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General rogue waves in (1+1)-dimensional three-wave resonant interaction systems are derived by the bilinear method. These solutions are divided into three families, which correspond to a simple root, two simple roots and a double root of a certain quartic equation arising from the dimension reduction, respectively. It is shown that while the first family of solutions associated with a simple root exists for all signs of the nonlinear coefficients in the three-wave interaction equations, the other two families of solutions associated with two simple roots and a double root can only exist in the so-called soliton-exchange case, where the nonlinear coefficients have certain signs. Many of these rogue wave solutions, such as those associated with two simple roots, the ones generated by a $2\times 2$ block determinant in the double-root case, and higher-order solutions associated with a simple root, are new solutions which have not been reported before. Technically, our bilinear derivation of rogue waves for the double-root case is achieved by a generalization to the previous dimension reduction procedure in the bilinear method, and this generalized procedure allows us to treat roots of arbitrary multiplicities. Dynamics of the derived rogue waves is also examined, and new rogue wave patterns are presented. Connection between these bilinear rogue waves and those derived earlier by Darboux transformation is also explained.

Recently, a number of nonlocal integrable equations, such as the PT -symmetric nonlinear Schrödinger (NLS) equation and PT -symmetric Davey-Stewartson equations, were proposed and studied. Here we show that many of such nonlocal integrable equations can be converted to local integrable equations through simple variable transformations. Examples include these nonlocal NLS and Davey-Stewartson equations, a nonlocal derivative NLS equation, the reverse space-time complex modified Korteweg-de Vries (CMKdV) equation, and many others. These transformations not only establish immediately the integrability of these nonlocal equations, but also allow us to construct their analytical solutions from solutions of the local equations. These transformations can also be used to derive new nonlocal integrable equations. As applications of these transformations, we use them to derive rogue wave solutions for the partially PT -symmetric Davey-Stewartson equations and the nonlocal derivative NLS equation. In addition, we use them to derive multi-soliton and quasi-periodic solutions in the reverse space-time CMKdV equation. Furthermore, we use them to construct many new nonlocal integrable equations such as nonlocal short pulse equations, nonlocal nonlinear diffusion equations, and nonlocal Sasa-Satsuma equations.

In this paper, we investigate higher-order rogue wave solutions of the Kundu-Eckhaus equation, which contains quintic nonlinearity and the Raman effect in nonlinear optics. By means of a gauge transformation, the Kundu-Eckhaus equation is converted to an extended nonlinear Schrödinger equation. We derive the Lax pair, the generalized Darboux transformation, and the Nth-order rogue wave solution for the extended nonlinear Schrödinger equation. Then, by using the gauge transformation between the two equations, a concise unified formula of the Nth-order rogue wave solution with several free parameters for the Kundu-Eckhaus equation is obtained. In particular, based on symbolic computation, explicit rogue wave solutions to the Kundu-Eckhaus equation from the first to the third order are presented. Some figures illustrate dynamic structures of the rogue waves from the first to the fourth order. Moreover, through numerical calculations and plots, we show that the quintic and Raman-effect nonlinear terms affect the spatial distributions of the humps in higher-order rogue waves, although the amplitudes and the time of appearance of the humps are unchanged.

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BackgroundDexmedetomidine (Dex) was reported to exhibit anti-inflammatory effect in the nervous system. However, the mechanism by which Dex exhibits anti-inflammation effects on LPS-stimulated BV2 microglia cells remains unclear. Thus, this study aimed to investigate the role of Dex in LPS-stimulated BV2 cells.MethodsThe BV2 cells were stimulated by lipopolysaccharides (LPS). BV2 cells were infected with short-hairpin RNAs targeting NF-κB (NF-κB-shRNAs) and NF-κB overexpression lentivirus, respectively. In addition, miR-340 mimics or miR-340 inhibitor was transfected into BV2 cells, respectively. Meanwhile, the dual-luciferase reporter system assay was used to explore the interaction of miR-340 and NF-κB in BV2 cells. CCK-8 was used to detect the viability of BV2 cells. In addition, Western blotting was used to detect the level of NF-κB in LPS-stimulated BV2 cells. The levels of TNF-α, IL-6, IL-1β, IL-2, IL-12, IL-10 and MCP-1 in LPS-stimulated BV2 cells were measured with ELISA.ResultsThe level of miR-340 was significantly upregulated in Dex-treated BV2 cells. Meanwhile, the level of NF-κB was significantly increased in BV2 cells following infection with lenti-NF-κB, which was markedly reversed by Dex. LPS markedly increased the expression of NF-κB and proinflammatory cytokines in BV2 cells, which were reversed in the presence of Dex. Moreover, miR-340 mimics enhanced the anti-inflammatory effects of Dex in LPS-stimulated BV2 cells via inhibiting NF-κB and proinflammatory cytokines. Furthermore, Dex obviously inhibited LPS-induced phagocytosis in BV2 cells.ConclusionTaken together, our results suggested that Dex might exert anti-inflammatory effects in LPS-stimulated BV2 cells via upregulation of miR-340. Therefore, Dex might serve as a potential agent for the treatment of neuroinflammation.

We derive general rogue wave solutions of arbitrary orders in the Boussinesq equation by the bilinear Kadomtsev-Petviashvili (KP) reduction method. These rogue solutions are given as Gram determinants with 2N − 2 free irreducible real parameters, where N is the order of the rogue wave. Tuning these free parameters, rogue waves of various patterns are obtained, many of which have not been seen before. Compared to rogue waves in other integrable equations, a new feature of rogue waves in the Boussinesq equation is that the rogue wave of maximum amplitude at each order is generally asymmetric in space. On the technical aspect, our contribution to the bilinear KP-reduction method for rogue waves is a new judicious choice of differential operators in the procedure, which drastically simplifies the dimension reduction calculation as well as the analytical expressions of rogue wave solutions.