arXiv subjects

Agissilaos Athanassoulis

Publications and source records attributed to Agissilaos Athanassoulis.

13 recordsLinked to original sources

arXiv2026

Global solutions for the Alber equation in $H^1\mathfrak{S}^1(\mathbb{T})$

The Alber equation is the mixed-state nonlinear Schrödinger equation with singular ($δ$-interaction) kernel. It is used in the modeling of stochastic ocean waves, where it appears with the focusing sign in the nonlinearity, on $d=1.$ The main result of the paper is global well-posedness for self-adjoint, non-negative data in the Schatten-Sobolev space $H^1\mathfrak{S}^1(\mathbb{T}),$ for both the focusing and defocusing cases. The Schatten class norms achieve control of the position density without derivative loss, and a systematic Fourier-Galerkin argument tailored to the $δ$ kernel allows us to establish several qualitative properties of the solution, including energy conservation. In the focusing case, Hoffmann-Ostenhof and Gagliardo-Nirenberg estimates yield a global a priori $H^1\mathfrak{S}^1$ bound with no smallness condition. Non-negativity is a structural requirement for the energy argument to work. The propagation of higher Sobolev regularity $H^s\mathfrak{S}^1$ follows. As an application, small perturbations around Penrose-stable backgrounds are shown to grow at most polynomially in $H^1\mathfrak{S}$ over long timescales.

math.AP↗

arXiv2026

A second-order-in-time scheme for the von Neumann equation with singular self-interaction and simulation of the onset of instability

The von Neumann equation with delta self-interaction kernel serves as a statistical model for nonlinear waves, and it exhibits a bifurcation between stable and unstable regimes. In oceanography it is known as the Alber equation, and its bifurcation is important for understanding rogue waves, a key problem in marine safety. Despite its significance, only one first-order-in-time numerical method exists in the literature. In this paper, we propose a structure-preserving, linearly implicit, second-order-in-time scheme for its numerical solution. We employ fourth-order finite differences for the spatial discretization. As an illustrative example, we explore the onset of modulation instability. We verify that the linear stability analysis accurately predicts the initial growth phase, but fails to forecast the maximum amplitude, the formation of a coherent structure in the nonlinear regime, or the relevant timescales. Monte Carlo simulations with Gaussian background spectra reveal that the maximum amplitude depends mainly on the homogeneous background rather than the initial inhomogeneity. For weak instabilities, the inhomogeneity grows substantially from its initial condition, but remains small compared to the background. On the other hand, strong instability leads to recurrent hotspots of increased variance. This provides a possible explanation of how modulation instability makes rogue waves more likely in unidirectional sea states.

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arXiv2024

Efficient numerical approximations for a non-conservative Nonlinear Schrodinger equation appearing in wind-forced ocean waves

We consider a non-conservative nonlinear Schrodinger equation (NCNLS) with time-dependent coefficients, inspired by a water waves problem. This problem does not have mass or energy conservation, but instead mass and energy change in time under explicit balance laws. In this paper we extend to the particular NCNLS two numerical schemes which are known to conserve energy and mass in the discrete level for the cubic NLS. Both schemes are second oder accurate in time, and we prove that their extensions satisfy discrete versions of the mass and energy balance laws for the NCNLS. The first scheme is a relaxation scheme that is linearly implicit. The other scheme is a modified Delfour-Fortin-Payre scheme and it is fully implicit. Numerical results show that both schemes capture robustly the correct values of mass and energy, even in strongly non-conservative problems. We finally compare the two numerical schemes and discuss their performance.

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arXiv2022

A novel, structure-preserving, second-order-in-time relaxation scheme for Schrödinger-Poisson systems

We introduce a new structure preserving, second order in time relaxation-type scheme for approximating solutions of the Schrödinger-Poisson system. More specifically, we use the Crank-Nicolson scheme as a time stepping mechanism, whilst the nonlinearity is handled by means of a relaxation approach in the spirit of \cite{Besse, KK} for the nonlinear Schrödinger equation. For the spatial discretisation we use the standard conforming finite element scheme. The resulting scheme is explicit with respect to the nonlinearity, i.e. it requires the solution of a linear system for each time-step, and satisfies discrete versions of the system's mass conservation and energy balance laws for constant meshes. The scheme is seen to be second order in time. We conclude by presenting some numerical experiments, including an example from cosmology and an example with variable time-steps which demonstrate the effectiveness and robustness of the new scheme.

math.NA↗

arXiv2021

Modelling of ocean waves with the Alber equation: application to non-parametric spectra and generalization to crossing seas

The Alber equation is a phase-averaged second-moment model for the statistics of a sea state, which recently has been attracting renewed attention. We extend it in two ways: firstly, we derive a generalized Alber system starting from a system of nonlinear Schrödinger equations, which contains the classical Alber equation as a special case but can also describe crossing seas, i.e. two wavesystems with different wavenumbers crossing. (These can be two completely independent wavenumbers, i.e. in general different directions and different moduli.) We also derive the associated 2-dimensional scalar instability condition. This is the first time that a modulation instability condition applicable to crossing seas has been systematically derived for general spectra. Secondly, we use the classical Alber equation and its associated instability condition to quantify how close a given non-parametric spectrum is to being modulationally unstable. We apply this to a dataset of 100 non-parametric spectra provided by the Norwegian Meteorological Institute, and find the vast majority of realistic spectra turn out to be stable, but three extreme sea states are found to be unstable (out of 20 sea states chosen for their severity). Moreover, we introduce a novel "proximity to instability" (PTI) metric, inspired by the stability analysis. This is seen to correlate strongly with the steepness and Benjamin-Feir Index (BFI) for the sea states in our dataset (>85% Spearman rank correlation). Furthermore, upon comparing with phase-resolved broadband Monte Carlo simulations, the kurtosis and probability of rogue waves for each sea state are also seen to correlate well with its PTI (>85% Spearman rank correlation).

