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1. Non-perturbative Solution of the 1d Schrödinger Equation Describing Photoemission from a Sommerfeld Model Metal by an Oscillating Field

   https://doi.org/10.1007/s00220-023-04771-0  

   Costin, Ovidiu; Costin, Rodica; Jauslin, Ian; Lebowitz, Joel L (September 2023, Communications in Mathematical Physics)

   Springer (Ed.)

   We analyze non-perturbatively the one-dimensional Schrödinger equation describing the emission of electrons from a model metal surface by a classical oscillating electric field. Placing the metal in the half-space x 0, the Schrödinger equation of the system is i∂t ψ = − 2 1 ∂x 2 ψ + (x)(U − E x cos ωt)ψ, t > 0, x ∈ R, where (x) is the Heaviside function and U > 0 is the effective confining potential (we choose units so that m = e = = 1). The amplitude E of the external electric field and the frequency ω are arbitrary. We prove existence and uniqueness of classical solutions of the Schrödinger equation for general initial conditions ψ(x, 0) = f (x), x ∈ R. When 2the initial condition is in L the evolution is unitary and the wave function goes to zero at any fixed x as t → ∞. To show this we prove a RAGE type theorem and show that the discrete spectrum of the quasienergy operator is empty. To obtain positive electron current we consider non-L 2 initial conditions containing an incoming beam from the left. The beam is partially reflected and partially transmitted for all t > 0. For these initial conditions we show that the solution approaches in the large t limit a periodic state that satisfies an infinite set of equations formally derived, under the assumption that the solution is periodic by Faisal et al. (Phys Rev A 72:023412, 2005). Due to a number of pathological features of the Hamiltonian (among which unboundedness in the physical as well as the spatial Fourier domain) the existing methods to prove such results do not apply, and we introduce new, more general ones. The actual solution exhibits a very complex behavior, as seen both analytically and numerically. It shows a steep increase in the current as the frequency passes a threshold value ω = ωc , with ωc depending on the strength of the electric field. For small E, ωc represents the threshold in the classical photoelectric effect, as described by Einstein’s theory. 

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2. Uniqueness and Symmetry for the Mean Field Equation on Arbitrary Flat Tori

   https://doi.org/10.1093/imrn/rnaa109  

   Gu, Guangze; Gui, Changfeng; Hu, Yeyao; Li, Qinfeng (May 2020, International Mathematics Research Notices)

   Abstract We study the following mean field equation on a flat torus $$T:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau )$$: $$\begin{equation*} \varDelta u + \rho \left(\frac{e^{u}}{\int_{T}e^u}-\frac{1}{|T|}\right)=0, \end{equation*}$$where $$ \tau \in \mathbb{C}, \mbox{Im}\ \tau>0$$, and $|T|$ denotes the total area of the torus. We first prove that the solutions are evenly symmetric about any critical point of $$u$$ provided that $$\rho \leq 8\pi $$. Based on this crucial symmetry result, we are able to establish further the uniqueness of the solution if $$\rho \leq \min{\{8\pi ,\lambda _1(T)|T|\}}$$. Furthermore, we also classify all one-dimensional solutions by showing that the level sets must be closed geodesics. 

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3. Reduced models for electron magnetohydrodynamics: Well-posedness and singularity formation

   https://doi.org/10.1090/tran/9427  

   Dai, Mimi (June 2025, Transactions of the American Mathematical Society)

   We propose one-dimensional reduced models for the three-dimensional electron magnetohydrodynamics which involves a highly nonlinear Hall term with intricate structure. The models contain nonlocal nonlinear terms which are more singular than that of the one-dimensional models for the Euler equation and the surface quasi-geostrophic equation. Local well-posedness is obtained in certain circumstances. Moreover, for a model with nonlocal transport term, we show that singularity develops in finite time for a class of initial data. 

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4. Stabilization of the response of cyclically loaded lattice spring models with plasticity

   https://doi.org/10.1051/cocv/2020043  

   Gudoshnikov, Ivan; Makarenkov, Oleg (January 2021, ESAIM: Control, Optimisation and Calculus of Variations)

   null (Ed.)

   This paper develops an analytic framework to design both stress-controlled and displacement-controlled T -periodic loadings which make the quasistatic evolution of a one-dimensional network of elastoplastic springs converging to a unique periodic regime. The solution of such an evolution problem is a function t ↦( e ( t ), p ( t )), where e i ( t ) is the elastic elongation and p i ( t ) is the relaxed length of spring i , defined on [ t 0 , ∞ ) by the initial condition ( e ( t 0 ), p ( t 0 )). After we rigorously convert the problem into a Moreau sweeping process with a moving polyhedron C ( t ) in a vector space E of dimension d , it becomes natural to expect (based on a result by Krejci) that the elastic component t ↦ e ( t ) always converges to a T -periodic function as t → ∞ . The achievement of this paper is in spotting a class of loadings where the Krejci’s limit doesn’t depend on the initial condition ( e ( t 0 ), p ( t 0 )) and so all the trajectories approach the same T -periodic regime. The proposed class of sweeping processes is the one for which the normals of any d different facets of the moving polyhedron C ( t ) are linearly independent. We further link this geometric condition to mechanical properties of the given network of springs. We discover that the normals of any d different facets of the moving polyhedron C ( t ) are linearly independent, if the number of displacement-controlled loadings is two less the number of nodes of the given network of springs and when the magnitude of the stress-controlled loading is sufficiently large (but admissible). The result can be viewed as an analogue of the high-gain control method for elastoplastic systems. In continuum theory of plasticity, the respective result is known as Frederick-Armstrong theorem. 

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5. Bubble pinch-off in turbulence

   https://doi.org/10.1073/pnas.1909842116  

   Ruth, Daniel J.; Mostert, Wouter; Perrard, Stéphane; Deike, Luc (December 2019, Proceedings of the National Academy of Sciences)

   Although bubble pinch-off is an archetype of a dynamical system evolving toward a singularity, it has always been described in idealized theoretical and experimental conditions. Here, we consider bubble pinch-off in a turbulent flow representative of natural conditions in the presence of strong and random perturbations, combining laboratory experiments, numerical simulations, and theoretical modeling. We show that the turbulence sets the initial conditions for pinch-off, namely the initial bubble shape and flow field, but after the pinch-off starts, the turbulent time at the neck scale becomes much slower than the pinching dynamics: The turbulence freezes. We show that the average neck size, d ¯ , can be described by d ¯ ∼ ( t − t 0 ) α , where t 0 is the pinch-off or singularity time and α ≈ 0.5 , in close agreement with the axisymmetric theory with no initial flow. While frozen, the turbulence can influence the pinch-off through the initial conditions. Neck shape oscillations described by a quasi–2-dimensional (quasi-2D) linear perturbation model are observed as are persistent eccentricities of the neck, which are related to the complex flow field induced by the deformed bubble shape. When turbulent stresses are less able to be counteracted by surface tension, a 3-dimensional (3D) kink-like structure develops in the neck, causing d ¯ to escape its self-similar decrease. We identify the geometric controlling parameter that governs the appearance of these kink-like interfacial structures, which drive the collapse out of the self-similar route, governing both the likelihood of escaping the self-similar process and the time and length scale at which it occurs. 

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