Numerical inverse scattering transform for the derivative nonlinear Schrodinger equation

In this paper, we develop the numerical inverse scattering transform (NIST) for solving the derivative nonlinear Schrodinger (DNLS) equation. The key technique involves formulating a Riemann-Hilbert problem (RHP) that is associated with the initial value problem and solving it numerically. Before solving the RHP, two essential operations need to be carried out. Firstly, high-precision numerical calculations are performed on the scattering data. Secondly, the RHP is deformed using the Deift-Zhou nonlinear steepest descent method. The DNLS equation has a continuous spectrum consisting of the real and imaginary axes and features three saddle points, which introduces complexity not encountered in previous NIST approaches. In our numerical inverse scattering method, we divide the $(x,t)$-plane into three regions and propose specific deformations for each region. These strategies not only help reduce computational costs but also minimize errors in the calculations. Unlike traditional numerical methods, the NIST does not rely on time-stepping to compute the solution. Instead, it directly solves the associated Riemann-Hilbert problem. This unique characteristic of the NIST eliminates convergence issues typically encountered in other numerical approaches and proves to be more effective, especially for long-time simulations

Suppression of blow-up in multi-species Patlak-Keller-Segel-Navier-Stokes system via the Poiseuille flow in a finite channel

In this paper, we consider the multi-species parabolic-elliptic Patlak-Keller-Segel system coupled with the Navier-Stokes equations near the 2-D Poiseuille flow $(\ A(1-y^2), 0\ )$ in a finite channel $\Omega=\mathbb{T}\times\mathbb{I}$ with $\mathbb{I}=(-1,1)$. Furthermore, the Navier-slip boundary condition is imposed on the perturbation of velocity $u$. We show that if the Poiseuille flow is sufficiently strong ($A$ is large enough), the solutions to the system are global in time without any smallness restriction on the initial cell mass

Efficient method for calculating the eigenvalue of the Zakharov-Shabat system

In this paper, a direct method is proposed to calculate the eigenvalue of the Zakharov-Shabat system. The main tools of our method are Chebyshev polynomials and the QR algorithm. After introducing the hyperbolic tangent mapping, the eigenfunctions and potential function defined in the real field can be represented by Chebyshev polynomials. Using Chebyshev nodes, the Zakharov-Shabat eigenvalue problem is transformed into a matrix eigenvalue problem. The matrix eigenvalue problem is solved by the QR algorithm. Our method is used to calculate eigenvalues of the Zakharov-Shabat equation with three potentials, the rationality of our method is verified by comparison with analytical results

A deep learning method for solving high-order nonlinear soliton equation

We propose effective scheme of deep learning method for high-order nonlinear soliton equation and compare the activation function for high-order soliton equation. The neural network approximates the solution of the equation under the conditions of differential operator, initial condition and boundary condition. We apply this method to high-order nonlinear soliton equation, and verify its efficiency by solving the fourth-order Boussinesq equation and the fifth-order Korteweg de Vries equation. The results show that deep learning method can solve the high-order nonlinear soliton equation and reveal the interaction between solitons

Application of Ion Exchange Resin in the Advanced Treatment of Condensate Water

The advanced treatment of condensate water is important for efficient reuse of water resources, especially in confined space. In this work, a novel integrated process of ion exchange resins and activated carbon is proposed to remove various pollutants in condensate water. A fixed bed column of pre-treated basic anion exchange resin, acidic cation exchange resin, mixed ion exchange resins and modified activated carbon was applied to remove ionic pollutants, organic pollutants and adjust the pH value of output water. The effects of the types, amount ratios and the sequence of ion exchange resins were investigated using two types of condensate water. The results showed that the output water of the fixed bed column had an average TOC of 30~70 ppm, conductivity under 5 μS/cm, pH value of 5~8, which could meet the requirements of sanitary water. The saturated adsorption capacities of the basic anion exchange resin and the acidic cation exchange resin were calculated to be 0.87 mol/L and 1.82 mol/L, respectively. Under the actual operating conditions, continuous dynamic test was carried out over a condensate water treatment module consisting of two adsorption columns and four exchange columns to evaluate its real service life

