A stable and conservative high order multi-block method for the compressible Navier–Stokes equations

A stable and conservative high order multi-block method for the time-dependent compressible Navier–Stokes equations has been developed. Stability and conservation are proved using summation-by-parts operators, weak interface conditions and the energy method. This development makes it possible to exploit the efficiency of the high order finite difference method for non-trivial geometries. The computational results corroborate the theoretical analysis.     

Analytical travelling wave solutions and parameter analysis for the (2+1)-dimensional Davey–Stewartson-type equations

By using dynamical system method, this paper considers the (2+1)-dimensional Davey–Stewartson-type equations. The analytical parametric representations of solitary wave solutions, periodic wave solutions as well as unbounded wave solutions are obtained under different parameter conditions. A few diagrams corresponding to certain solutions illustrate some dynamical properties of the equations.     

Positon and hybrid solutions for the(2+1)-dimensional complex modified Korteweg–de Vries equations

Solving nonlinear partial differential equations have attracted intensive attention in the past few decades. In this paper, the Darboux transformation method is used to derive several positon and hybrid solutions for the(2+1)-dimensional complex modified Korteweg–de Vries equations. Based on the zero seed solution, the positon solution and the hybrid solutions of positon and soliton are constructed. The composition of positons is studied, showing that multi-positons of(2+1)-dimensional equations...     

The Fe1−xCoxS system (x < 0.25); transition and the high temperature phase

The α-transition in Fe_1−xCo_xS materials is examined based on electrical, magnetic and structural properties. A decrease of the transition temperature is observed with increasing alloying rate up to x_co = 0.24, where the transition is completely cancelled. Beyond this critical concentration the materials exhibit a high-temperature FeS type behavior.The existence of hk0 reflections, with h and/or k = 2n + 1 in the cobalt-stabilized superstructure, the high-temperature phase of FeS, establishes that the structure is hexagonal, space group P6₃mc, with two independent metal positions, in contrast to the previously proposed tri-twinned MnP type model.The physical and structural results are discussed within the framework of conductivity by polarons. The α-transition occurs when the polaron concentration, thermally created or introduced by impurities, reaches the critical value x_p ≅ 0.08. The driving mechanism for the α-transition is a reduction of the polaron dissociation energy by a temperature increase or by cobalt alloying.     

Relationship between (2 + 1) and (3 + 1)-Friedmann–Robertson–Walker cosmologies; linear,non-linear,and polytropic state equations

It is shown that Friedmann–Robertson–Walker (FRW) cosmological models coupled to a single scalar field and to a perfect fluid fitting a wide class of matter perfect fluid state equations, determined in (3+1) dimensional gravity can be related to their (2+1) cosmological counterparts, and vice-versa, by using simple algebraic rules relating gravitational constants, state parameters, perfect fluid and scalar field characteristics. It should be pointed out that the demonstration of these relations for the scalar fields and potentials does not require the fulfilment of any state equation for the scalar field energy density and pressure. As far as to the perfect fluid is concerned, one has to demand the fulfilment of state equations of the form p+ = f(). If the considered cosmologies contain the inflation field alone, then any (3+1) scalar field cosmology possesses a (2+1) counterpart, and vice-versa. Various families of solutions are derived, and we exhibited their correspondence; for instance, solutions for pure matter perfect fluids and single scalar field fulfilling linear state equations, solutions for scalar fields coupled to matter perfect fluids, a general class of solutions for scalar fields subjected to a state equation of the form p + = are reported, in particular Barrow–Saich, and Barrow–Burd–Lancaster–Madsen solutions are exhibited explicitly, and finally perfect fluid solutions for polytropic state equations are given.     

Room temperature precipitation in quenched Al–Cu–Mg alloys: a model for the reaction kinetics and yield strength development

The microstructural evolution during low temperature ageing of two commercial purity alloys (Al–1.2Cu–1.2Mg–0.2Mn and Al–1.9Cu–1.6Mg–0.2Mn?at.%) was investigated. The initial stage of hardening in these alloys is very rapid, with the alloys nearly doubling in hardness during 20?h ageing at room temperature. The microstructural evolution during this stage of hardening was investigated using differential scanning calorimetry (DSC), isothermal calorimetry and three–dimensional atom probe analysis (3DAP). It is found that, during the hardening, a substantial exothermic heat evolution occurs and that the only microstructural change involves the formation of Cu–Mg co–clusters. The kinetics of cluster formation is analysed and the magnitude of the hardening is discussed on the basis of a model incorporating solid solution hardening and modulus hardening originating from the difference in modulus between Al and clusters.     

Iterative Construction of U q (s ℓ(n+1)) Representations and Lax Matrix Factorisation

The iterative construction of a generic representation of g?(n + 1) or of the trigonomentric deformation of its enveloping algebra is conveniently formulated in terms of Lax matrices. The Lax matrix of the constructed representation factorises into parts determined by the Lax matrix of a generic representation of the algebra with reduced rank and others appearing in the factorised expression of the Lax matrix of the special Jordan–Schwinger representation.     

