New Exact Periodic Solitary-Wave Solutions for New (2+1)-Dimensional KdV Equation

The extended homoclinic test function method is a kind of classic, efficient and well-developed method to solve nonlinear evolution equations. In this paper, with the help of this approach, we obtain new exact solutions （including kinky periodic solitary-wave solutions, periodic soliton solutions, and cross kink-wave solutions） for the new （2＋1）-dimensional KdV equation. These results enrich the variety of the dynamics of higher-dimensionai nonlinear wave field.

Exact Periodic and Solitary-Wave Solutions of Multi-component Modified Korteweg-de Vries Equations

The present work extends the search of Jacobi elliptic function solutions for the multi-component modified Korteweg-de Vries equations. When the modulum m →1, those periodic solutions degenerate as the corresponding solitary wave and shock wave ones. Especially, exact solutions for the three-component system are presented in detail and graphically.

Nonautonomous solitary-wave solutions of the generalized nonautonomous cubic-quintic nonlinear Schrdinger equation with time-and space-modulated coefficients

A class of analytical solitary-wave solutions to the generalized nonautonomous cubic–quintic nonlinear Schrdinger equation with time-and space-modulated coefficients and potentials are constructed using the similarity transformation technique. Constraints for the dispersion coefficient, the cubic and quintic nonlinearities, the external potential, and the gain （loss） coefficient are presented at the same time. Various shapes of analytical solitary-wave solutions which have important applications of physical interest are studied in detail, such as the solutions in Feshbach resonance management with harmonic potentials, Faraday-type waves in the optical lattice potentials, and localized solutions supported by the Gaussian-shaped nonlinearity. The stability analysis of the solutions is discussed numerically.

Exact Solitary-wave Solutions and Periodic Wave Solutions for Generalized Modified Boussinesq Equation and the Effect of Wave Velocity on Wave Shape

By means of the undetermined assumption method, we obtain some new exact solitary-wave solutions with hyperbolic secant function fractional form and periodic wave solutions with cosine function form for the generalized modified Boussinesq equation. We also discuss the boundedness of these solutions. More over, we study the correlative characteristic of the solitary-wave solutions and the periodic wave solutions along with the travelling wave velocity＇s variation.

SOLITARY-WAVE SOLUTIONS OF GRAVITATIONAL WAVE IN BACKGROUND OF STATIC, CENTRAL SYMMETRIC GRAVITATIONAL FIELD

Ⅰ. INTRODUCTION Early in 1917, Einstein proved that there is a wave solution of gravitational field in vacuum as an approximate solution of weak gravitational field, if these fields satisfy some conditions. This solution corresponds to a plane gravitational wave. When the wave propagates along x-axis, the polarized wave will be determined by 2-rank symmetric tensors in yz-plane. So this is a transversal wave. The approximate conditions of weak field

EXPLICIT SOLITARY-WAVE SOLUTIONS TO GENERALIZED POCHHAMMER-CHREE EQUATIONS

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GLOBAL SOLUTIONS TO 1D COMPRESSIBLE NAVIER-STOKES/ALLEN-CAHN SYSTEM WITH DENSITY-DEPENDENT VISCOSITY AND FREE-BOUNDARY

This paper is concerned with the Navier-Stokes/Allen-Cahn system,which is used to model the dynamics of immiscible two-phase flows.We consider a 1D free boundary problem and assume that the viscosity coefficient depends on the density in the form ofη(ρ)=ρ^(α).The existence of unique global H^(2m)-solutions(m∈N)to the free boundary problem is proven for when 0<α<1/4.Furthermore,we obtain the global C^(∞)-solutions if the initial data is smooth.

Design of low-alloying and high-performance solid solution-strengthened copper alloys with element substitution for sustainable development

Solid solution-strengthened copper alloys have the advantages of a simple composition and manufacturing process,high mechanical and electrical comprehensive performances,and low cost;thus,they are widely used in high-speed rail contact wires,electronic component connectors,and other devices.Overcoming the contradiction between low alloying and high performance is an important challenge in the development of solid solution-strengthened copper alloys.Taking the typical solid solution-strengthened alloy Cu-4Zn-1Sn as the research object,we proposed using the element In to replace Zn and Sn to achieve low alloying in this work.Two new alloys,Cu-1.5Zn-1Sn-0.4In and Cu-1.5Zn-0.9Sn-0.6In,were designed and prepared.The total weight percentage content of alloying elements decreased by 43%and 41%,respectively,while the product of ultimate tensile strength(UTS)and electrical conductivity(EC)of the annealed state increased by 14%and 15%.After cold rolling with a 90%reduction,the UTS of the two new alloys reached 576 and 627MPa,respectively,the EC was 44.9%IACS and 42.0%IACS,and the product of UTS and EC(UTS×EC)was 97%and 99%higher than that of the annealed state alloy.The dislocations proliferated greatly in cold-rolled alloys,and the strengthening effects of dislocations reached 332 and 356 MPa,respectively,which is the main reason for the considerable improvement in mechanical properties.

