicate the resonance. During the preparatory phase of the FAMU experiment, several measurements have been performed both to validate the methodology and to prepare the best configuration of target and detectors for the spectroscopic measurement. We present here the crucial study of the energy dependence of the transfer rate from muonic hydrogen to oxygen ($\Lambda_{\mu p \rightarrow \mu O}$), precisely measured for the first time., Comment: 11 pages, 4 figures, published on Journal of Physics: Conference Series, proc. of International Conference on Precision Physics of Simple Atomic Systems - PSAS2018. arXiv admin note: text overlap with arXiv:1708.03172

A multidomain spectral approach for Painlev\'e transcendents on unbounded domains is presented. This method is designed to study solutions determined uniquely by a, possibly divergent, asymptotic series valid near infinity in a sector and approximates the solution on straight lines lying entirely within said sector without the need of evaluating truncations of the series at any finite point. The accuracy of the method is illustrated for the example of the tritronqu\'ee solution to the Painlev\'e I equation.

In a recent work, we demonstrated that a modified gravity model in which a scalar "darkon" field is coupled to both the standard Riemannian metric and to another non-Riemannian volume form is compatible with observational data from Supernovae Type Ia. Here, we investigate a more complicated model with an additional "inflaton" scalar field. We demonstrate numerically that the model can qualitatively reproduce the Universe inflation epoch, matter dominated epoch, and present accelerating expansion in a seamless way. We show that such solutions occur only when the model parameters are within a very particular range. The main numerical problem we are faced with is reproducing the extremely small time of the inflation epoch. Here, we present how the variation of some parameters affects this time., Comment: 11 pages, 4 figures, new figures and comments added. Final version published in Symmetry

Recently, few cosmological models with additional non-Riemannian volume form(s) have been proposed. In this article we use Supernovae type Ia experimental data to test one of these models which provides a unified description of both dark energy via dynamically generated cosmological constant and dark matter as a "dust" fluid due to a hidden nonlinear Noether symmetry. It turns out that the model allows various scenarios of the future Universe evolution and in the same time perfectly fits contemporary observational data. Further, we investigate the influence of an additional inflaton field with a step like potential. With its help we can reproduce the Universe inflation epoch, matter dominated epoch and present accelerating expansion in a seamless way. Interesting feature is that inflaton undergoes a finite change during its evolution. It can be speculated that the inflaton asymptotic value is connected to the vacuum expectation value of the Higgs field., Comment: 10 pages, 4 figures, prepared for the Proceedings of the XII. International Workshop Lie Theory And Its Applications In Physics (2017)

We consider a simplified model of the optical multi-pass cavity that is being currently developed by the FAMU collaboration for the measurement of the hyperfine splitting in the ground state of muonic hydrogen and of the Zemach radius of the proton. The model is focused on the time distribution of the events of laser-stimulated hyperfine transitions in the muonuc atom and may be helpful in the preliminary design of the FAMU experimental set-up and, more generally, in the optimization of multi-pass optical cavities for experiments with pulsed lasers., Comment: Talk at the EXA-2017 Conference, Vienna, 2017

We present a detailed numerical study of various blow-up issues in the context of the focusing Davey-Stewartson II equation. To this end we study Gaussian initial data and perturbations of the lump and the explicit blow-up solution due to Ozawa. Based on the numerical results it is conjectured that the blow-up in all cases is self similar, and that the time dependent scaling is as in the Ozawa solution and not as in the stable blow-up of standard $L^{2}$ critical nonlinear Schr\"odinger equations. The blow-up profile is given by a dynamically rescaled lump.

At present, the Boer-Mulders (BM) functions are extracted from asymmetry data using the simplifying assumption of their proportionality to the Sivers functions for each quark flavour. Here we present two independent tests for this assumption. We subject COMPASS data on semi-inclusive deep inelastic scattering on the $\langle\cos\phi_h\rangle$, $\langle\cos 2\phi_h\rangle$ and Sivers asymmetries to these tests. Our analysis shows that the tests are satisfied with the available data if the proportionality constant is the same for all quark flavours, which does not correspond to the flavour dependence used in existing analyses. This suggests that the published information on the BM functions may be unreliable. The $\langle\cos\phi_h\rangle$ and $\langle\cos 2 \phi_h\rangle$ asymmetries receive contributions also from the, in principle, calculable Cahn effect. We succeed in extracting the Cahn contributions from experiment (we believe for the first time) and compare with their calculated values, with interesting implications., Comment: 6 pages , a talk presented at the XVII workshop DSPIN-17 in JINR-Dubna, Russia