We study the Cauchy problem for the integrable nonlocal modified Korteweg–deVries (mKdV) equation with a step-like initial data: q(x,0)=q0(x), where q0(x)=o(1) as x→−∞ and q0(x)=A+o(1), as x→∞, A>0. We construct the solution of the nonlocal mKdV equation via the solution of a 2 × 2 matrix Riemann–Hilbert problem in the complex plane. Further, The explicit form of the one-soliton solution in a special

In this paper, we present some useful necessary and sufficient conditions for the unique solvability of a class of the new generalized absolute value equation (NGAVE) Ax−|Bx|=d with A,B∈Rn×n from the optimization field. Incidentally, some sufficient conditions for the unique solvability of a class of the NGAVE are obtained as well.

We show that the Euler system of gas dynamics in Rd, d=2,3, with positive far field density and arbitrary far field entropy, admits infinitely many steady solutions with compactly supported velocity. The same proof yields a similar result for the incompressible Euler system with variable density. In particular, these are examples of global in time smooth (non-trivial) solutions for the corresponding

This paper deals with the study of combined effects of logarithmic and critical nonlinearities for the following class of fractional p–Kirchhoff equations: M([u]s,pp)(−Δ)psu=λ|u|q−2uln|u|2+|u|ps∗−2uinΩ,u=0inRN∖Ω,where Ω⊂RN is a bounded domain with Lipschitz boundary, N>sp with s∈(0,1), p≥2, ps∗ = Np/(N - ps) is the fractional critical Sobolev exponent, and λ is a positive parameter. The main result

This paper is concerned with the global attractivity of the positive steady state for a time-delayed viral infection model when R0>1. Our study solves the open problem left in a recent work of Yang and Wei (2020) by Lyapunov functional method.

This paper is concerned with the existence and uniqueness of solutions for a nonlinear fractional-order coupled system involving Caputo fractional derivatives of different orders equipped with a new kind of coupled boundary conditions. We transform the given system into an equivalent fixed point problem and solve it by applying the standard fixed point theorems. Examples are constructed for the illustration

In this paper, a meshless method in reproducing kernel space is proposed for solving VOTFA-DE on arbitrary domain. Advantages of the meshless method proposed could avoid effectively difficulties of constructing shape functions using known Mercer kernel and deal with arbitrary domains. And the accuracy is verified by two examples.

By estimating the nonlinear term in an innovative way, we could obtain the optimal regularity criterion for the shear thinning fluids via the velocity field.

In this paper, we discuss the k-convex solution to the following boundary blow-up k-Hessian equation with singular weights: SkD2u=H(x)up,inΩ,u=∞,on∂Ω, where k∈{1,2,…,n}, SkD2u is the k-Hessian operator, Ω is a smooth, bounded and strictly convex domain in Rn (n≥2),p>0.

A Kirchhoff-type parabolic equation with logarithmic nonlinearity is studied in this paper. By employing the potential well theory and some differential inequality techniques, a new blow-up condition, the upper bound of the blow-up time, and the lower bound of the growth rate of blow-up solutions are obtained. The effect of the parameters a,b,θ in the assumption of Kirchhoff function M is simultaneously

A new fully discrete scheme of stochastic space-fractional nonlinear damped wave equations respectively driven by additive and multiplicative noise is presented. Based on the spatial discretization done via a fourth-order central difference scheme, the semi-implicit Crank–Nicolson scheme is developed for the temporal approximation. The trace formulas for the energy are given for both additive noise

In this paper, we consider a stochastic Kawasaki disease model, which is perturbed by both white noise and color noise. The sufficient condition for the existence of a unique ergodic stationary distribution is established by using Khasminskii’s theorem and constructing stochastic Lyapunov functions with regime switching. A brief summary is given at the end of this paper.

A fast, easily implemented and high efficiency algorithm for time fractional Maxwell’s system is constructed. The algorithm is based on recently developed the sum-of-exponentials (SOE) approximation and Finite-Difference Time-Domain (FDTD) method. A particular feature of our proposed algorithm is that it can achieve high efficiency with no loss in accuracy. The computing process of our algorithm in

We construct two rational approximate solutions to the Thomas–Fermi (TF) nonlinear differential equation. These expressions follow from an application of the principle of dynamic consistency. In addition to examining differences in the predicted numerical values of the two approximate solutions, we compare these values with an accurate numerical solution obtained using a fourth-order Runge–Kutta method

In this paper, we perform sensitivity analysis to a stochastic two-prey one-predator model and investigate the stationary probability distributions of its population densities. The semi-relative and logarithmic sensitivity functions are utilized to evaluate the effects of system parameters on the population of each species, and to conduct the uncertainty quantification of the model. By numerically

