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1. On the Whitham system for the radial nonlinear Schrödinger equation

   https://doi.org/10.1111/sapm.12254  

   Ablowitz, Mark J.; Cole, Justin T.; Rumanov, Igor (January 2019, Studies in Applied Mathematics)

   Abstract Dispersive shock waves (DSWs) of the defocusing radial nonlinear Schrödinger (rNLS) equation in two spatial dimensions are studied. This equation arises naturally in Bose‐Einstein condensates, water waves, and nonlinear optics. A unified nonlinear WKB approach, equally applicable to integrable or nonintegrable partial differential equations, is used to find the rNLS Whitham modulation equation system in both physical and hydrodynamic type variables. The description of DSWs obtained via Whitham theory is compared with direct rNLS numerics; the results demonstrate very good quantitative agreement. On the other hand, as expected, comparison with the corresponding DSW solutions of the one‐dimensional NLS equation exhibits significant qualitative and quantitative differences. 

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2. Surface crossing and energy flow in many-dimensional quantum systems

   https://doi.org/10.1073/pnas.2221690120  

   Zhang, Chenghao; Gruebele, Martin; Logan, David E.; Wolynes, Peter G. (February 2023, Proceedings of the National Academy of Sciences)

   Energy flow in molecules, like the dynamics of other many-dimensional finite systems, involves quantum transport across a dense network of near-resonant states. For molecules in their electronic ground state, the network is ordinarily provided by anharmonic vibrational Fermi resonances. Surface crossing between different electronic states provides another route to chaotic motion and energy redistribution. We show that nonadiabatic coupling between electronic energy surfaces facilitates vibrational energy flow and, conversely, anharmonic vibrational couplings facilitate nonadiabatic electronic state mixing. A generalization of the Logan–Wolynes theory of quantum energy flow in many-dimensional Fermi resonance systems to the two-surface case gives a phase diagram describing the boundary between localized quantum dynamics and global energy flow. We explore these predictions and test them using a model inspired by the problem of electronic excitation energy transfer in the photosynthetic reaction center. Using an explicit numerical solution of the time-dependent Schrödinger equation for this ten-dimensional model, we find quite good agreement with the expectations from the approximate analytical theory. 

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3. Relation between solutions of the Schrödinger equation with transitioning resonance solutions of the gravitational three-body problem

   https://doi.org/10.1088/2399-6528/ab693f  

   Belbruno, Edward (January 2020, Journal of Physics Communications)

   Abstract It is shown that a class of approximate resonance solutions in the three-body problem under the Newtonian gravitational force are equivalent to quantized solutions of a modified Schrödinger equation for a wide range of masses that transition between energy states. In the macroscopic scale, the resonance solutions are shown to transition from one resonance type to another through weak capture at one of the bodies, while in the Schrödinger equation, one obtains quantized wave solutions transitioning between different energies. The resonance transition dynamics provides a classical model of a particle moving between different energy states in the Schrödinger equation. This methodology provides a connection between celestial and quantum mechanics. 

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4. On the Wave Turbulence Theory for the Nonlinear Schrödinger Equation with Random Potentials

   https://doi.org/https://doi.org/10.3390/e21090823  

   Nazarenko, Sergy; Soffer, Avy; Tran, Minh-Binh (August 2019, Entropy)

   We derive new kinetic and a porous medium equations from the nonlinear Schrödinger equation with random potentials. The kinetic equation has a very similar form compared to the four-wave turbulence kinetic equation in the wave turbulence theory. Moreover, we construct a class of self-similar solutions for the porous medium equation. These solutions spread with time, and this fact answers the “weak turbulence” question for the nonlinear Schrödinger equation with random potentials. We also derive Ohm’s law for the porous medium equation. 

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5. The Laplace Method for Energy Eigenvalue Problems in Quantum Mechanics

   https://doi.org/10.3390/quantum5020024  

   Canfield, Jeremy; Galler, Anna; Freericks, James K. (June 2023, Quantum Reports)

   Quantum mechanics has about a dozen exactly solvable potentials. Normally, the time-independent Schrödinger equation for them is solved by using a generalized series solution for the bound states (using the Fröbenius method) and then an analytic continuation for the continuum states (if present). In this work, we present an alternative way to solve these problems, based on the Laplace method. This technique uses a similar procedure for the bound states and for the continuum states. It was originally used by Schrödinger when he solved the wave functions of hydrogen. Dirac advocated using this method too. We discuss why it is a powerful approach to solve all problems whose wave functions are represented in terms of confluent hypergeometric functions, especially for the continuum solutions, which can be determined by an easy-to-program contour integral. 

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