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This paper explores the reverse space-time mKdV equation through the application of the Riemann–Hilbert problem. Under the zero boundary condition, we derive the Jost solutions, examine their the analytic and symmetry properties alongside those of the scattering matrix, and formulate the corresponding Riemann–Hilbert problem. By assuming that the scattering coefficient has multiple simple zero points and one higher-order zero point, we obtain explicit solutions to the Riemann–Hilbert problem in a reflection-less situation and display two types of formulae for the N-th order solutions of the reverse space-time nonlocal mKdV equation, which correspond to multiple simple poles and one higher-order pole, respectively. As applications, we display two kinds of double-periodic solutions explicitly and graphically. Additionally, we display the conversation laws for the reverse space-time nonlocal mKdV equation. [ABSTRACT FROM AUTHOR]

With a non-vanishing boundary condition, we study the Kaup–Newell (KN) equation (or the derivative nonlinear Schrödinger equation) using the Riemann–Hilbert approach. Our study yields four types of Nth order solutions of the KN equation that corresponding to simple poles on or not on the ρ circle (ρ related to the non-vanishing boundary condition), and higher-order poles on or not on the ρ circle of the Riemann–Hilbert problem (RHP). We make revisions to the usual RHP by introducing an integral factor that ensures the RHP satisfies the normalization condition. This is important because the Jost solutions go to an integral factor rather than the unit matrix when the spectral parameter goes to infinity. To consider the cases of higher-order poles, we study the parallelization conditions between the Jost solutions without assuming that the potential has compact support, and present the generalizations of residue conditions of the RHP, which play crucial roles in solving the RHP with higher-order poles. We provide explicit closed-form formulae for four types of Nth order solutions, display the explicit first-order and double-pole solitons as examples and study their properties in more detail, including amplitude, width, and exciting collisions. [ABSTRACT FROM AUTHOR]

Heavy metal accumulation in agricultural products has become a major concern. Previous studies have focused on the transport of heavy metals from the soil and their accumulation in crops. However, recent studies revealed that wheat leaves, ears, and awns can also transport and accumulate heavy metals. Wheat grains can be influenced by two sources of heavy metals: soil contamination and atmospheric deposition. To comprehend the transport characteristics of heavy metals in soil, atmospheric deposition, and wheat, 37 samples each for wheat rhizosphere soil, wheat roots, stems, leaves, and grains were collected. Fifteen samples of atmospheric dry deposition and atmospheric wet deposition were collected from Linshu County (northern area), China. Based on the test data, the characteristics of heavy metals and their distribution in the study area were analyzed. Migration patterns of heavy metals in crops from different sources were investigated using Pearson correlation and redundancy analysis. Finally, a predictive model for heavy metals in wheat grains was developed using multiple linear regression analysis. Significant disparities in the distribution of heavy metals existed among wheat roots, stems, leaves, and grains. The coefficient of variation of heavy metals in atmospheric deposition was relatively high, indicating discernible spatial patterns influenced by human activities. Notably, a positive correlation was observed between the concentration of heavy metals in wheat grains and atmospheric deposition of Hg, Cd, and Pb. Conversely, Zn and Ni levels in wheat grains were significantly negatively associated with soil Zn, Ni, pH, and OM content. The contribution of heavy metal elements from different sources varied in their impact on the grain's heavy metal content. Specifically, atmospheric deposition was the primary source of Hg and Pb in wheat grains, while Cd, Ni, Cu, and Zn were predominantly derived from soil. Using a multiple linear regression model, we could accurately predict Hg, Pb, Cd, Ni, Zn, and As concentrations in crop grains. This model can facilitate quantitative evaluation of ecological risk of heavy metals accumulation in crops in the study area. [ABSTRACT FROM AUTHOR]