In this paper, we investigate the beak-shaped rogue waves for a higher-order coupled nonlinear Schrödinger system. Firstly, we construct a 4 × 4 Lax pair and the Nth-order Darboux transformation. Secondly, with the non-zero seed solutions, we derive the Nth-order beak-shaped rogue wave solutions and beak-shaped rogue wave pair solutions. Then, via such solutions, we graphically illustrate the nature

In this paper, we investigate a class of impulsive differential equations with state-dependent delay under analytic semigroup in Banach spaces. The existence and uniqueness of classical solution is obtained under the assumptions that nonlinear function and impulsive map are Lipschitz continuous. On these assumptions, we further study the existence and smoothness of solution manifold of impulsive differential

Nonlinear ordinary differential equations of the new hierarchy are considered. We demonstrate that although the equations of this hierarchy are similar in appearance to the equations of the first Painlevé hierarchy, they have the different Lax pairs. An algorithm for constructing the Lax pairs of equations of a new hierarchy is proposed. This algorithm is illustrated for the first, second, third, and

In this paper, based on our recent work (Zhang and Zhang, 2021), we introduce multiple perturbations in an SIS epidemic model with isolation and varying total population size. We present the threshold Rs of the model. When Rs is less than 1, we prove that the disease will die out. When Rs is greater than 1, we construct an appropriate stochastic Lyapunov function and using the well-known Khasminskii’s

This paper addresses the finite-time stability of generalized neutral fractional systems with time delays. Firstly, the bivariate Mittag-Leffler type matrix function is introduced and some new estimations for the Mittag-Leffler function in terms of the exponential function are deduced. Additionally, based on the ρ-Laplace transform, the properties of the Mittag-Leffler function and the generalized

Based on N-soliton solutions, fusion and fission waves are obtained through a new constraint among parameters. With this new constraint, fusion and fission phenomena can be reduced from N-soliton solutions in most (2+1)-dimensional integrable systems. This paper takes the (2+1)-dimensional fifth-order KdV equation as an example to introduce how to use this constraint to generate these fusion and fission

In this paper, a linearized energy-stable Galerkin scheme is investigated for a semilinear wave equation. Based on the special property of the bilinear element on the rectangular mesh, the superconvergence error estimate in L∞(H1(Ω)) is obtained in terms of a suitable post-processing approach. Finally, a numerical example is presented to support the theoretical analysis.

We consider the 3D ideal MHD equations on T3 with partial dampings. The dampings, which act as partial dissipations, are defined by the fractional Laplacian removing Fourier modes in certain symmetric subsets Kj⊂Z3 (j=1,2). This model is motivated by Elgindi et al. (2017) and Kim and Dubrulle (2002) where the phenomenon of engergy cascade is analyzed for the 2D inviscid fluid flow with partial dampings

This paper is concerned with the long time asymptotic stability of global weak solution for 3D incompressible Navier–Stokes equations with nonlinear damping, namely, the convergence of solutions between Navier–Stokes systems and its corresponding generalized elliptic equation.

In Morosi and Pizzocchero (2015) and previous papers by the same authors, a general smooth setting was proposed for the incompressible Navier–Stokes (NS) Cauchy problem on a torus of any dimension d⩾2, and the a posteriori analysis of its approximate solutions. In this note, using the same setting I propose an elementary proof of the following statement: global existence and time decay of the NS solutions

We study Lp-strong convergence for coupled stochastic differential equations (SDEs) driven by Lévy noise with non-Lipschitz coefficients. Utilizing Khasminkii’s time discretization technique, the Kunita’s first inequality and Bihari’s inequality, we show that the slow solution processes converge strongly in Lp to the solution of the corresponding averaged equation.

The numerical solution of a quasilinear singularly perturbed problem with an integral boundary condition is considered. For discretizing the differential equation, we use the backward Euler formula and for the integral boundary condition the left rectangle formula is constructed on an arbitrary nonuniform mesh. Based on the truncation error analysis and mesh equidistribution principle, a suitable monitor

In this paper, we derive several differential Harnack estimates (also known as Li–Yau–Hamilton type estimates) for nonnegative solutions and positive solutions of the porous medium equation ft=ffxx+fx2+f2, which is a nonlinear parabolic equation. Our derivations rely on an idea related to the parabolic maximum principle. We prove these estimates by different methods. As the applications of the estimates

A class of singularly perturbed fourth order nonlinear differential equation is considered in this article. The existence of solution to boundary value problem (BVP) without singularly perturbed is obtained via nonlinear analysis method and upper and lower solution theory firstly. In addition, based upon some differential inequality techniques, as well as appropriate upper–lower solutions, the existence

This paper considers the structure of basis functions in the bilinear immersed finite element space for two dimensional elliptic interface problems. On a rectangular interface element, each immersed basis function can be decomposed into a standard bilinear basis function and a corresponding bubble function, which provides another perspective on the nature of immersed basis functions. Detailed expressions

Image restoration is an ill-conditioned problem, so the regularization method is often applied to stabilize the solution. In this paper, we consider an additive half-quadratic (HQ) regularized image restoration problem and use the Newton method to solve it. At each Newton iteration step, a structured system of linear equations with symmetric positive definite coefficient matrix is needed to be solved

The existence of a solution is proved for a nonlinear finite element flux-corrected-transport (FEM-FCT) scheme with arbitrary time steps for evolutionary convection–diffusion–reaction equations and transport equations.