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arXiv2017

Semiclassical regularization of Vlasov equations and wavepackets for nonlinear Schrödinger equations

We consider the semiclassical limit of nonlinear Schrödinger equations with wavepacket initial data. We recover the Wigner measure of the problem, a macroscopic phase-space density which controls the propagation of the physical observables such as mass, energy and momentum. Wigner measures have been used to create effective models for wave propagation in random media, quantum molecular dynamics, mean field limits, and the propagation of electrons in graphene. In nonlinear settings, the Vlasov-type equations obtained for the Wigner measure are often ill-posed on the physically interesting spaces of initial data. In this paper we are able to select the measure-valued solution of the 1+1 dimensional Vlasov-Poisson equation which correctly captures the semiclassical limit, thus finally resolving the non-uniqueness in the seminal result of [Zhang, Zheng \& Mauser, Comm. Pure Appl. Math. (2002) 55, doi:10.1002/cpa.3017]. The same approach is also applied to the Vlasov-Dirac-Benney equation with small wavepacket initial data, extending several known results.

math.AP↗

arXiv2015

Regularized semiclassical limits: linear flows with infinite Lyapunov exponents

Semiclassical asymptotics for linear Schrödinger equations with non-smooth potentials give rise to ill-posed formal semiclassical limits. These problems have attracted a lot of attention in the last few years, as a proxy for the treatment of eigenvalue crossings, i.e. general systems. It has recently been shown that the semiclassical limit for conical singularities is in fact well-posed, as long as the Wigner measure (WM) stays away from singular saddle points. In this work we develop a family of refined semiclassical estimates, and use them to derive regularized transport equations for saddle points with infinite Lyapunov exponents, extending the aforementioned recent results. In the process we answer a related question posed by P. L. Lions and T. Paul in 1993. If we consider more singular potentials, our rigorous estimates break down. To investigate whether conical saddle points, such as $-|x|$, admit a regularized transport asymptotic approximation, we employ a numerical solver based on posteriori error control. Thus rigorous upper bounds for the asymptotic error in concrete problems are generated. In particular, specific phenomena which render invalid any regularized transport for $-|x|$ are identified and quantified. In that sense our rigorous results are sharp. Finally, we use our findings to formulate a precise conjecture for the condition under which conical saddle points admit a regularized transport solution for the WM.

math.AP↗

arXiv2014

Numerical simulations of X-rays Free Electron Lasers (XFEL)

We study a nonlinear Schrödinger equation which arises as an effective single particle model in X-ray Free Electron Lasers (XFEL). This equation appears as a first-principles model for the beam-matter interactions that would take place in an XFEL molecular imaging experiment in \cite{frat1}. Since XFEL is more powerful by several orders of magnitude than more conventional lasers, the systematic investigation of many of the standard assumptions and approximations has attracted increased attention. In this model the electrons move under a rapidly oscillating electromagnetic field, and the convergence of the problem to an effective time-averaged one is examined. We use an operator splitting pseudo-spectral method to investigate numerically the behaviour of the model versus its time-averaged version in complex situations, namely the energy subcritical/mass supercritical case, and in the presence of a periodic lattice. We find the time averaged model to be an effective approximation, even close to blowup, for fast enough oscillations of the external field. This work extends previous analytical results for simpler cases \cite{xfel1}.

math.NA↗

arXiv2012

On the selection of the classical limit for potentials with BV derivatives

We consider the classical limit of the quantum evolution, with some rough potential, of wave packets concentrated near singular trajectories of the underlying dynamics. We prove that under appropriate conditions, even in the case of BV vector fields, the correct classical limit can be selected.

math.AP↗

arXiv2011

Strong and weak semiclassical limits for some rough Hamiltonians

We present several results concerning the semiclassical limit of the time dependent Schrödinger equation with potentials whose regularity doesn't guarantee the uniqueness of the underlying classical flow. Different topologies for the limit are considered and the situation where two bicharateristics can be obtained out of the same initial point is emphasized.

math.AP↗

arXiv2011

Semiclassical Propagation of Coherent States for the Hartree equation

In this paper we consider the nonlinear Hartree equation in presence of a given external potential, for an initial coherent state. Under suitable smoothness assumptions, we approximate the solution in terms of a time dependent coherent state, whose phase and amplitude can be determined by a classical flow. The error can be estimated in $L^2$ by $C \sqrt {\var}$, $\var$ being the Planck constant. Finally we present a full formal asymptotic expansion.

math-ph↗

arXiv2010

Smoothed Affine Wigner Transform

We study a generalization of Husimi function in the context of wavelets. This leads to a nonnegative density on phase-space for which we compute the evolution equation corresponding to a SchrÄodinger equation.

math.AP↗

arXiv2010

Strong phase-space semiclassical asymptotics

Wigner and Husimi transforms have long been used for the phase-space reformulation of Schrödinger-type equations, and the study of the corresponding semiclassical limits. Most of the existing results provide approximations in appropriate weak topologies. In this work we are concerned with semiclassical limits in the strong topology, i.e. approximation of Wigner functions by solutions of the Liouville equation in $L^2$ and Sobolev norms. The results obtained improve the state of the art, and highlight the role of potential regularity, especially through the regularity of the Wigner equation. It must be mentioned that the strong convergence can be shown up to $O(log \frac{1}ε)$ time-scales, which is well known to be, in general, the limit of validity of semiclassical asymptotics.

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