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1. Quasi-diabatic propagation scheme for simulating polariton chemistry

   https://doi.org/10.1063/5.0127118  

   Hu, Deping; Mandal, Arkajit; Weight, Braden M.; Huo, Pengfei (November 2022, The Journal of Chemical Physics)

   We generalize the quasi-diabatic (QD) propagation scheme to simulate the non-adiabatic polariton dynamics in molecule–cavity hybrid systems. The adiabatic-Fock states, which are the tensor product states of the adiabatic electronic states of the molecule and photon Fock states, are used as the locally well-defined diabatic states for the dynamics propagation. These locally well-defined diabatic states allow using any diabatic quantum dynamics methods for dynamics propagation, and the definition of these states will be updated at every nuclear time step. We use several recently developed non-adiabatic mapping approaches as the diabatic dynamics methods to simulate polariton quantum dynamics in a Shin–Metiu model coupled to an optical cavity. The results obtained from the mapping approaches provide very accurate population dynamics compared to the numerically exact method and outperform the widely used mixed quantum-classical approaches, such as the Ehrenfest dynamics and the fewest switches surface hopping approach. We envision that the generalized QD scheme developed in this work will provide a powerful tool to perform the non-adiabatic polariton simulations by allowing a direct interface between the diabatic dynamics methods and ab initio polariton information. 

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2. Rotating spirals in oscillatory media with nonlocal interactions and their normal form

   https://doi.org/10.3934/dcdss.2022085  

   Jaramillo, Gabriela (January 2022, Discrete and Continuous Dynamical Systems - S)

   Biological and physical systems that can be classified as oscillatory media give rise to interesting phenomena like target patterns and spiral waves. The existence of these structures has been proven in the case of systems with local diffusive interactions. In this paper the more general case of oscillatory media with nonlocal coupling is considered. We model these systems using evolution equations where the nonlocal interactions are expressed via a diffusive convolution kernel, and prove the existence of rotating wave solutions for these systems. Since the nonlocal nature of the equations precludes the use of standard techniques from spatial dynamics, the method we use relies instead on a combination of a multiple-scales analysis and a construction similar to Lyapunov-Schmidt. This approach then allows us to derive a normal form, or reduced equation, that captures the leading order behavior of these solutions. 

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3. Global-in-time semiclassical regularity for the Hartree–Fock equation

   https://doi.org/10.1063/5.0089741  

   Chong, J_J; Lafleche, L.; Saffirio, C. (August 2022, Journal of Mathematical Physics)

   For arbitrarily large times T > 0, we prove the uniform-in-ℏ propagation of semiclassical regularity for the solutions to the Hartree–Fock equation with singular interactions of the form V(x)=±x−a with a∈(0,12). As a by-product of this result, we extend to arbitrarily long times the derivation of the Hartree–Fock and the Vlasov equations from the many-body dynamics provided in the work of Chong et al. [arXiv:2103.10946 (2021)]. 

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4. The time-dependent Hartree–Fock–Bogoliubov equations for Bosons

   https://doi.org/10.1007/s00028-022-00799-2  

   Bach, Volker; Breteaux, Sebastien; Chen, Thomas; Fröhlich, Jürg; Sigal, Israel Michael (June 2022, Journal of Evolution Equations)

   Abstract We introduce the map of dynamics of quantum Bose gases into dynamics of quasifree states, which we call the “nonlinear quasifree approximation”. We use this map to derive the time-dependent Hartree–Fock–Bogoliubov (HFB) equations describing the dynamics of quantum fluctuations around a Bose–Einstein condensate. We prove global well-posedness of the HFB equations for pair potentials satisfying suitable regularity conditions, and we establish important conservation laws. We show that the space of solutions of the HFB equations has a symplectic structure reminiscent of a Hamiltonian system. This is then used to relate the HFB equations to the HFB eigenvalue equations discussed in the physics literature. We also construct Gibbs equilibrium states at positive temperature associated with the HFB equations, and we establish criteria for the appearance of Bose–Einstein condensation. 

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5. Six wave interaction equations in finite-depth gravity waves with surface tension

   https://doi.org/10.1017/jfm.2023.128  

   Ablowitz, Mark J.; Luo, Xu-Dan; Musslimani, Ziad H. (April 2023, Journal of Fluid Mechanics)

   Three wave resonant triad interactions in two space and one time dimensions form a well-known system of first-order quadratically nonlinear evolution equations that arise in many areas of physics. In deep water waves, they were first derived by Simmons in 1969 and later shown to be exactly solvable by Ablowitz & Haberman in 1975. Specifically, integrability was established by introducing a system of six wave interactions whose symmetry reduction leads to the well-known three wave equations. Here, it is shown that the six wave interaction and classical three wave equations satisfying triad resonance conditions in finite-depth gravity waves can be derived from the non-local integro-differential formulation of the free surface gravity wave equation with surface tension. These quadratically nonlinear six wave interaction equations and their reductions to the classical and non-local complex as well as real reverse space–time three wave interaction equations are integrable. Limits to infinite and shallow water depth are also discussed. 

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