We are concerned with numerical approximation of the linear Klein–Gordon equation with time- and space-dependent mass. We propose an asymptotic-numerical approach as a powerful tool in the presence of very high-frequency oscillations. As we shall show, the proposed method achieves high accuracy with a very low computational cost. A numerical example and comparisons are presented to confirm the efficiency

We consider the Zakharov–Kuznetsov-type limit for ion dynamics system in R3 with external magnetic field and derive the 3D ZK-type equation from ion dynamics system by Gardner–Morikawa transforms. By making energy estimate for the remainders system we prove that the solutions of the Euler–Poisson system converge globally to ZK-type equation when ε→0 in 3D.

The oscillatory properties of solutions to the nth order delay differential equations Lny(t)+p(t)y(τ(t))=0,where Ln is a disconjugate strongly noncanonical differential operator, p(t)>0, τ(t)≤t, and limt→∞τ(t)=∞, are studied. The main idea is to show that, unlike Trench original, rather theoretical result, simple closed formulas appearing in a uniquely determined canonical representation of Ln are

In this letter, we propose a hierarchy of new generalized Korteweg–de Vries equations and find that the generalized Korteweg–de Vries equation has a class of pseudo-peakons. The so-called pseudo-peakon looks like a peakon, but it is continuously differentiable everywhere and its second-order derivative goes to infinity at some point. The infinite sequence of conserved quantities of the generalized

Singular perturbation problem cannot be solved without oscillations by the classical finite difference methods on the uniform grid unless sufficient resolution is employed. In this note we use the Gaussian radial basis function (RBF) finite difference method and show that it is possible for the finite difference method to yield non-oscillatory and accurate solutions to the problem for any degree of

The purpose of this article is to establish a kind of stricter Hyers–Ulam stability of second order linear differential equation of Carathéodory type. More explicitly, we prove that if x is an approximate solution satisfying a kind of stricter condition of the differential equation x′′(t)+b1x′(t)+b0x(t)=f(t) without the assumption of continuity of f(t), then there exists an exact solution of the differential

We study necessary conditions for stability of a Numerov-type compact higher-order finite-difference scheme for the 1D homogeneous wave equation in the case of non-uniform spatial meshes. We first show that the uniform in time stability cannot be valid in any spatial norm provided that the complex eigenvalues appear in the associated mesh eigenvalue problem. Moreover, we prove that then the solution

In this paper, we consider a weighted integral equation involving Wolff potentials in Rn u(x)=Wβ,p(|y|auq)(x)and a related differential form −div(|∇u|p−2∇u)=|x|auq(x).where p∈1,2, q>p−1, β>0 and 0≤−a

In this paper, we study existence of stable standing waves for the following Lee–Huang–Yang corrected dipolar Gross–Pitaevskii equation with a partial harmonic confine i∂tψ=−Δψ+(x12+x22)ψ+λ1|ψ|2ψ+λ2(K∗|ψ|2)ψ+λ3|ψ|pψ,(t,x)∈[0,T∗)×R3.When 0

The letter presents a penalized version of the time-discretization local minimization scheme first proposed by Efendiev and Mielke in 2006 to resolve time discontinuities in rate-independent systems with nonconvex energies. In order to penalize inequality constrains enforcing the local minimality, the Moreau–Yosida approximation is employed. We prove the convergence of time-discrete solutions to functions

The wave propagation dynamics of bistable lattice dynamical systems was extensively studied in literatures. It is a challenging problem if the wave profiles are unique up to translation. In this note, we establish a new uniqueness theorem of wave profiles for a 2-D bistable lattice dynamical system with global interaction. The obtained result greatly extends the known results in the literature, which

In this paper we study some key effects of a discontinuous forcing term in a fourth order wave equation on a bounded domain, modeling the adhesion of an elastic beam with a substrate through an elastic-breakable interaction. By using a spectral decomposition method we show that the main effects induced by the nonlinearity at the transition from attached to detached states can be traced in a loss of

We study the effects of two-dimensional chaotic advection on a chemical system characterised by competitive exothermic and endothermic reactions. In previous studies, in which advective flow and reaction processes were assumed to dominate weak diffusive effects, two distinct behaviours were observed in the system. The first, when the stirring is fast and the reaction is slow. In this case, flame quenching