Spatiotemporal Rogue Waves for the Variable-Coefficient (3+1)-Dimensional Nonlinear Schrdinger Equation

With the help of the similarity transformation connected the variable-coefficient (3+1)-dimensional nonlinear Schrdinger equation with the standard nonlinear Schrdinger equation, we firstly obtain first-order and second-order rogue wave solutions. Then, we investigate the controllable behaviors of these rogue waves in the hyperbolic dispersion decreasing profile. Our results indicate that the integral relation between the accumulated time T and the real time t is the basis to realize the control and manipulation of propagation behaviors of rogue waves, such as sustainment and restraint. We can modulate the value T 0 to achieve the sustained and restrained spatiotemporal rogue waves. Moreover, the controllability for position of sustainment and restraint for spatiotemporal rogue waves can also be realized by setting different values of X 0 .     

Nonlocal symmetries,consistent Riccati expansion integrability,and their applications of the (2+1)-dimensional Broer–Kaup–Kupershmidt system

For the (2+1)-dimensional Broer–Kaup–Kupershmidt(BKK) system, the nonlocal symmetries related to the Schwarzian variable and the corresponding transformation group are found. Moreover, the integrability of the BKK system in the sense of having a consistent Riccati expansion(CRE) is investigated. The interaction solutions between soliton and cnoidal periodic wave are explicitly studied.     

Adjoint-based error estimation and adaptive mesh refinement for the RANS and k–ω turbulence model equations

In this article we present the extension of the a posteriori error estimation and goal-oriented mesh refinement approach from laminar to turbulent flows, which are governed by the Reynolds-averaged Navier–Stokes and k–ω turbulence model (RANS-kω) equations. In particular, we consider a discontinuous Galerkin discretization of the RANS-kω equations and use it within an adjoint-based error estimation and adaptive mesh refinement algorithm that targets the reduction of the discretization error in single as well as in multiple aerodynamic force coefficients. The accuracy of the error estimation and the performance of the goal-oriented mesh refinement algorithm is demonstrated for various test cases, including a two-dimensional turbulent flow around a three-element high lift configuration and a three-dimensional turbulent flow around a wing-body configuration.     

A Study of the Behavioral Controllability for Superposed Kuznetsov–Ma Soliton of the (3+1)-Dimensional Generalized Nonlinear Schr?dinger Equation

The(3+1)-dimensional variable-coefficient nonlinear Schr?dinger equation with linear and parabolic traps is studied, and an exact Kuznetsov–Ma soliton solution in certain parameter conditions is derived. These precise expressions indicate that diffraction and chirp factors influence phase, center and widths, while the gain/loss parameter only affects peaks. By adjusting the relation between the maximum accumulated time Tm and the accumulated time T0 based on maximum amplitude of Kuznetsov–Ma soliton, postpone, maintenance and restraint of superposed Kuznetsov–Ma solitons are investigated.     

Complex solutions and novel complex wave localized excitations for the(2+1)-dimensional Boiti–Leon–Pempinelli system

With the help of the symbolic computation system Maple, the Riccati equation mapping approach and a linear variable separation approach, a new family of complex solutions for the （2＋ 1）-dimensional Boiti-Leon-Pempinelli system （BLP） is derived. Based on the derived solitary wave solution, some novel complex wave localized excitations are obtained.     

Shape-changed propagations and interactions for the (3+1)-dimensional generalized Kadomtsev–Petviashvili equation in fluids

In this article, we consider the(3+1)-dimensional generalized Kadomtsev–Petviashvili(GKP)equation in fluids. We show that a variety of nonlinear localized waves can be produced by the breath wave of the GKP model, such as the(oscillating-) W-and M-shaped waves, rational W-shaped waves, multi-peak solitary waves,(quasi-) Bell-shaped and W-shaped waves and(quasi-) periodic waves. Based on the characteristic line analysis and nonlinear superposition principle, we give the transition conditions analytically. We find the interesting dynamic behavior of the converted nonlinear waves, which is known as the time-varying feature. We further offer explanations for such phenomenon. We then discuss the classification of the converted solutions. We finally investigate the interactions of the converted waves including the semi-elastic collision, perfectly elastic collision, inelastic collision and one-off collision. And the mechanisms of the collisions are analyzed in detail. The results could enrich the dynamic features of the high-dimensional nonlinear waves in fluids.     

Kink degeneracy and rogue potential solution for the (3+1)-dimensional B-type Kadomtsev–Petviashvili equation

In this paper, we obtained the exact breather-type kink soliton and breather-type periodic soliton solutions for the (3 + 1)-dimensional B-type Kadomtsev–Petviashvili (BKP) equation using the extended homoclinic test technique. Some new nonlinear phenomena, such as kink and periodic degeneracies, are investigated. Using the homoclinic breather limit method, some new rational breather solutions are found as well. Meanwhile, we also obtained the rational potential solution which is found to be just a rogue wave. These results enrich the variety of the dynamics of higher-dimensional nonlinear wave field.     

Bound states of the Klein－Gordon and Dirac equations for potential V(r)=Ar-2-Br-1

The exact normalized bound-state wavefunctions and energy equations of Klein－Gordon and Dirac equations are given with equal scalar and vector potentials s(r)=v(r)=V(r)/2=(Ar^-2-Br^-1)/2.     

Application of the (G^{\prime}/G)-expansion method for the Burgers, Burgers–Huxley and modified Burgers–KdV equations

In this work, we present travelling wave solutions for the Burgers, Burgers–Huxley and modified Burgers–KdV equations. The (G′/G)-expansion method is used to determine travelling wave solutions of these sets of equations. The travelling wave solutions are expressed by the hyperbolic functions, the trigonometric functions and the rational functions. It is shown that the proposed method is direct, effective and can be used for many other nonlinear evolution equations in mathematical physics.