On the Coupled Maxwell–Bloch System of Equations With Nondecaying Fields at InfinityLi, Sitai; Biondini, Gino; Kovačič, Gregor
doi: 10.1111/sapm.70055pmid: N/A
We study an initial‐boundary‐value problem (IBVP) for a system of coupled Maxwell–Bloch equations (CMBE) that model two colors or polarizations of light resonantly interacting with a degenerate, two‐level, active optical medium with an excited state and a pair of degenerate ground states. We assume that the electromagnetic field approaches nonvanishing plane waves in the far past and future. This type of interaction has been found to underlie nonlinear optical phenomena including electromagnetically induced transparency, slow light, stopped light, and quantum memory. Under the assumptions of unidirectional, lossless propagation of slowly modulated plane waves, the resulting CMBE become completely integrable in the sense of possessing a Lax pair. In this paper, we formulate an inverse scattering transform (IST) corresponding to these CMBE and their Lax pair, allowing for the spectral line of the atomic transitions in the active medium to have a finite width. The scattering problem for this Lax pair is the same as for the Manakov system. The main advancement in this IST for CMBE is calculating the nontrivial spatial propagation of the spectral data and determining the state of the optical medium in the distant future from that in the distant past, which is needed for the complete formulation of the IBVP. The Riemann–Hilbert problem is used to extract the spatio‐temporal dependence of the solution from the evolving spectral data. We further derive and analyze several types of solitons and determine their velocity and stability, as well as find dark states of the medium, which fail to interact with a given pulse.
A Theory of Generalized Coordinates for Stochastic Differential EquationsDa Costa, Lancelot; Da Costa, Nathaël; Heins, Conor; Medrano, Johan; Pavliotis, Grigorios A.; Parr, Thomas; Meera, Ajith Anil; Friston, Karl
doi: 10.1111/sapm.70062pmid: N/A
Stochastic differential equations are ubiquitous modeling tools in applied mathematics and the sciences. In most modeling scenarios, random fluctuations driving dynamics or motion have some nontrivial temporal correlation structure, which renders the SDE non‐Markovian; a phenomenon commonly known as ‘colored’’ noise. Thus, an important objective is to develop effective tools for mathematically and numerically studying (possibly non‐Markovian) SDEs. In this paper, we formalize a mathematical theory for analyzing and numerically studying SDEs based on so‐called “generalized coordinates of motion.” Like the theory of rough paths, we analyze SDEs pathwise for any given realization of the noise, not solely probabilistically. Like the established theory of Markovian realization, we realize non‐Markovian SDEs as a Markov process in an extended space. Unlike the established theory of Markovian realization however, the Markovian realizations here are accurate on short timescales and may be exact globally in time, when flows and fluctuations are analytic. This theory is exact for SDEs with analytic flows and fluctuations, and is approximate when flows and fluctuations are differentiable. It provides useful analysis tools, which we employ to solve linear SDEs with analytic fluctuations. It may also be useful for studying rougher SDEs, as these may be identified as the limit of smoother ones. This theory supplies effective, computationally straightforward methods for simulation, filtering and control of SDEs; among others, we rederive generalized Bayesian filtering, a state‐of‐the‐art method for time‐series analysis. Looking forward, this paper suggests that generalized coordinates have far‐reaching applications throughout stochastic differential equations.
Derivation of the Bacterial Run‐and‐Tumble Kinetic Model: Quantitative and Strong Convergence ResultsBlaustein, Alain
doi: 10.1111/sapm.70060pmid: N/A
During the past century, biologists and mathematicians investigated two mechanisms underlying bacteria motion: the run phase during which bacteria move in straight lines and the tumble phase in which they change their orientation. When surrounded by a chemical attractant, experiments show that bacteria increase their run time as moving up concentration gradients, leading to a biased random walk toward favorable regions. This observation raises the following question, which has drawn intense interest from both biological and mathematical communities: what cellular mechanisms enable bacteria to feel concentration gradients? In this article, we investigate an asymptotic regime that was proposed to explain this ability thanks to internal mechanisms. More precisely, we derive the run‐and‐tumble kinetic equation with concentration's gradient‐dependent tumbling rate from a more comprehensive model, which incorporates internal cellular mechanisms. Our result improves on previous investigations, as we obtain strong convergence toward the gradient‐dependent kinetic model with quantitative and formally optimal convergence rates. The main ingredient consists in identifying a set of coordinates for the internal cellular dynamics in which concentration gradients arise explicitly. Then, we use relative entropy methods in order to capture quantitative measurement of the distance between the model incorporating cellular mechanisms and the one with concentration‐gradient‐dependent tumbling rate.