Localized wave solutions and interactions of the (2+1)-dimensional Hirota-Satsuma-Ito equation

This paper studies the(2+1)-dimensional Hirota-Satsuma-Ito equation.Based on an associated Hirota bilinear form,lump-type solution,two types of interaction solutions,and breather wave solution of the(2+1)-dimensional Hirota-Satsuma-Ito equation are obtained,which are all related to the seed solution of the equation.It is interesting that the rogue wave is aroused by the interaction between one-lump soliton and a pair of resonance stripe solitons,and the fusion and fission phenomena are also found in the interaction between lump solitons and one-stripe soliton.Furthermore,the breather wave solution is also obtained by reducing the two-soliton solutions.The trajectory and period of the one-order breather wave are analyzed.The corresponding dynamical characteristics are demonstrated by the graphs.

PDE Standardization Analysis and Solution of TypicalMechanics Problems

A numerical approach is an effective means of solving boundary value problems(BVPs).This study focuses on physical problems with general partial differential equations(PDEs).It investigates the solution approach through the standard forms of the PDE module in COMSOL.Two typical mechanics problems are exemplified:The deflection of a thin plate,which can be addressed with the dedicated finite element module,and the stress of a pure bending beamthat cannot be tackled.The procedure for the two problems regarding the three standard forms required by the PDE module is detailed.The results were in good agreement with the literature,indicating that the PDE module provides a promising means to solve complex PDEs,especially for those a dedicated finite element module has yet to be developed.

THE LIMITING PROFILE OF SOLUTIONS FOR SEMILINEAR ELLIPTIC SYSTEMS WITH A SHRINKING SELF-FOCUSING CORE

In this paper,we consider the semilinear elliptic equation systems{△u+u=αQ_(n)(x)|u|^(α-2)|v|^(β)u in R^(N),-△v+v=βQ(x)|u|^(α)|v|^(β-2)v in R^(N),where N≥3,α,β>1,α+β<2^(*),2^(*)=2N/N-2 and Q_(n) are bounded given functions whose self-focusing cores{x∈R^(N)|Q_(n)(x)>0} shrink to a set with finitely many points as n→∞.Motivated by the work of Fang and Wang[13],we use variational methods to study the limiting profile of ground state solutions which are concentrated at one point of the set with finitely many points,and we build the localized concentrated bound state solutions for the above equation systems.

Solution approximations for a mathematical model of relativistic electrons with beta derivative

The main aim of this paper is to obtain the exact and semi-analytical solutions of the nonlinear Klein-Fock-Gordon(KFG)equation which is a model of relativistic electrons arising in the laser thermonuclear fusion with beta derivative.For this purpose,both the modified extended tanh-function(mETF)method and the homotopy analysis method(HAM)are used.While applying the mETF the chain rule for beta derivative and complex wave transform are used for obtaining the exact solution.The advantage of this procedure is that discretization or normalization is not required.By applying the mETF,the exact solutions are obtained.Also,by applying the HAM semi-analytical results for the considered equation are acquired.In HAM?curve gives us a chance to find the suitable value of the for the convergence of the solution series.Also,comparative graphical representations are given to show the effectiveness,reliability of the methods.The results show that the m ETF and HAM are reliable and applicable tools for obtaining the solutions of non-linear fractional partial differential equations that involve beta derivative.This study can bring a new perspective for studies on fractional differential equations.On the other hand,it can be said that scientists can apply the considered methods for different mathematical models arising in physics,chemistry,engineering,social sciences and etc.which involves fractional differentiation.Briefly the results may cause a new insight who studies on relativistic electron modelling.

Bifurcations, Analytical and Non-Analytical Traveling Wave Solutions of (2 + 1)-Dimensional Nonlinear Dispersive Boussinesq Equation

For the (2 + 1)-dimensional nonlinear dispersive Boussinesq equation, by using the bifurcation theory of planar dynamical systems to study its corresponding traveling wave system, the bifurcations and phase portraits of the regular system are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of analytical and non-analytical solutions of the singular system are given by using singular traveling wave theory. For certain special cases, some explicit and exact parametric representations of traveling wave solutions are derived such as analytical periodic waves and non-analytical periodic cusp waves. Further, two-dimensional wave plots of analytical periodic solutions and non-analytical periodic cusp wave solutions are drawn to visualize the dynamics of the equation.

Analytic Solutions for Hindered Rotation Quantum Problem

Normalizable analytic solutions of the quantum rotor problem with divergent potential are presented here as solution of the Schrödinger equation. These solutions, unknown to the literature, represent a mathematical advance in the description of physical phenomena described by the second derivative operator associated with a divergent interaction potential and, being analytical, guarantee the optimal interpretation of such phenomena.

