Riemann–Hilbert problem for the defocusing Lakshmanan–Porsezian–Daniel equation with fully asymmetric nonzero boundary conditions

The Riemann–Hilbert approach is demonstrated to investigate the defocusing Lakshmanan–Porsezian–Daniel equation under fully asymmetric nonzero boundary conditions.In contrast to the symmetry case,this paper focuses on the branch points related to the scattering problem rather than using the Riemann surfaces.For the direct problem,we analyze the Jost solution of lax pairs and some properties of scattering matrix,including two kinds of symmetries.The inverse problem at branch points can be presented,corresponding to the associated Riemann–Hilbert.Moreover,we investigate the time evolution problem and estimate the value of solving the solutions by Jost function.For the inverse problem,we construct it as a Riemann–Hilbert problem and formulate the reconstruction formula for the defocusing Lakshmanan–Porsezian–Daniel equation.The solutions of the Riemann–Hilbert problem can be constructed by estimating the solutions.Finally,we work out the solutions under fully asymmetric nonzero boundary conditions precisely via utilizing the Sokhotski–Plemelj formula and the square of the negative column transformation with the assistance of Riemann surfaces.These results are valuable for understanding physical phenomena and developing further applications of optical problems.

具有骨样微环境的压电催化可控矿化支架实现内源性骨再生

Orderly hierarchical structure with balanced mechanical,chemical,and electrical properties is the basis of the natural bone microenvironment.Inspired by nature,we developed a piezocatalytically-induced controlled mineralization strategy using piezoelectric polymer poly-L-lactic acid(PLLA)fibers with ordered micro-nano structures to prepare biomimetic tissue engineering scaffolds with a bone-like microenvironment(pcm-PLLA),in which PLLA-mediated piezoelectric catalysis promoted the in-situ polymerization of dopamine and subsequently regulated the controllable growth of hydroxyapatite crystals on the fiber surface.PLLA fibers,as analogs of mineralized collagen fibers,were arranged in an oriented manner,and ultimately formed a bone-like interconnected pore structure;in addition,they also provided bone-like piezoelectric properties.The uniformly sized HA nanocrystals formed by controlled mineralization provided a bone-like mechanical strength and chemical environment.The pcm-PLLA scaffold could rapidly recruit endogenous stem cells,and promote their osteogenic differentiation by activating cell membrane calcium channels and PI3K signaling pathways through ultrasound-responsive piezoelectric signals.In addition,the scaffold also provided a suitable microenvironment to promote macrophage M2 polarization and angiogenesis,thereby enhancing bone regeneration in skull defects of rats.The proposed piezocatalytically-induced controllable mineralization strategy provides a new idea for the development of tissue engineering scaffolds that can be implemented for multimodal physical stimulation therapy.