Asymptotic Behaviors of Chandrasekhar Variational Problem for Neutron Stars With Slater‐Type ModificationLi, Deke; Wang, Qingxuan
doi: 10.1111/sapm.70058pmid: N/A
In this paper, we consider the Chandrasekhar variational model for neutron stars with defocusing Slater‐type modifications. First, we show the existence and nonexistence of the ground state ρε$\rho _{\varepsilon }$ by concentration–compactness method, and particularly use two auxiliary functions to prove the strongly binding inequality. Second, we characterize perturbation limit behaviors of ground states ρε$\rho _{\varepsilon }$ as ε→0+$\varepsilon \rightarrow 0^+$ and obtain two different blow‐up profiles corresponding to two limit equations for N=N∗$N= N_*$ and N>N∗$N> N_*$, where ε$\varepsilon$ is a parameter corresponding to Slater‐type modifications, and N∗$N_*$ is a threshold value related to the Chandrasekhar limit. Finally, we study the limit behaviors for N≥N∗$N\ge N_*$ as ε→+∞$\varepsilon \rightarrow +\infty$, using some iterate arguments, we obtain a vanishing rate for ρε$\rho _{\varepsilon }$ that ∥ρε∥L∞≲ε−1α−1$\Vert \rho _\varepsilon \Vert _{L^\infty }\lesssim \varepsilon ^{-\frac{1}{\alpha -1}}$ as ε→+∞$\varepsilon \rightarrow +\infty$ for any 4/3<α<+∞$4/3<\alpha <+\infty$. Moreover, we characterize the limit behaviors of the energy Eε(N)$E_\varepsilon (N)$ with respect to ε$\varepsilon$, and show that limε→+∞Eε(N)=mN$\lim _{\varepsilon \rightarrow +\infty }E_\varepsilon (N)=mN$, Eε(N)$E_\varepsilon (N)$ is concave and strictly monotonically increasing with respect to ε>0$\varepsilon >0$ in some case.
Analyticity and Stable Computation of Dirichlet–Neumann Operators for Laplace's Equation Under Quasiperiodic Boundary Conditions in Two and Three DimensionsNicholls, David P.; Wilkening, Jon; Zhao, Xinyu
doi: 10.1111/sapm.70059pmid: N/A
Dirichlet–Neumann operators (DNOs) are important to the formulation, analysis, and simulation of many crucial models found in engineering and the sciences. For instance, these operators permit moving‐boundary problems, such as the classical water wave problem (free‐surface ideal fluid flow under the influence of gravity and capillarity), to be restated in terms of interfacial quantities, which not only eliminates the boundary tracking problem, but also reduces the problem's dimension. While these DNOs have been the object of much recent study regarding their numerical simulation and rigorous analysis, they have yet to be examined in the setting of laterally quasiperiodic boundary conditions. The purpose of this contribution is to begin this investigation with a particular focus on the more realistic simulation of two‐ and three‐dimensional free‐surface water waves. Here, we not only carefully define the DNO with respect to these boundary conditions for Laplace's equation, but we also show the rigorous analyticity of these operators with respect to sufficiently smooth boundary perturbations. These theoretical developments suggest a novel algorithm for the stable and high‐order simulation of the DNO, which we implement and extensively test.
Separation of the Initial Conditions in the Inverse Problem for One‐Dimensional Nonlinear Tsunami Wave Run‐Up TheoryRybkin, Alexei; Bobrovnikov, Oleksandr; Palmer, Noah; Abramowicz, Daniel; Pelinovsky, Efim
doi: 10.1111/sapm.70054pmid: N/A
We investigate the inverse tsunami wave problem within the framework of the one‐dimensional (1D) nonlinear shallow water equations (SWE). Specifically, we show that the initial displacement η0(x)$\eta _0(x)$ and velocity u0(x)$u_0(x)$ of the wave can be recovered, given the known motion of the shoreline R(t)$R(t)$ (the wet/dry free boundary), in terms of the Abel transform. We demonstrate that for power‐shaped inclined bathymetries, this problem admits a complete solution for any η0$\eta _0$ and u0$u_0$, provided the wave does not break.