The five-component Manakov system (alias 5-Manakov system) appears in many fields such as nonlinear optics, Bose–Einstein condensates (alias the spin-2 Gross–Pitaevskii equation), and ocean. In this Letter, we investigate the semi-rational vector rogon-soliton and soliton of the 5-Manakov system with mixed zero and non-zero backgrounds by using the modified Darboux transform. The key point is to explicitly

In this paper, we introduce a pseudo-symmetry hypothesis, under which the N-fold binary Darboux transformation for the nth-order (n=−1,0,1,…) Ablowitz-Kaup-Newell-Segur (AKNS) system is presented in a unified form. Through the N-fold (N=1,2,3,…) binary Darboux transformation, several high-order analytical solutions of certain nonlinear evolution equations can be obtained based on simple seed solutions

This paper is concerned with nonlinear Choquard equations with doubly critical growth. We apply the Pohozăev-type identity to overcome loss of compactness caused by the doubly critical nonlinearities and obtain the existence of a ground state solution.

This work considers the chemotaxis model with density-dependent motility ut=Δ(ϕ(v)u),x∈Ω,t>0,vt=Δv−uv,x∈Ω,t>0under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂R2, with the nonnegative initial data (u0,v0)∈[W1,∞(Ω)]2. The motility function ϕ(v) satisfies ϕ(v)∈C3([0,∞)) with ϕ(v)>0 and ϕ′(v)<0 for all v≥0. For all suitably regular initial data, we prove that this system has a

In this article, we propose a new projection-type algorithm for solving ℓ1−ℓ2 minimization. Compared with the proposed method by Feng and Wang (2018), a new stepsize rule is introduced which has allowed the algorithms to work without any prior information about operator norms. Under suitable conditions, the convergence and linear convergence rate of the designed algorithm are established. Several examples

We consider a continuum mechanical model for the migration of multiple cell populations through parts of tissue separated by thin membranes. In this model, cells belonging to different populations may be characterised by different proliferative abilities and mobility, which may vary from part to part of the tissue, as well as by different invasion potentials within the membranes. The original transmission

In this paper, we consider the averaging principle for a class of stochastic differential equations (SDEs) with the nonlinear terms only satisfying the local Lipschitz and monotone conditions. Compared with the classical Lipschitz and linear growth conditions, conditions given in this paper are relatively weak, so the conclusions obtained in this paper are suitable for a wider class of SDEs.

In this paper we improve the cubature rules discussed in Sommariva and Vianello (2021) for the computation of integrals by radial basis functions (RBFs). More precisely, we introduce in the context of meshless cubature a leave-one-out cross validation criterion for the optimization of the RBF shape parameter. This choice allows us to get highly reliable and accurate results for any kind of both infinity

The constrained mock-Chebyshev least squares interpolation is a univariate polynomial interpolation technique exploited to cut-down the Runge phenomenon. It takes advantage of the optimality of the interpolation on the mock-Chebyshev nodes, i.e. the subset of the uniform grid formed by nodes that mimic the behavior of Chebyshev-Lobatto nodes. The other nodes of the grid are not discarded, rather they

We propose a Kansa-radial basis function (RBF) collocation method for the solution of 2D and 3D high order (i.e. of order 2N where N∈N,N≥3) boundary value problems (BVPs) in multiply connected domains. These problems include N boundary conditions (BCs), and thus we need to select N distinct sets of boundary centres. One set is positioned on the physical boundary of the problem while the rest are placed

The paper deals with nonlinear reaction–diffusion systems of two PDEs of a rather general form, which contain arbitrary functions and several delays. Reductions of such non-stationary nonlinear reaction–diffusion systems to simpler systems of stationary PDEs or delayed ODEs are described using a new modification of the functional constraints method. New exact solutions in elementary functions of a

This research focuses on optimal impulsive harvesting strategy of a stochastic Gompertz model with periodic coefficients. By investigating the associated Hamilton function, the explicit expressions of the optimal impulsive effort, the maximum annual yield and the expectation of the optimal population level are provided. The theoretical results are also utilized to investigate the optimal impulsive

In this article, we formulate a determinant representation of the Darboux transformation for the Kulish-Sklyanin (KS) model, which can be viewed as a generalization of the nonlinear Schrödinger equation, and obtain a new compact formula of n-soliton solution for the KS system. As applications of the formula, the abundant dynamics of one-, two- and three-soliton solutions for the case m=2 are discussed

This paper is concerned with Caputo-Hadamard fractional Halanay inequality. An useful fractional derivative inequality regarding on x2 in the Caputo-Hadamard sense is obtained, which makes Halanay inequality more applicable to choose the Lyapunov function and to study the stability of fractional system.

In this paper, we prove the existence, uniqueness and regularity of global solution to the bistable reaction–diffusion equation with degenerate diffusion and nonlocal time-delay. The adopted approach is the Holmgren’s approximation scheme combining the compactness analysis.

