Dynamics of some coupled nonlinear Schrödinger systems in

In this paper, we show the scattering and blow up result of the solution for some coupled nonlinear Schrödinger system with static energy less than that of the ground state in , where . We first use the Nehari manifold approach and the Schwarz symmetrization technique to construct the ground state and obtain the threshold energy of scattering solution, then use Payne–Sattinger's potential well argument and Kenig–Merle's compactness‐rigidity argument to show the aforementioned dichotomy result. As we know, it is the first attempt to show the scattering result for the coupled nonlinear Schrödinger system. Copyright © 2013 John Wiley & Sons, Ltd.     

ψ‐Hyperholomorphic functions and a Kolosov–Muskhelishvili formula

Holomorphic function theory is an effective tool for solving linear elasticity problems in the complex plane. The displacement and stress field are represented in terms of holomorphic functions, well known as Kolosov–Muskhelishvili formulae. In , similar formulae were already developed in recent papers, using quaternionic monogenic functions as a generalization of holomorphic functions. However, the existing representations use functions from to , embedded in . It is not completely appropriate for applications in . In particular, one has to remove many redundancies while constructing basis solutions. To overcome that problem, we propose an alternative Kolosov–Muskhelishvili formula for the displacement field by means of a (paravector‐valued) monogenic, an anti‐monogenic and a ψ‐hyperholomorphic function. Copyright © 2015 John Wiley & Sons, Ltd.     

On decay estimates of the 3D nematic liquid crystal flows in critical Besov spaces

In this paper, we develop the energy argument in homogeneous Besov space framework to study the large time behavior of global‐in‐time strong solutions to the Cauchy problem of the three‐dimensional incompressible nematic liquid crystal flows with low regularity assumptions on initial data. More precisely, if the small initial data with 1 < p < ∞ and further assume that with 1 < q≤p and , then the global‐in‐time strong solution (u,d) to the nematic liquid crystal flows admits the following temporal decay rate: Here, is a constant unit vector. The highlight of our argument is to show that the ‐norms (with ) of solution are preserved along time evolution. Copyright © 2016 John Wiley & Sons, Ltd.     

Homoclinic solutions for second‐order discrete Hamiltonian systems with asymptotically quadratic potentials

In this paper, we study the existence of infinitely many homoclinic solutions for the second‐order self‐adjoint discrete Hamiltonian system , where , and are unnecessarily positive definites for all . By using the variant fountain theorem, we obtain an existence criterion to guarantee that the aforementioned system has infinitely many homoclinic solutions under the assumption that W(n,x) is asymptotically quadratic as | x | → + ∞ . Copyright © 2013 John Wiley & Sons, Ltd.     

Infinitely many solutions for a class of fractional Hamiltonian systems via critical point theory

In this paper, we are concerned with the existence of infinitely many solutions for the following fractional Hamiltonian systems where , , and . The novelty of this paper is that, relaxing the conditions on the potential function W(t,x), we obtain infinitely many solutions via critical point theory. Our results generalize and improve some existing results in the literature. Copyright © 2015 John Wiley & Sons, Ltd.     

Asymptotic behavior of solutions to a class of nonlocal non‐autonomous diffusion equations

In this paper, we consider the nonlocal non‐autonomous evolution problems where Ω is a bounded smooth domain in , N≥1, β is a positive constant, the coefficient a is a continuous bounded function on , and K is an integral operator with symmetric kernel , being J a non‐negative function continuously differentiable on and . We prove the existence of global pullback attractor, and we exhibit a functional to evolution process generated by this problem that decreases along of solutions. Assuming the parameter β is small enough, we show that the origin is locally pullback asymptotically stable. Copyright © 2014 John Wiley & Sons, Ltd.     

Linear statistics of matrix ensembles in classical background

Given a joint probability density function of N real random variables, , obtained from the eigenvector–eigenvalue decomposition of N × N random matrices, one constructs a random variable, the linear statistics, defined by the sum of smooth functions evaluated at the eigenvalues or singular values of the random matrix, namely, . For the joint PDFs obtained from the Gaussian and Laguerre ensembles, we compute, in this paper, the moment‐generating function , where denotes expectation value over the orthogonal (β = 1) and symplectic (β = 4) ensembles, in the form one plus a Schwartz function, none vanishing over for the Gaussian ensembles and for the Laguerre ensembles. These are ultimately expressed in the form of the determinants of identity plus a scalar operator, from which we obtained the large N asymptotic of the linear statistics from suitably scaled F(·). Copyright © 2016 John Wiley & Sons, Ltd.     

