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1. Analytical construction of soliton families in one‐ and two‐dimensional nonlinear Schrödinger equations with nonparity‐time‐symmetric complex potentials

   https://doi.org/10.1111/sapm.12383  

   Yang, Jianke (April 2021, Studies in Applied Mathematics)

   Abstract The existence of soliton families in nonparity‐time‐symmetric complex potentials remains poorly understood, especially in two spatial dimensions. In this article, we analytically investigate the bifurcation of soliton families from linear modes in one‐ and two‐dimensional nonlinear Schrödinger equations with localized Wadati‐type nonparity‐time‐symmetric complex potentials. By utilizing the conservation law of the underlying non‐Hamiltonian wave system, we convert the complex soliton equation into a new real system. For this new real system, we perturbatively construct a continuous family of low‐amplitude solitons bifurcating from a linear eigenmode to all orders of the small soliton amplitude. Hence, the emergence of soliton families in these nonparity‐time‐symmetric complex potentials is analytically explained. We also compare these analytically constructed soliton solutions with high‐accuracy numerical solutions in both one and two dimensions, and the asymptotic accuracy of these perturbation solutions is confirmed. 

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2. Cooling and instabilities in colliding flows

   https://doi.org/10.1093/mnras/stab2577  

   Markwick, R N; Frank, A; Carroll-Nellenback, J; Liu, B; Blackman, E G; Lebedev, S V; Hartigan, P M (October 2021, Monthly Notices of the Royal Astronomical Society)

   ABSTRACT Collisional self-interactions occurring in protostellar jets give rise to strong shocks, the structure of which can be affected by radiative cooling within the flow. To study such colliding flows, we use the AstroBEAR AMR code to conduct hydrodynamic simulations in both one and three dimensions with a power-law cooling function. The characteristic length and time-scales for cooling are temperature dependent and thus may vary as shocked gas cools. When the cooling length decreases sufficiently and rapidly, the system becomes unstable to the radiative shock instability, which produces oscillations in the position of the shock front; these oscillations can be seen in both the one- and three-dimensional cases. Our simulations show no evidence of the density clumping characteristic of a thermal instability, even when the cooling function meets the expected criteria. In the three-dimensional case, the nonlinear thin shell instability (NTSI) is found to dominate when the cooling length is sufficiently small. When the flows are subjected to the radiative shock instability, oscillations in the size of the cooling region allow NTSI to occur at larger cooling lengths, though larger cooling lengths delay the onset of NTSI by increasing the oscillation period. 

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3. Polarization-diverse soliton transitions and deterministic switching dynamics in strongly-coupled and self-stabilized microresonator frequency combs

   https://doi.org/10.1038/s42005-024-01773-9  

   Wang, Wenting; Aldhafeeri, Alwaleed; Zhou, Heng; Melton, Tristan; Jiang, Xinghe; Vinod, Abhinav_Kumar; Yu, Mingbin; Lo, Guo-Qiang; Kwong, Dim-Lee; Wong, Chee_Wei (August 2024, Communications Physics)

   Abstract Dissipative Kerr soliton microcombs in microresonators have enabled fundamental advances in chip-scale precision metrology, communication, spectroscopy, and parallel signal processing. Here we demonstrate polarization-diverse soliton transitions and deterministic switching dynamics of a self-stabilized microcomb in a strongly-coupled dispersion-managed microresonator driven with a single pump laser. The switching dynamics are induced by the differential thermorefractivity between coupled transverse-magnetic and transverse-electric supermodes during the forward-backward pump detunings. The achieved large soliton existence range and deterministic transitions benefit from the switching dynamics, leading to the cross-polarized soliton microcomb formation when driven in the transverse-magnetic supermode of the single resonator. Secondly, we demonstrate two distinct polarization-diverse soliton formation routes – arising from chaotic or periodically-modulated waveforms via pump power selection. Thirdly, to observe the cross-polarized supermode transition dynamics, we develop a parametric temporal magnifier with picosecond resolution, MHz frame rate and sub-ns temporal windows. We construct picosecond temporal transition portraits in 100-ns recording length of the strongly-coupled solitons, mapping the transitions from multiple soliton molecular states to singlet solitons. This study underpins polarization-diverse soliton microcombs for chip-scale ultrashort pulse generation, supporting applications in frequency and precision metrology, communications, spectroscopy and information processing. 

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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. On the propulsion of a rigid body in a viscous liquid under the action of a time-periodic force

   https://doi.org/10.1007/s40687-025-00510-0  

   Galdi, Giovanni P; Karakouzian, Mher M (June 2025, Research in the Mathematical Sciences)

   Arigid body B moves in an otherwise quiescent viscous liquid filling the whole space outside B, under the action of a time-periodic force f of period T applied to a given point of B and of fixed direction. We assume that the average of f over an interval of length T does not vanish, and that the amplitude, δ,off is sufficiently small. Our goal is to investigate when B executes a nonzero net motion; that is, B is able to cover any prescribed distance in a finite time. We show that, at the order δ, this happens if and only if f and B satisfy a certain condition. We also show that this is always the case if B is prevented from spinning. Finally, we provide explicit examples where the condition above is satisfied or not. All our analysis is performed in a general class of weak solutions to the coupled system body-liquid problem. 

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