A (3+1)-dimensional variable-coefficient partially nonlocal coupled Gross–Pitaevskii equation trapped in a harmonic potential becomes a focus of this paper. A counterpart of this variable-coefficient coupled equation is found as a (2+1)-dimensional constant-coefficient single nonlinear Schrödinger equation via the reduction procedure. By solutions of constant-coefficient single equation via the Hirota

This paper is concerned with the existence and uniqueness of positive solution for tensor equations. Under suitable assumptions, we show that the concerned tensor equations have a unique positive solution. In particular, we show that our main result includes the existing result in literature as its special case. Finally, we illustrate one example to verify the obtained results.

The present study formulates a stochastic predator–prey model with infected prey and regime-switching, and shows that the model has a unique stable non-boundary invariant measure which is ergodic. The theoretical results are also utilized to explore the evolution of red grouse Lagopus lagopus scoticus and fox in Scotland.

Randomized block coordinate descent type methods have been demonstrated to be efficient for solving large-scale optimization problems. Linear convergence to the unique solution is established if the objective function is strongly convex. In this paper we propose a randomized block coordinate descent algorithm for solving the matrix least squares problem minX∈Rm×n‖C − AXB‖F with A∈Rp×m, B∈Rn×q, and

This paper is concerned with a class of reaction–diffusion systems with cross-diffusion which have various applications such as autocatalytic chemical reaction, predator–prey interactions, combustion and so on. By imposing some suitable structure assumptions, we establish the global existence and asymptotic behavior of solutions of the system in a two-dimensional bounded domain with Neumann boundary

In this letter, we study energy-preserving (EP) splitting methods for solving charged-particle dynamics in a normal or strong magnetic field. Two novel EP splitting methods are formulated, and their energy-preserving property and accuracy are analysed. A numerical experiment is carried out to demonstrate the error and energy behaviour of the splitting methods.

The nonlocal Boussinesq equations (NLBEs) are investigated in this work. The general forms of soliton solutions of the equations are firstly derived via Hirota bilinear method. Subsequently, the first- to fourth-order soliton solutions are obtained by taking auxiliary function in the bilinear form. According to the system parameter, we classify the multiple solitons into two types: stripe-like solitons

In this paper, we study the existence of nontrivial solutions to the nonlinear differential equation with derivative term u′′(t)+a(t)u(t)=f(t,u(t),u′(t)),t∈[0,ω],u(0)=u(ω),u′(0)=u′(ω), where a: [0,ω]→R+(R+=[0,+∞)) is a continuous function with a(t)⁄≡0, and f: [0,ω]×R×R→R is continuous and may be sign-changing and unbounded from below. Without making any nonnegative assumption on the nonlinearity, by

We show that a viscous steady water flow of constant non-vanishing vorticity situated below a free surface (assumed to have the shape of a two-dimensional time dependent graph) and above a flat bottom has to be two-dimensional; that is, while satisfying the viscous three-dimensional water wave equations, it turns out that the free surface, the velocity field and the pressure present no variation in

We consider the following Kirchhoff type problem: −ɛ2a+ɛb∫R3|∇u|2Δu+V(x)u=|u|p−2u,x∈R3,where a,b>0, p∈(4,6), V∈C1(R,[0,∞)) and ɛ>0 is a parameter. Under some local assumptions on V, we prove the uniqueness of multi-bump solutions concentrating around zero points of V by a local Pohozaev identity.

In this article, we consider the following system involving fully nonlinear nonlocal operators Fα(u(x))+a1(x)v(x)=f(v(x)),Gβ(v(x))+a2(x)u(x)=g(u(x)).Compared to results in Zhang et al. (2020), we apply a narrow region principle and a direct method of moving planes to prove that all positive solutions are radially symmetric about some point in RN under the weaker condition on ai(x) (i=1,2). Furthermore

In this paper, we are concerned with the radial solutions for the eigenvalue problem of a singular k-Hessian equation. By constructing the upper and lower solutions of the k-Hessian equation, the existence of a radial solution for the eigenvalue problem is established via Schauder’s fixed point theorem under the case where the nonlinearity possesses a singularity with respect to the space variable

In the scalar case, the nondegeneracy of heteroclinic orbits is a well-known property, commonly used in problems involving nonlinear elliptic, parabolic or hyperbolic P.D.E. On the other hand, Schatzman proved that in the vector case this assumption is generic, in the sense that for any potential W:Rm→R, m≥2, there exists an arbitrary small perturbation of W, such that for the new potential minimal

This paper establishes the basic structure of the exponential Euler difference form for Caputo–Fabrizio fractional-order differential equations (CF-FODEs) with multiple lags. The research shows that the acquired difference form (i.e., discrete-time CF-FODEs) belongs to the scope of implicit Euler differences and then the fractional PECE algorithm is proposed to solve this implicit difference. At last

This paper aims to study a degenerate parabolic problem for a solenoidal vector field in which the time derivative acts on a moving body. We propose a fully-discrete finite element scheme combined with backward Euler’s method for the saddle-point variational formulation. The convergence of this numerical scheme is proved and error estimates for some stable finite element pairs are also established

We study a class of Schrödinger–Kirchhoff equations in R3 with subquadratic nonlinearity and indefinite potential. By variational methods, we obtain the existence of multiple solutions for this problem.

