Decomposition solutions and Bäcklund transformations of the B-type and C-type Kadomtsev-Petviashvili equations

This paper introduces a modified formal variable separation approach,showcasing a systematic and notably straightforward methodology for analyzing the B-type Kadomtsev-Petviashvili(BKP)equation.Through the application of this approach,we successfully ascertain decomposition solutions,Bäcklund transformations,the Lax pair,and the linear superposition solution associated with the aforementioned equation.Furthermore,we expand the utilization of this technique to the C-type Kadomtsev-Petviashvili(CKP)equation,leading to the derivation of decomposition solutions,Bäcklund transformations,and the Lax pair specific to this equation.The results obtained not only underscore the efficacy of the proposed approach,but also highlight its potential in offering a profound comprehension of integrable behaviors in nonlinear systems.Moreover,this approach demonstrates an efficient framework for establishing interrelations between diverse systems.

In situ regulating intimately connected heterostructure by decomposition of solid solution oxides toward high-efficient water oxidation

Heterogeneous interfaces produced by interdomain interactions on a nanoscale performs a crucial role in boosting the properties of an electrocatalyst toward oxygen evolution reaction(OER)process.Herein,a series of dual-phase electrodes with intimately connected heterointerfaces are prepared by in situ decomposing solid solution oxide of Ni_(x)Co_(y)Fe_(100-x-y)O,which grew on Ni foam massively via an ultrafast combustion approach.Particularly,with high-reaction kinetics caused by the reduction treatment at 450℃,the less electronegative Fe and Co are more oxyphilic than Ni,which facilitated their co-exsolution and formation of CoFe_2O_4/NiO oxide with enriched oxygen vacancies.Benefiting from the nanoporous framework,heterojunction structure,and oxygen defects,the self-supporting electrodes present rapid charge/mass transmission and provide abundant active sites for OER.The optimized sample(R-SNCF4.5)shows low overpotentials of 226 and 324 mV at 10 and100 mA·cm^(-2),a small Tafel slope(46.7 mV·dec^(-1)),and excellent stability.The assembled R-SNCF4.5//Pt/C/NF electrolyzer demonstrates continuous electrolysis over 50 h at a current density of 10 mA·cm^(-2),under 1.51 V.Density functional theory(DFT)calculations verify that the strong electronic modulation plays a critical part in the CoFe_2O_4/NiO hybrid by lowering the energy barriers for the ratedetermining steps,and Fe sites are the most active OER sites.

An iterative algorithm for solving ill-conditioned linear least squares problems

Linear Least Squares（LLS） problems are particularly difficult to solve because they are frequently ill-conditioned, and involve large quantities of data. Ill-conditioned LLS problems are commonly seen in mathematics and geosciences, where regularization algorithms are employed to seek optimal solutions. For many problems, even with the use of regularization algorithms it may be impossible to obtain an accurate solution. Riley and Golub suggested an iterative scheme for solving LLS problems. For the early iteration algorithm, it is difficult to improve the well-conditioned perturbed matrix and accelerate the convergence at the same time. Aiming at this problem, self-adaptive iteration algorithm（SAIA） is proposed in this paper for solving severe ill-conditioned LLS problems. The algorithm is different from other popular algorithms proposed in recent references. It avoids matrix inverse by using Cholesky decomposition, and tunes the perturbation parameter according to the rate of residual error decline in the iterative process. Example shows that the algorithm can greatly reduce iteration times, accelerate the convergence,and also greatly enhance the computation accuracy.

INFLUENCE OF Fe^(3+) AND Fe^(2+) ON THE STRUCTURE OF C_3S LATTICE

Analytical reagents have been used for preparation of pure C_3S and its solid solutions containing Fe_2O_3 or FeO. The data from M(?)ssbauer Spectra of samples suggest that 4Si^(4+) are replaced by 4Fe^(3+), 1Ca^(2+) by 1Fe^(3+), and 1Fe^(3+) occupies one octahedron hole; Fe^(2+) mainly replaced Ca^(2+). Spectroscopic Study has indicated there may be extream distortion and higher orientation disorder in (SiO_4) tetrahedron, the reduction of lattice order and translation symmetry by results of ionic replacements of Fe^(3+) or Fe^(2+) in C_3S crystal lattice.