Inverse Identification of Virtual Material Parameters Using Surface Response Methodology

The virtual material model is now widely applied for modeling the dynamical performance of assembled structures since it can effectively represent the complicated contact behavior of joint interfaces despite being relatively simple to create. In this study, a virtual material model is adopted for modeling the dominant physics of a bolted joint subject to a set of pretightening conditions. The unknown virtual material parameters are acquired by an inverse identification procedure that uses the surface response methodology. The greatest advantage of this approach is the ease with which it acquires the joint parameters without taking apart a built-up structure to do special measurements on each separated component. Intricate theoretical calculations can also be avoided when this method is used. This study addresses the responses of virtual material parameters under different pretightening considerations. Predictions based on the identified virtual material parameters are compared with the corresponding results obtained using the analytical method. The correlation between the two sets of results at all preload levels is promising, which indicates the successful identification of the virtual material parameters

Application of Ion Exchange Resin in the Advanced Treatment of Condensate Water

The advanced treatment of condensate water is important for efficient reuse of water resources, especially in confined space. In this work, a novel integrated process of ion exchange resins and activated carbon is proposed to remove various pollutants in condensate water. A fixed bed column of pre-treated basic anion exchange resin, acidic cation exchange resin, mixed ion exchange resins and modified activated carbon was applied to remove ionic pollutants, organic pollutants and adjust the pH value of output water. The effects of the types, amount ratios and the sequence of ion exchange resins were investigated using two types of condensate water. The results showed that the output water of the fixed bed column had an average TOC of 30~70 ppm, conductivity under 5 μS/cm, pH value of 5~8, which could meet the requirements of sanitary water. The saturated adsorption capacities of the basic anion exchange resin and the acidic cation exchange resin were calculated to be 0.87 mol/L and 1.82 mol/L, respectively. Under the actual operating conditions, continuous dynamic test was carried out over a condensate water treatment module consisting of two adsorption columns and four exchange columns to evaluate its real service life

Dietary sodium acetate (SA) improves the growth performance, intestinal health, and carbohydrate metabolism of juvenile common carp (Cyprinus carpio)

This experiment investigated the effects of dietary sodium acetate (SA) on the growth performance, intestinal health, and carbohydrate metabolism of juvenile Cyprinus carpio L. Four experimental diets were evaluated, including diets supplemented with 0.5 %, 1.0 % and 1.5 % SA and a control diet without supplementation. Triplicate repetitions (n = 30) of common carp (initial weight of 17.24 ± 0.40 g) per group were fed the respective diets for 8 weeks. Dietary SA significantly improved final weight, specific growth rate (SGR), weight gain (WG) and condition factor (CF) and a lower feed conversion ratio (FCR) compared to control group (P  0.05). For intestinal health, dietary SA significantly promoted digestive enzyme activities, including amylase, lipase and protease, and increased villus length and muscle thickness. With respect to carbohydrate metabolism, dietary SA significantly decreased blood glucose levels and hepatopancreas glycogen content, downregulated sglt1 and gult2 gene expression in the intestine (P < 0.05), and increased muscle glycogen content in comparison to the control group. In addition, dietary SA increased expression of glucose-6-phosphatase (g6pase) and phosphoenolpyruvate kinase (pepck) in gluconeogenesis, pyruvate kinase (pk) and glucokinase (gk) in glycolysis, and glucose transporter gult2 in the hepatopancreas (P < 0.05) but decreased expression of glycogen phosphorylase (pygl) in glycogenolysis and glycogen synthase (gys) in glycogen synthesis (P < 0.05). Overall, these findings indicate that dietary SA improves growth performance, intestinal health and carbohydrate metabolism, and that SA may potentially be used as a feed additive for common carp. According to the results, dietary 1.0 % SA is recommended for the formulation of common carp feed