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In this study, we present the rigorous theory of the robust inverse scattering method for the discrete high-order nonlinear Schrödinger (HNLS) equation with a nonzero boundary condition (NZBC). Using the direct scattering problem, we deduce the analyticity, symmetries, and asymptotic behaviors of the Jost solutions and scattering matrix. We also formulate the inverse scattering problem using the matrix Riemann–Hilbert problem (RHP). Furthermore, utilizing the loop group theory, we construct the multi-fold Darboux transformation (DT) within the framework of the robust inverse scattering transform. Additionally, we develop the corresponding Bäcklund transformation (BT) to obtain the multi-fold lattice soliton solutions. To derive the high-order rational solutions, we further construct the high-order DT. Finally, we theoretically and graphically analyze these solutions, which exhibit lattice breather waves, W-shape lattice solitons, high-order lattice rogue waves (RW), and their interactions.

An understanding of dendrite growth is required in order to improve the properties of castings. For this reason, cellular automaton−finite difference (CA−FD) method was used to investigate the dendrite growth during directional solidification (DS) process. The solute diffusion model combined with macro temperature field model was established for predicting the dendrite growth behavior. Model validation was performed by the DS experiment, and the cooling curves and grain structures obtained by the experiment presented a reasonable agreement with the simulation results. The competitive growth of dendrites was also simulated by the proposed model, and the competitive behavior of dendrites with different misalignment angles was also discussed in detail. Subsequently, 3D dendrites growth was also investigated by experiment and simulation, and both were in good accordance. The influence on dendrites growth of initial nucleus was investigated by three simulation cases, and the results showed that the initial nuclei just had an effect on the initial growth stage of columnar dendrites, but had little influence on the final dendritic morphology and the primary dendrite arm spacing.

We consider the generalised dispersive modified Benjamin–Bona–Mahony equation, which describes an approximation status for long surface wave existed in the non-linear dispersive media. By employing the truncated Painlevé expansion method, we derive its non-local symmetry and Bäcklund transformation. The non-local symmetry is localised by a new variable, which provides the corresponding non-local symmetry group and similarity reductions. Moreover, a direct method can be provided to construct a kind of finite symmetry transformation via the classic Lie point symmetry of the normal prolonged system. Finally, we find that the equation is a consistent Riccati expansion solvable system. With the help of the Jacobi elliptic function, we get its interaction solutions between solitary waves and cnoidal periodic waves.

Isolation of single circulating tumor cells (CTCs) from patients is a very challenging technique that may promote the process of individualized antitumor therapies. However, there exist few systems capable of highly efficient capture and release of single CTCs with high viability for downstream analysis and culture. Herein, we designed a near-infrared (NIR) light-responsive substrate for highly efficient immunocapture and biocompatible site-release of CTCs by a combination of the photothermal effect of gold nanorods (GNRs) and a thermoresponsive hydrogel. The substrate was fabricated by imprinting target cancer cells on a GNR-pre-embedded gelatin hydrogel. Micro/nanostructures generated by cell imprinting produce artificial receptors for cancer cells to improve capture efficiency. Temperature-responsive gelatin dissolves rapidly at 37 °C; this allows bulk recovery of captured CTCs at physiological temperature or site-specific release of single CTCs by NIR-mediated photothermal activation of embedded GNRs. Furthermore, the system has been applied to capture, individually release, and genetically analyze CTCs from the whole blood of cancer patients. The multifunctional NIR-responsive platform demonstrates excellent performance in capture and site-release of CTCs with high viability, which provides a robust and versatile means toward individualized antitumor therapies and also shows promising potential for dynamically manipulating cell–substrate interactions in vitro.

As a new method, liquid-metal cooling (LMC) process is used in manufacturing industrial gas turbines (IGT) blades. Numerical simulation is an effective way to investigate the grain’s growth and morphology, and optimize the process. In this paper, mathematical models for heat dynamic radiation and convection boundary of LMC process is established to simulate the temperature fields. Cellular Automaton (CA) method and KGT growth model are used to describe the nucleation and growth. Simulation results and experimental results are compared. The mushy zone and microstructure evolution are studied in detail. This study indicates that simulation and experimental results agree very well with each other. The withdrawal rate has an important influence on the shape of mushy zone and growth rate of the grain directly. A concave mushy zone is formed and the grain tends to convergent under an excessive high of withdrawal rate. But, the mushy zone has a convex shape and the grain is divergent under a smaller withdrawal rate. A variation withdrawal rate (from 2mm/min to 9mm/min) is found to obtain smooth mushy zone, which improves the parallelism of grain and produces high quality IGT blades.