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1. Riemann-Hilbert problems and N-soliton solutions for a coupled mKdV system

   Ma, Wen-Xiu (October 2018, Journal of geometry and physics)

   A 3×3 matrix spectral problem is introduced and its associated AKNS integrable hierarchy with four components is generated. From this spectral problem, a kind of Riemann–Hilbert problems is formulated for a system of coupled mKdV equations in the resulting AKNS integrable hierarchy. N-soliton solutions to the coupled mKdV system are presented through a specific Riemann–Hilbert problem with an identity jump matrix. 

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2. Generalized matrix exponential solutions to the AKNS hierarchy

   Zhang, Jian-bing; Gu, Canyuan; Ma, Wen-Xiu (February 2018, Advances in mathematical physics)

   Generalized matrix exponential solutions to the AKNS equation are obtained by the inverse scattering transformation (IST). The resulting solutions involve six matrices, which satisfy the coupled Sylvester equations. Several kinds of explicit solutions including soliton, complexiton, and Matveev solutions are deduced from the generalized matrix exponential solutions by choosing different kinds of the six involved matrices. Generalized matrix exponential solutions to a general integrable equation of the AKNS hierarchy are also derived. It is shown that the general equation and its matrix exponential solutions share the same linear structure. 

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3. An integrable generalization of the D-Kaup–Newell soliton hierarchy and its bi-Hamiltonian reduced hierarchy

   McAnally, Morgan; Ma, Wen-Xiu (April 2018, Applied mathematics and computation)

   We present a new spectral problem, a generalization of the D-Kaup–Newell spectral problem, associated with the Lie algebra sl(2,R). Zero curvature equations furnish the soliton hierarchy. The trace identity produces the Hamiltonian structure for the hierarchy and shows its Liouville integrability. Lastly, a reduction of the spectral problem is shown to have a different soliton hierarchy with a bi-Hamiltonian structure. The major motivation of this paper is to present spectral problems that generate two soliton hierarchies with infinitely many conservation laws and high-order symmetries. 

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4. Tau‐function formulation for bright, dark soliton and breather solutions to the massive Thirring model

   https://doi.org/10.1111/sapm.12532  

   Chen, Junchao; Feng, Bao‐Feng (August 2022, Studies in Applied Mathematics)

   Abstract In the present paper, we are concerned with the link between the Kadomtsev–Petviashvili–Toda (KP–Toda) hierarchy and the massive Thirring (MT) model. First, we bilinearize the MT model under both the vanishing and nonvanishing boundary conditions. Starting from a set of bilinear equations of two‐component KP–Toda hierarchy, we derive multibright solution to the MT model. Then, considering a set of bilinear equations of the single‐component KP–Toda hierarchy, multidark soliton and multibreather solutions to the MT model are constructed by imposing constraints on the parameters in two types of tau function, respectively. The dynamics and properties of one‐ and two‐soliton for bright, dark soliton and breather solutions are analyzed in details. 

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5. 3D super-resolution imaging using a generalized and scalable progressive refinement method on sparse recovery (PRIS)

   https://doi.org/10.1117/12.2506827  

   Yi, Xiyu; Piestun, Rafael; Weiss, Shimon; Gregor, Ingo; Gryczynski, Zygmunt K.; Koberling, Felix (February 2019, Single Molecule Spectroscopy and Superresolution Imaging XII)

   Within the family of super-resolution (SR) fluorescence microscopy, single-molecule localization microscopies (PALM[1], STORM[2] and their derivatives) afford among the highest spatial resolution (approximately 5 to 10 nm), but often with moderate temporal resolution. The high spatial resolution relies on the adequate accumulation of precise localizations, which requires a relatively low density of bright fluorophores. Several methods have demonstrated localization at higher densities in both two dimensions (2D)[3, 4] and three dimensions (3D)[5-7]. Additionally, with further advancements, such as functional super-resolution[8, 9] and point spread function (PSF) engineering with[8-11] or without[12] multi-channel observations, extra information (spectra, dipole orientation) can be encoded and recovered at the single molecule level. However, such advancements are not fully extended for high-density conditions in 3D. In this work, we adopt sparse recovery using simple matrix/vector operations, and propose a systematic progressive refinement method (dubbed as PRIS) for 3D high-density condition. We also generalized the method for PSF engineering, multichannel and multi-species observations using different forms of matrix concatenations. Specifically, we demonstrate reconstructions with both double-helix and astigmatic PSFs, for both single and biplane settings. We also demonstrate the recovery capability for a mixture of two different color species. 

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