Solutions of the Camassa-Holm Equation Near the Soliton

In this paper,solutions of the Camassa-Holm equation near the soliton Q is decomposed by pseudoconformal transformation as follows:λ^(1/2)(t)u(t,λ(t)y+x(t))=Q(y)+ε(t,y),and the estimation formula with respect toε(t,y)is obtained:|ε(t,y)|≤Ca_(3)Te^(-θ)|y|+|λ^(1/2)(t)ε0|.For the CH equation,we prove that the solution of the Cauchy problem and the soliton Q is sufficiently close as y→∞,and the approximation degree of the solution and Q is the same as that of initial data and Q,besides the energy distribution ofεis consistent with the distribution of the soliton Q in H^(2).

PARAMETER-UNIFORM FINITE DIFFERENCE SCHEME FOR A SYSTEM OF COUPLED SINGULARLY PERTURBED CONVECTION-DIFFUSION EQUATIONS

A coupled system of singularly perturbed convection-diffusion equations is considered. The leading term of each equation is multiplied by a small positive parameter, but these parameters may have different magnitudes. The solutions to the system have boundary layers that overlap and interact. The structure of these layers is analyzed, and this leads to the construction of a piecewise-uniform mesh that is a variant of the usual Shishkin mesh. On this mesh an upwind difference scheme is proved to be almost first- order accurate, uniformly in both small parameters. We present the results of numerical experiments to confirm our theoretical results.

ROBUSTNESS OF AN UPWIND FINITE DIFFERENCE SCHEME FOR SEMILINEAR CONVECTION-DIFFUSION PROBLEMS WITH BOUNDARY TURNING POINTS

We consider a singularly perturbed semilinear convection-diffusion problem with a boundary layer of attractive turning-point type. It is shown that its solution can be decomposed into a regular solution component and a layer component. This decomposition is used to analyse the convergence of an upwinded finite difference scheme on Shishkin meshes.

Nanostructures formation in ferroelectrics in the process of phase transformation

Received 26 June 2014;Revised 13 October 2014;Accepted 20 October 2014;Published 12 November 2014 Inhomogeneous states caused by the coexistence of the ferroelectric(FE)and antiferroelectric(AFE)phases in lead–zirconate–titanate based solid solutions have been investigated.It has been found that the domains of the FE and AFE phases with sizes of the order of 20 nm to 30 nm coexist in the bulk of the samples due to a small difference in the free energies of these phases.The coherent character of the interphase boundaries(IPBs)leads to the concentration of the elastic stresses along these boundaries.These elastic stresses cause the local decomposition of the solid solution and formation of segregates near the IPBS due to the condition that equivalent positions of the crystal lattice are occupied by the ions with different sizes.The sizes of the segregates formed in this way are of the order 8 nm to 15 nm.Some physical effects caused by the presence of these segregate nanostructures are analyzed and discussed.

Effects caused by antiferroelectric nanodomains in PZT-based coarse-grained ceramics with compositions from the morphotropic boundary region

Presented results demonstrate importance of taking into account such a phenomenon as the solid solution decomposition at the boundaries separating coexisting phases in lead zirconate-titanate-based solid solutions with compositions belonging to the morphotropic boundary region of the"temperature–composition"phase diagram.It is shown that in the local decomposition of solid solutions in the vicinity of the boundaries separating the tetragonal and rhombohedral phases in lead zirconate-titanate-based solid solutions lead to the changes of the solid solution's chemical composition and to the formation of segregates.It is also shown that the proper thermoelectric treatment of samples containing these segregates can give substantially higher values of piezoelectric parameters in the lead zirconate-titanate-based compounds.