In this paper, we propose a stochastic predator–prey model with hunting cooperation and nonlinear perturbation of white noise. The sufficient criteria for the existence of a unique ergodic stationary distribution are established by constructing suitable Lyapunov functions. It reveals that the white noise has a significant impact on the dynamical behavior of the model.

We analyze the consensus based optimization method proposed in Pinnau et al. (2017) in one dimension. We rigorously provide a quantitative error estimate between the consensus point and global minimizer of a given objective function. Our analysis covers general objective functions; we do not require any structural assumption on the objective function.

This work proposes a meshless generalized finite difference method (GFDM) with supplementary nodes for the solution of thin elastic plate bending under dynamic loading. The first- and second-order time derivatives in equilibrium equation are firstly discretized with the Houbolt method. The numerical solution of the resulting spatial problem at each time step is then calculated by the GFDM. The supplementary

In this paper, we study the b-family of Fokas–Olver–Rosenau–Qiao (bFORQ) model. We discuss the persistence property in weighted Sobolev spaces. The infinite propagation speed is also investigated. We prove that the strong solution u(x,t) does not have compact x-support for any t>0 in its lifespan, although the corresponding u0(x) is compactly supported. We also present a special property that if the

The existence of solitary wave solutions for a perturbed generalized mKdV equation with a cubic evolution term is investigated. All possible solitary waves for the corresponding unperturbed equation are firstly explored by dynamical system analysis. Then by using geometric singular perturbation theory and Melnikov’s method, we prove that two solitary wave solutions of the unperturbed generalized mKdV

In this paper we study the uniqueness of positive solutions of the nonlinear Kirchhoff type equations in arbitrary dimension N: (P)−(a+b∫RN|∇u|2dx)Δu+u=|u|p−1uinRN,where a≥0,b>0, 1

In recent papers, some fractional Newton-type methods have been proposed by using the Riemann–Liouville and Caputo fractional derivatives in their iterative schemes, with order 2α or 1+α. In this manuscript, we introduce the Conformable fractional Newton-type method by using the so-called fractional derivative. The convergence analysis is made, proving its quadratic convergence, and the numerical results

The time and spatial fractional constitutive models are employed to examine the flow, heat and mass transfer of a viscoelastic fluid flowing in a vertical porous media square channel accompanied by Soret and Dufour effects. The fractional modified Fourier’s law and Fick’s model are utilized to characterize the heat and mass transfer. The finite difference algorithm is used to calculate the governing

In this Letter we extend the proof, by Faraco and Lindberg (2020), of Taylor’s conjecture in multiply connected domains to cover arbitrary vector potentials and remove the need to impose restrictions on the magnetic field to ensure gauge invariance of the helicity integral. This extension allows us to treat general magnetic fields in closed domains that are important in laboratory plasmas and brings

The goal of this paper is to prove the theory that the L1 scheme for solving time fractional partial differential equations with nonsmooth data has the uniform l1 optimal order error estimate. For the L1 scheme combined with the first-order convolution quadrature scheme, by using Laplace transform rules we obtain the uniform l1 long time convergence of the L1 scheme for smooth and nonsmooth initial

Some special travelling wave solutions of the (2+1)-dimensional Sawada–Kotera (SK) equation are constructed by a new travelling wave method related to the bilinear transformation. It is found the SK equation possesses plenty of novel travelling wave solutions, such as the few-cycle-pulse solitons (FCPs), the asymmetric soliton, the soliton molecules, the three-peak soliton, the plat soliton, the double-peak

The Broyden–Fletcher–Goldfarb–Shanno (BFGS) method plays an important role among the quasi-Newton algorithms for nonconvex and unconstrained optimization problems. However, in the proof of global convergence, BFGS-type methods generally need to assume that the gradient of the objective function is Lipschitz continuous. This issue prompts us to try to find quasi-Newton method for gradient non-Lipschitz

In this paper, we consider the energy measure E for the Euler equations in the endpoint case u∈L2,∞(BMO(Ω)) for any Ω⊆R3. We establish the lower bounds of the lower local dimension and concentration dimension of the energy measure E at the first blow-up time.