Boundedness of solutions to parabolic–elliptic Keller–Segel systems with signal‐dependent sensitivity

This paper deals with the parabolic–elliptic Keller–Segel system with signal‐dependent chemotactic sensitivity function, under homogeneous Neumann boundary conditions in a smooth bounded domain , with initial data satisfying u₀ ≥ 0 and . The chemotactic sensitivity function χ(v) is assumed to satisfy The global existence of weak solutions in the special case is shown by Biler (Adv. Math. Sci. Appl. 1999; 9:347–359). Uniform boundedness and blow‐up of radial solutions are studied by Nagai and Senba (Adv. Math. Sci. Appl. 1998; 8:145–156). However, the global existence and uniform boundedness of classical nonradial solutions are left as an open problem. This paper gives an answer to the problem. Namely, it is shown that the system possesses a unique global classical solution that is uniformly bounded if , where γ > 0 is a constant depending on Ω and u₀. Copyright © 2014 John Wiley & Sons, Ltd.     

Asymptotics for the third‐order nonlinear Schrödinger equation in the critical case

We consider the Cauchy problem for the third‐order nonlinear Schrödinger equation where and is the Fourier transform. Our purpose in this paper is to prove the large time asymptoitic behavior of solutions for the defocusing case λ > 0 with a logarithmic correction under the non zero mass condition Copyright © 2016 John Wiley & Sons, Ltd.     

Infinitely many peak solutions for a biharmonic equation involving critical exponent

In this paper, we study the following biharmonic equation where , K(1) > 0,K′(1) > 0, B₁(0) is the unit ball in (N≥6). We show that the aforementioned problem has infinitely many peak solutions, whose energy can be made arbitrarily large. Copyright © 2016 John Wiley & Sons, Ltd.     

Infinitely many solutions for a Hardy–Sobolev equation involving critical growth

In this paper, by an approximating argument, we obtain infinitely many solutions for the following Hardy–Sobolev equation with critical growth: provided N > 6 + t, where and Ω is an open bounded domain in , which contains some points x₀ = (0,z₀). Copyright © 2013 John Wiley & Sons, Ltd.     

Ground state solutions for asymptotically periodic fractional Schrödinger–Poisson problems with asymptotically cubic or super‐cubic nonlinearities

In this paper, we consider the following fractional Schrödinger–Poisson problem: where s,t∈(0,1],4s+2t>3,V(x),K(x), and f(x,u) are periodic or asymptotically periodic in x. We use the non‐Nehari manifold approach to establish the existence of the Nehari‐type ground state solutions in two cases: the periodic one and the asymptotically periodic case, by introducing weaker conditions uniformly in with and with constant θ₀∈(0,1), instead of uniformly in and the usual Nehari‐type monotonic condition on f(x,τ)/|τ|³. Our results unify both asymptotically cubic or super‐cubic nonlinearities, which are new even for s=t=1. Copyright © 2017 John Wiley & Sons, Ltd.     

Attractors for the strongly damped wave equation with p‐Laplacian

This paper is concerned with the initial‐boundary value problem for one‐dimensional strongly damped wave equation involving p‐Laplacian. For p > 2 , we establish the existence of weak local attractors for this problem in . Under restriction 2 < p < 4, we prove that the semigroup, generated by the considered problem, possesses a strong global attractor in , and this attractor is a bounded subset of . Copyright © 2017 John Wiley & Sons, Ltd.     

Infinitely many solutions for nonlinear Schrödinger systems with magnetic potentials in

We study the following nonlinear Schrödinger system with magnetic potentials in : where μ₁>0, μ₂>0, and is a coupling constant. Under some weak symmetry conditions on A(y), P(y), and Q(y), which are given in the introduction, we prove that the nonlinear Schrödinger system has infinitely many non‐radial complex‐valued segregated and synchronized solutions. Copyright © 2015 John Wiley & Sons, Ltd.     