In this paper, two new sufficient conditions for the unique solution of a Sylvester-like absolute value equation AXB+C|X|D=E with A,C∈Rm×m, B,D∈Rn×n and E∈Rm×n are given, where A and B are nonsingular, which are distinct from the published work by Hashemi (2021). When the involved matrices are square, two useful necessary and sufficient conditions for the unique solution of the Sylvester-like absolute

A family of travelling wave solutions to the Fisher–KPP equation with speeds c=±5∕6 can be expressed exactly using Weierstraß elliptic functions. The well-known solution for c=5∕6, which decays to zero in the far-field, is exceptional in the sense that it can be written simply in terms of an exponential function. This solution has the property that the phase-plane trajectory is a heteroclinic orbit

In this paper, we deal with the existence and multiplicity of solutions of Kirchhoff-type problem (0.1)−a+b∫RN|∇u|2dx△u+u=f(x,u),inRN,u∈H1(RN),where N≥3. Under more relaxed assumptions on f, we establish some existence criteria to guarantee that the above problem has at least one or infinitely many nontrival solutions by using the genus properties in critical point theory.

The optical fiber communication system is one of the components of the supporting system in the modern Internet fields. Under investigation in this paper is a coupled fourth-order nonlinear Schrödinger system, which describes the ultrashort optical pluses in a birefringent optical fiber. By virtue of the existing Lax pair, generalized Darboux transformation, two- and three-soliton solutions are derived

This paper presents a pseudopotential lattice Boltzmann model for multicomponent flows involving large viscosity ratios. Firstly, a rigorous Chapman–Enskog analysis is conducted to show that previous models cannot recover the correct governing equation with a single-relaxation-time scheme when the viscosity ratio is not unity. Then, based on this analysis, we established a modified model to obtain

In the present paper, we consider the following discrete Schrödinger equations m(∑k∈Z(|Δuk−1|2+Vk|uk|2))(−Δ2uk−1+Vkuk)−ωuk=fk(uk)k∈Z,where m is a continuous function and V={Vk} is a positive potential, ω∈R, Δuk−1=uk−uk−1 and Δ2=Δ(Δ) is the one dimensional discrete Laplacian operator. Under some suitable assumptions on fk, we prove the existence of nontrivial solutions for this nonlocal problems by

In this paper, a three-species food web stochastic model with generalist predator is proposed and studied. We obtain sufficient criteria for the existence and uniqueness of an ergodic stationary distribution of positive solutions to the system by establishing a series of suitable Lyapunov functions. In a biological viewpoint, the existence of a stationary distribution implies that all species can coexist

In this paper, we prove global existence and moment propagation of weak solutions for the 3-D Vlasov–Poisson system in the presence of a point charge with large initial data, which improves the results in Desvillettes et al. (2015).

Different from existing researches, under non-Lipschitz condition, we establish an averaging principle for a class of fractional neutral stochastic differential equations (FNSDEs) involving delayed impulses without periodic assumptions of coefficients and impulses. To overcome the difficulties brought by the impulsive terms, we develop a new estimating approach, which can be used to improve some recent

In this study, the space–time (ST) generalized finite difference method (GFDM) was combined with Newton’s method to stably and accurately solve two-dimensional unsteady Burgers’ equations. In the coupled ST approach, the time axis is selected as a spatial axis; thus, the temporal derivative in governing equations is treated as a spatial derivative. In general, the GFDM is an optimal meshless collocation

We establish the local existence and uniqueness of weak solutions for fractional pseudo-parabolic equation with singular potential by means of Galërkin method and contraction mapping theorem.

This short note is concerned with Nicholson’s blowflies equation with Dirichlet boundary. For the challenging non-monotone case, we show the exponential convergence rate by the energy method, where the key observation is that the energy integrations for the perturbations on the inner boundary will be disappeared automatically, even the solutions at the boundary are non-zero.

This paper considers a modified Leslie–Gower delayed reaction–diffusion predator–prey model with prey harvesting of Michaelis–Menten type and subject to homogeneous Neumann boundary condition. The global asymptotic stability of the positive constant steady state of the model is analyzed further and an existing global stability result is improved.

We give the global existence results for the isentropic gas dynamics system with large BV initial data, assuming that the initial data are far from vacuum. The distance of the initial data from vacuum may be so large depending on the initial BV bounds. A new scaling framework is introduced, and we obtain the crucial BV bounds estimates of the vanishing viscosity solutions with the help of this framework

In this paper a hybrid difference method is used to approximate a parameterized singular perturbation problem. An improved a posteriori error estimation for the difference scheme on an arbitrary mesh is given. A solution-adaptive algorithm based on the a posteriori error estimation is designed by equidistributing a monitor function. Numerical experiments show that the scheme is second-order uniformly