Vandermonde-like determinants’ representations of Darboux transformations and explicit solutions for the modified Kadomtsev-Petviashvili equation

Recently, the (2+1)-dimensional modified Kadomtsev-Petviashvili (mKP) equation was decomposed into two known (1+1)-dimensional soliton equations by Dai and Geng [H.H. Dai, X.G. Geng, J. Math. Phys. 41 (2000) 7501]. In the present paper, a systematic and simple method is proposed for constructing three kinds of explicit N-fold Darboux transformations and their Vandermonde-like determinants’ representations of the two known (1+1)-dimensional soliton equations based on their Lax pairs. As an application of the Darboux transformations, three explicit multi-soliton solutions of the two (1+1)-dimensional soliton equations are obtained; in particular six new explicit soliton solutions of the (2+1)-dimensional mKP equation are presented by using the decomposition. The explicit formulas of all the soliton solutions are also expressed by Vandermonde-like determinants which are remarkably compact and transparent.     

The ASBM-SA turbulence closure: Taking advantage of structure-based modeling in current engineering CFD codes

Structure-based turbulence models (SBM) carry information about the turbulence structure that is needed for the prediction of complex non-equilibrium flows. SBM have been successfully used to predict a number of canonical flows, yet their adoption rate in engineering practice has been relatively low, mainly because of their departure from standard closure formulations, which hinders easy implementation in existing codes. Here, we demonstrate the coupling between the Algebraic Structure-Based Model (ASBM) and the one-equation Spalart–Allmaras (SA) model, which provides an easy route to bringing structure information in engineering turbulence closures. As the ASBM requires correct predictions of two turbulence scales, which are not taken into account in the SA model, Bradshaw relations and numerical optimizations are used to provide the turbulent kinetic energy and dissipation rate. Attention is paid to the robustness and accuracy of the hybrid model, showing encouraging results for a number of simple test cases. An ASBM module in Fortran-90 is provided along with the present paper in order to facilitate the testing of the model by interested readers.     

Explicit finite difference simulations for accelerating cross diffusion flow over a perpendicular shield with parabolic motion

The heat transfer phenomenon subject to thermos-diffusion effects convey important applications in the heating processes, extrusion systems, chemical processes and various engineering systems. The objective of current work is to observe the contribution of Soret and Dufour effects in oscillating shield for cross diffusion flow. The perpendicular shield with oscillating motion induced the flow. The magnitude of oscillations is assumed to be small so that laminar flow due to oscillating shield has been resulted. The motivations for addressing the thermos-diffusion phenomenon due to oscillating of shield are due to applications in oscillatory pumps, moving surface, metal detectors, power systems etc. The dimensionless problem is obtained via introducing the appropriate set of variables. The numerical outcomes are suggested by using the most interesting explicit finite difference scheme. The physical illustration for flow parameters is presented. Moreover, the aspect of physical quantities involving the flow are graphically reported.     

四维抛物型方程的高精度分支稳定的显式差分格式

对四维抛物型方程构造了一个高精度显格式,格式的稳定性条件为r=Δt/Δx2=△t/Δy2=△t/△z2=Δt/Δw2<1/2,截断误差阶达到O(Δt2 Δx4),通过数值实验,表明本文理论分析的正确性和文中格式较同类格式的优越性.     

Schläfli modular equations for generalized Weber functions

Sets of appropriately normalized eta quotients, that we call level n Weber functions, are defined, and certain identities generalizing Weber function identities are proved for these functions. Schläfli type modular equations are explicitly obtained for Generalized Weber Functions associated with a Fricke group Γ⁰(n)⁺, for n=2,3,5,7,11,13 and 17.     

解模守恒微分方程的显式平方守恒格式

对具有模守恒的微分方程,经典的显式Runge—Kutta方法和线性多步方法不能保微分方程的模守恒特性．我们利用李群算法和Cayley变换构造了高阶显式平方守恒格式,应用到模守恒的微分方程如Euler方程,Landau—Lifshitz方程,并且与相同阶的显式Runge—Kutta方法在保模守恒和精度方面进行了比较,数值结果表明用李群算法构造的新的显式平方守恒格式能保微分方程模守恒的特性且它和相应Runge—Kutta方法有相同的精度．     

Rayleigh waves in anisotropic porous media and the polarization vector method

In this paper, the propagation of Rayleigh waves in orthotropic non-viscous fluid-saturated porous half-spaces with sealed surface-pores and with impervious surface is investigated. The main aim of the investigation is to derive explicit secular equations and based on them to examine the effect of the material parameters and the boundary conditions on the propagation of Rayleigh waves. By employing the method of polarization vector the explicit secular equations have been derived. These equations recover the ones corresponding to Rayleigh waves propagating in purely elastic half-spaces. It is shown from numerical examples that the Rayleigh wave velocity depends strongly on the porosity, the elastic constants, the anisotropy, the boundary conditions and it differs considerably from the one corresponding to purely elastic half-spaces. Remarkably, in the fluid saturated porous half-spaces, Rayleigh waves may travel with a larger velocity than that of the shear wave, a fact that is impossible for the purely elastic half-spaces.     

Hopf bifurcation of the third-order Hénon system based on an explicit criterion

In this paper, Hopf bifurcation of the third-order Hénon system is studied via a simple explicit criterion, which is derived from the Schur–Cohn Criterion. Moreover stability of Hopf bifurcation is also investigated by using the normal form method and center manifold theory for the discrete time system developed by Kuznetsov. Test results containing simulations and circuit measurement are shown to demonstrate that the criterion is correct and feasible.     

Vector measure as controls for explicit nonlinear impulsive system of fed-batch culture

In this paper, we consider an optimal control problem of microbial fermentation process in which glycerol is converted to 1,3-propanediol by Klebsiella pneumoniae in fed-batch culture. During the period of reaction, the variation of pH value is monitored to determine glycerol replenishment quantity, guaranteeing that microorganism can always keep growing fast under enough nutrition. Every time pH value is lower than seven, the quantity of glycerol added is such that pH value returns seven again. Glycerol is poured into reactor at discrete time instant and the quantity is controllable. The problem is to determine for each discrete time instant the glycerol quantity to add and maximize the final concentration of 1,3-propanediol. We present a controlled explicit nonlinear impulsive dynamical system of fed-batch culture with state independent vector measures as controls and study the existence, uniqueness, boundedness, continuous dependence and Gâteaux differentiability of its solution with respect to controls. We then propose a multiple objective programming model and demonstrate the regularity of cost functionals and weak compactness of admissible control set. Finally we discuss the existence of optimal control and implement a hybrid particle swarm optimization algorithm to solve the model optimally. Computational results are presented on a numerical example.