The periodic solution bifurcated from homoclinic orbit for coupled ordinary differential equations

We consider the problem of the periodic solutions bifurcated from a homoclinic orbit for a pair of coupled ordinary differential equations in . Assume that the autonomous system has a degenerate homoclinic solution γ in . A functional analytic approach is used to consider the existence of periodic solution for the autonomous system with periodic perturbations. By exponential dichotomies and the method of Lyapunov–Schmidt, the bifurcation function defined between two finite dimensional subspaces is obtained, where the zeros correspond to the existence of periodic solutions for the coupled ordinary differential equations near . Copyright © 2016 John Wiley & Sons, Ltd.     

Simple eigenvalues for the Steklov problem in a domain with a small hole. A functional analytic approach

Let be a bounded open domain of . Let denote the outward unit normal of . We assume that the Steklov problem Δu = 0 in and on has a simple eigenvalue of . Then we consider an annular domain obtained by removing from a small‐cavity size of ε > 0, and we show that under proper assumptions there exists a real valued and real analytic function defined in an open neighborhood of (0,0) in and such that is a simple eigenvalue for the Steklov problem Δu = 0 in and on for all ε > 0 small enough, and such that . Here denotes the outward unit normal of , and δ_2,2 ≡ 1 and δ_2,n ≡ 0 if n ≥ 3. Then related statements have been proved for corresponding eigenfunctions. Copyright © 2013 John Wiley & Sons, Ltd.     

Global dynamics of a SEIR model with information dependent vaccination and periodically varying transmission rate

This paper is devoted to the investigation of the global dynamics of a SEIR model with information dependent vaccination. The basic reproduction number is derived for the model, and it is shown that gives the threshold dynamics in the sense that the disease‐free equilibrium is globally asymptotically stable and the disease dies out if , while there exists at least one positive periodic solution and the disease is uniformly persistent when . Further, we give the approximation formula of . This answers the concerns presented in [B. Buonomo, A. d'Onofrio, D. Lacitignola, Modeling of pseudo‐rational exemption to vaccination for SEIR diseases, J. Math. Anal. Appl. 404 (2013) 385–398]. Copyright © 2014 John Wiley & Sons, Ltd.     

Wave solutions for variants of the KdV–Burger and the K(n,n)–Burger equations by the generalized G′/G‐expansion method

An application of the ‐expansion method to search for exact solutions of nonlinear partial differential equations is analyzed. This method is used for variants of the Korteweg–de Vries–Burger and the K(n,n)–Burger equations. The generalized ‐expansion method was used to construct periodic wave and solitary wave solutions of nonlinear evolution equations. This method is developed for searching exact traveling wave solutions of nonlinear partial differential equations. It is shown that the generalized ‐expansion method, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving nonlinear problems. Copyright © 2017 John Wiley & Sons, Ltd.     

Mehler's formulas for the univariate complex Hermite polynomials and applications

We give 2 widest Mehler's formulas for the univariate complex Hermite polynomials , by performing double summations involving the products and . They can be seen as the complex analogues of the classical Mehler's formula for the real Hermite polynomials. The proof of the first one is based on a generating function giving rise to the reproducing kernel of the generalized Bargmann space of level m. The second Mehler's formula generalizes the one appearing as a particular case of the so‐called Kibble‐Slepian formula. The proofs we present here are direct and more simpler. Moreover, direct applications are given and remarkable identities are derived.     

Liouville theorems for the weighted Lane–Emden equation with finite Morse indices

In this paper, we study the nonexistence result for the weighted Lane–Emden equation: (0.1) and the weighted Lane–Emden equation with nonlinear Neumann boundary condition: (0.2) where f(|x|) and g(|x|) are the radial and continuously differential functions, is an upper half space in , and . Using the method of energy estimation and the Pohozaev identity of solution, we prove the nonexistence of the nontrivial solutions to problems 0.1 and 0.2 under appropriate assumptions on f(|x|) and g(|x|). Copyright © 2017 John Wiley & Sons, Ltd.