To reveal and analyze the dynamics of degenerate solitons, breathers and mixed breather-rogue waves, we consider an AB system in this manuscript, which models the marginally unstable baroclinic wave packets in geophysical fluids or ultra-short pulses in nonlinear optics. Based on the generalized Darboux transformation, we construct the degenerate soliton, breather and mixed solutions, then some figures

The integral inequality plays a key role in developing delay-dependent stability criteria for the linear system with multiple time delays. This paper investigates a novel integral inequality to bound the integral terms in the derivative of the Lyapunov–Krasovskii functional (LKF). Compared with the auxiliary function-based integral inequality, the presented integral inequality is more general. By removing

A basic problem in both pure and applied mathematics is solving various kinds of equations. In recent years, multilinear systems of equations have received a great deal of attention and active research. However, most studies are focused on those multilinear systems with the tensors involved being strong M-tensors. It is well known that many tensors from real problems are non-negative, including strong

Starting from the truncated Painlevé expansion, auto- and non-auto Bäcklund transformations are constructed for an extended (2+1)-dimensional Korteweg–de Vries equation. Then nonlocal residual symmetry is derived and localized to obtain the finite group transformation. By virtue of a simple connection between the truncated Painlevé expansion and the consistent tanh expansion (CTE) method, exact single

In this paper, the rheological properties of welan gum solutions with mass fractions of 2%, 3% and 4% are measured using an Anton paar MCR 302 rheometer. It is found that the liquids fit the five-parameter Jeffreys model of double fractional order with corresponding rheological parameters. The behaviors of these fluids over linear stretching surface are studied based on the experimental results. We

Consider plane steady viscous liquid flow around a translating and rotating obstacle. We derive the fundamental solution of the associated linearized problem, assuming a general 2D rigid body velocity V(x)=ζ+ωx⊥.

In this paper, a quasilinear shallow water model for moderate-amplitude waves with the effect of underlying shear flow is derived from the governing equations in the two-dimensional incompressible fluid. Such a model serves as a highly nonlinear generalized Camassa–Holm equation, which is based on the choice of depth and is proceeded for the case of a linear shear. Moreover, the effects of non-zero

In this paper, an analytical method on the motion of nonlinear oscillator with fractional-order restoring force and time variable mass is developed. The approximate solution for the periodic motion with the form of the trigonometric function form can describe the amplitude and phase of motion. The analytical solution is compared with the numerical solutions, which illustrate the accuracy and validity

We prove the wellposedness of a space-fractional diffusion equation governed by a hidden memory variably distributed-order fractional derivative, which provides a competitive means to describe the history memory property of anomalous diffusive transport of solutes in heterogeneous porous media.

In this paper, we study a non-autonomous delay mosquito population suppression model where the birth process of wild mosquitoes is density dependent due to the intra-specific competition of larvae. We only include the sexually active sterile mosquitoes in the model such that the number of sterile mosquitoes released is treated as a nonnegative function given in advance. We determine a release threshold

In this paper, the non-global solvability of the Cauchy problem for non-gauge invariant semilinear semirelativistic equations is considered. The lifespan estimate has been considered based on the analogy of semilinear Schrödinger equations and will be determined by the scaling property of semilinear semirelativistic equations. On the other hand, in this paper, in the one-dimensional case, a sharper

By introducing an attractive–repulsive force in Cucker–Smale model, the dynamic behaviour of multi-agent system is studied. The Cucker–Smale-type model considered in this paper has three properties: collision-avoidance, aggregation and velocity-matching, which are significant rules in dynamic behaviour of multi–agent systems. Results of collision-avoidance, flocking and swarming are given respectively

In this paper, we discuss the quasilinear Schrödinger equation with a Hardy potential −Δu+V(x)u−μu|x|2−Δ(u2)u=g(u),x∈RN∖{0},where N≥3, 0≤μ<μ̄=(N−2)24, 1|x|2 is called the Hardy potential. By using Jeanjean’s monotonicity trick, we admit a class of ground state solutions for the above problem, under the general Berestycki–Lions type assumptions on the nonlinearity g, as well as some weak assumptions

This study concerns a generalized class of biharmonic-type boundary value problems with nonstandard growth conditions over bounded irregular domains. We provide a unitary treatment for various types of applied problems with variable exponents, like the p(⋅)-biharmonic problems, mean curvature problems, and capillarity problems. The multiplicity result established here includes situations that cannot

A novel tensor product for third order quaternion tensors is proposed. With this tensor product, we further define the singular value decomposition and the rank of a third order quaternion tensor. It is also proved that the best rank-k approximation of a third order quaternion tensor exists. On the other hand, note that a color video can be expressed as a third order quaternion tensor, the proposed

In this paper, we study the existence and multiplicity of nontrivial solutions for equations driven by a nonlocal integrodifferential operator LK with homogeneous Dirichlet boundary conditions. Taking general potentials and combined nonlinearities into consideration, at least two weak solutions are obtained by abstract critical point results.