Existence of Entropy Solution for Degenerate Parabolic-Hyperbolic Problem Involving p(x)-Laplacian with Neumann Boundary Condition

We consider a strongly non-linear degenerate parabolic-hyperbolic problem with p(x)-Laplacian diffusion flux function. We propose an entropy formulation and prove the existence of an entropy solution.

Malargüe Site Remediation:A Successful Solution for Uranium Mill Tailings

Comisión Nacional de Energía Atómica (CNEA) has the responsibility for restoring uranium mining facilities once the operations have finished.CNEA,within its Environmental Program and in compliance with its legal responsibilities,decides to implement a restoration project for all sites related to the mining and processing of uranium ores.The Malargüe Site is located within the Province of Mendoza in the city of Malargüe.It is the first site to successfully complete its remediation.The activities consist of relocation of tailings to an engineering repository.The tailings management(encapsulation) and rehabilitation of the area was finished in June 2017.The remediation alternative for the ore tailings was selected after conducting comparative studies and submitted the project to the society for consideration.The objective of the encapsulation of the mineral tails is to isolate them from the environment,also proceeding with the decontamination and rehabilitation of the area (landscaping,post-closure monitoring and 20 years monitoring period).Encapsulation consisted of the construction of a containment cell for the mine tailings,to isolate them and prevent pollutants from entering the environment through the transfer routes.To clean the impacted areas,the soil was removed,it was incorporated into the encapsulation,and the filling was carried out with natural soils from the area.Remediation prevents radon transfer to the environment,as ^(222)Ra is an alpha emitter with a half-life of four days,which produces its own radioactive progeny.Radon progeny are solids,and when a ^(222)Ra nucleus emits an alpha particle into the air,the resulting ^(218)Po nucleus,momentarily electrically charged,adheres to any dust particle.Remediation prevents the discharge into the air containing radon and also containing dust particles charged with intensely radioactive radon progeny.The tasks mentioned make it possible to decrease radon emanation,reduce radiological risks to the public and prevent the entry of rainwater into the system.In addition,the containment system prevents the discharge of contaminated liquids into the environment,avoiding contamination of the groundwater.All these activities are according to the concepts of sustainability.

GLOBAL WEAK SOLUTIONS FOR AN ATTRACTION-REPULSION CHEMOTAXIS SYSTEM WITH p-LAPLACIAN DIFFUSION AND LOGISTIC SOURCE

This paper is concerned with the following attraction-repulsion chemotaxis system with p-Laplacian diffusion and logistic source:■The system here is under a homogenous Neumann boundary condition in a bounded domainΩ ■ R^(n)(n≥2),with χ,ξ,α,β,γ,δ,k_(1),k_(2)> 0,p> 2.In addition,the function f is smooth and satisfies that f(s)≤κ-μs~l for all s≥0,with κ ∈ R,μ> 0,l> 1.It is shown that(ⅰ)if l> max{2k_(1),(2k_(1)n)/(2+n)+1/(p-1)},then system possesses a global bounded weak solution and(ⅱ)if k_(2)> max{2k_(1)-1,(2k_(1)n)/(2+n)+(2-p)/(p-1)} with l> 2,then system possesses a global bounded weak solution.

Dancing bubble sonoluminescence in phosphoric acid solution

Sonoluminescence is more distinctly observed in phosphoric and sulfuric acid,which exhibit high viscosity and lower vapor pressures relative to water.Within an 85-wt%phosphoric acid solution saturated with argon(Ar),variations in the light-emitting regimes of bubbles were noted to correspond with increments in the driving acoustic intensity.Specifically,the bubbles were observed to perform a dance-like motion 2 cm below the multi-bubble sonoluminescence(MBSL)cluster,traversing a 25-mm^(2) grid during the camera exposure period.Spectral analysis conducted at the beginning of the experiment showed a gradual attenuation of CN(B^(2)S–X^(2)S)emission concurrent with a strengthening of Ar(4p–4s)atom emission lines.The application of a theoretical temperature model to the spectral data revealed that the internal temperature of the bubbles escalates swiftly upon their implosion.This study is instrumental in advancing the comprehension of the underlying mechanisms of sonoluminescence and in the formulation of a dynamic model for the behavior of the bubbles.

Hamiltonian system for the inhomogeneous plane elasticity of dodecagonal quasicrystal plates and its analytical solutions

A Hamiltonian system is derived for the plane elasticity problem of two-dimensional dodecagonal quasicrystals by introducing the simple state function. By using symplectic elasticity approach, the analytic solutions of the phonon and phason displacements are obtained further for the quasicrystal plates. In addition, the effectiveness of the approach is verified by comparison with the data of the finite integral transformation method.

On entire solutions of some Fermat type differential-difference equations

On one hand,we study the existence of transcendental entire solutions with finite order of the Fermat type difference equations.On the other hand,we also investigate the existence and growth of solutions of nonlinear differential-difference equations.These results extend and improve some previous in[5,14].