Solitary Wave Solutions and Kink Wave Solutions for a Generalized PC Equation

We present in this paper a generalised PC (GPC) equation which includes several known models. The corresponding traveling wave system is derived and we show that the homoclinic orbits of the traveling wave system correspond to the solitary waves of GPC equation, and the heteroclnic orbits correspond to the kink waves. Under some parameter conditions, the existence of above two types of orbits is demonstrated and the explicit expressions of the two solutions are worked out.     

On Fučik Spectra and Periodic Solutions

In this paper, we continue extending the theory of boundary-value problems to ordinary differential equations and inclusions with discontinuous right-hand side. To this end, we construct a new version of the method of shifts along trajectories. We compare the results obtained by the new approach and those obtained by the method of Fuik spectra.     

Solutions and Multiple Solutions for p（x）-Laplacian Equations with Nonlinear Boundary Condition

The authors study the p（x）-Laplacian equations with nonlinear boundary condition. By using the variational method, under appropriate assumptions on the perturbation terms f1 （x, u）, f2（x, u） and h1（x）, h2（x）, such that the associated functional satisfies the ＂mountain pass lemma＂ and ＂fountain theorem＂ respectively, the existence and multiplicity of solutions are obtained. The discussion is based on the theory of variable exponent Lebesgue and Sobolev spaces.     

Lump Solutions,Interaction Solutions and Breather Solutions of Generalized (3+1)-Dimensional KdV Equations北大核心CSCD

Based on the bilinear form of the generalized (3+1)-dimensional KdV equation, the lump solution, the interaction solution and the breather solution of the equation were obtained. The obtained lump solutions were proved to be rationally localized in all directions of the space, then the “fusion” and “fission” phenomena were observed during the interaction between the lump soliton wave and the one-stripe soliton. Finally, the breather solution of the equation was obtained. © 2023 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

Reflected and Doubly Reflected BSDEs for Levy Processes： Solutions and Comparison

In this paper we study reflected and doubly reflected backward stochastic differential equations （BSDEs, for short） driven by Teugels martingales associated with L~vy process satisfying some moment condi- tions and by an independent Brownian motion. For BSDEs with one reflecting barrier, we obtain a comparison theorem using the Tanaka-Meyer formula. For BSDEs with two reflecting barriers, we first prove the existence and uniqueness of the solutions under the Mokobodski＇s condition by using the Snell envelope theory and then we obtain a comparison result.     

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.     

Positive and Nontrivial Solutions for the Urysohn Integral Equation

We establish new criteria for the existence of either positive or nonzero solutions of the Urysohn integral equation. We also discuss the existence of an interval of positive eigenvalues and sufficient conditions for the existence of at least a positive eigenvalue with a nonzero or positive eigenfunction for the Urysohn integral operator. Among others, we employ techniques based on fixed point index theory for compact maps, which are new for this type of equation.     

η%-Superconvergence of Finite Element Solutions and Error Estimators

This paper is a summary of our study on the superconvergence of the finite element solutions and error estimators. We will persent the analysis of %-superconvergence for finite element solutions of the Poisson equation in the interior of meshes of triangles with straight edges, as well as the analysis at the boundary. The %-superconvergence via local averaging will also be presented, and the error estimators are compared in the sense of %-superconvergence.     

Involutive Solutions and Commutator Representations of the Dirac Hierarchy

ConsidertheDiracspectralproblemwherep,qaretwopotentials,Aisaspectralparameter.L*isaninjectivehomomorphism.ThefunctionalgradientVA=(2RR,ri-of)TofeigenvalueAwithrespecttop,qsatisfiesarecalledtheLenard'soperatorpairof(1).Theorem1LetG(1)(x),G(z)(x)betwoarbitarysmoothfunctions,G=(G(1),G(2))".ThenthefollowingoperatorequationwithrespecttoV=V(G),possessestheoperatorsolutionwhereL.'-jisthecommutator;L=L(p,q),K,Jaredefinedby(1)l(4)respectively.ProofSubstitute(6)into(5),directlycalculate.Defin…     

Existence and Uniqueness of Solutions to Random Impulsive Differential Systems

The existence and uniqueness in mean square of solutions to certain random impulsive differentialsystems is discussed in this paper.Cauchy-Schwarz inequality,Lipschtiz condition and techniques in stochasticanalysis are employed in achieve the desired results.     

On Comparison Theorem and Solutions of BSDEs for Levy Processes

In this paper, we consider backward stochastic differential equations driven by a Levy process. A comparison theorem and an existence and uniqueness theorem of BSDEs with non-Lipschitz coefficients are obtained.     

Exact Periodic Solutions to Generalized BBM Equation and Relevant Conclusions

In this paper, we consider generalized BBM equation with nonlinear terms of high order.In the case of p=1/2, p=1 and p=2, the exact periodic solutions to G-BBM equation are obtained by means of proper transformation, which degrades the order of nonlinear terms. And we prove that if p ≠ 1/2,p ≠ 1 or p ≠ 2, G-BBM equation does not exist this kind of periodic solution.     

Existence and Uniqueness of Weak Solutions for Quasilinear Parabolic Systems

Consider the following IBVP for quasilinear parabolic systemsWhereΩis a bounded domain in R~n with a sufficiently smooth boundary(?)Ω,u=     

Least-square Solutions of Inverse Problems for Anti-symmetric and Skew-symmetric Matrices

The least-square solutions of inverse problem for anti-symmetric and skew-symmetric matrices are studied. In addition, the problem of using anti-symmetric and skew-symmetric matrices to construct the optimal approximation to a given matrix is discussed, the necessary and sufficient conditions for the problem are derived, and the expression of the solution is provided. A numerical example is given to show the effectiveness of the proposed method.     

Positive Solutions to Singular Boundary Value Problems with Sign Changing Nonlinearities on the Half-Line via Upper and Lower Solutions

This paper presents a lower and upper solution technique for singular second order boundary value problems on the half line.     

Exact Solutions of Fisher and Generalized Fisher Equations with Variable Coefficients

In this work,we consider a Fisher and generalized Fisher equations with variable coefficients.Usingtruncated Painlevé expansions of these equations,we obtain exact solutions of these equations with a constrainton the coefficients a(t)and b(t).     

Existence and Stability of Positive Solutions for an Elliptic Cooperative System

By discussing the properties of a linear cooperative system, the necessary and sufficient conditions for the existence of positive solutions of an elliptic cooperative system in terms of the principal eigenvalue of the associated linear system are established, and some local stability results for the positive solutions are also obtained.     

Analytic Solutions of Klein-gordon Equation and Generalized Schrdinger Equation

Inmanyapproximationcases,wecansumlotsofphysicalphenomenonsuptoKlein_gor donequationutt- (uxx+uyy) +α2 u +g(uu )u =0 ,(1 )whereg(z)isafunctionofzandu iscojugatecomplexnumberofu .ManyscholarshavebeeninterestedinanalyticsolutionofEq .(1 ) .Sinceitisestablihed .Papers [1 ,2 ]and [3]viewedrespecrtivelyaccuratesolutionandanalyticsolutionofEq .(1 )wheng(z) =βz .Inpa per [4] ,weobtainedaclassofanalyticsolutionofEq .(1 )wheng(z) =βz1 /k,k∈R+ andaclassofanalyticsolutionofgeneralizedSchrodingerequ…