Modified ‐expansion method for finding exact wave solutions of the coupled Klein–Gordon–Schrödinger equation

The coupled Klein–Gordon–Schrödinger equation is reduced to a nonlinear ordinary differential equation (ODE) by using Lie classical symmetries, and various solutions of the nonlinear ODE are obtained by the modified ‐expansion method proposed recently. With the aid of solutions of the nonlinear ODE, more explicit traveling wave solutions of the coupled Klein–Gordon–Schrödinger equation are found out. The traveling wave solutions are expressed by the hyperbolic functions, trigonometric functions, and rational functions. Copyright © 2012 John Wiley & Sons, Ltd.     

Orbital stability of solitary waves of the coupled Klein–Gordon–Zakharov equations

This paper investigates the orbital stability of solitary waves for the coupled Klein–Gordon–Zakharov (KGZ) equations where α ≠ 0. Firstly, we rewrite the coupled KGZ equations to obtain its Hamiltonian form. And then, we present a pair of sech‐type solutions of the coupled KGZ equations. Because the abstract orbital stability theory presented by Grillakis, Shatah, and Strauss (1987) cannot be applied directly, we can extend the abstract stability theory and use the detailed spectral analysis to obtain the stability of the solitary waves for the coupled KGZ equations. In our work, α = 1,β = 0 are advisable. Hence, we can also obtain the orbital stability of solitary waves for the classical KGZ equations which was studied by Chen. Copyright © 2016 John Wiley & Sons, Ltd.     

Finite time blowup for Klein‐Gordon‐Schrödinger system

We investigate the blowup solutions to the Klein‐Gordon‐Schrödinger (KGS) system with power nonlinearity in spatial dimensions (N ≥ 2). Relying on a Lyapunov functional, we establish a perturbed virial‐type identity and prove the existence of blowup solutions for the system with a negative energy and small mass. Moreover, we obtain a new finite‐time blowup result of solutions to KGS system in the energy space by constructing a differential inequality.     

Stability and instability of the standing waves for the Klein‐Gordon‐Zakharov system in one space dimension

The orbital instability of standing waves for the Klein‐Gordon‐Zakharov system has been established in two and three space dimensions under radially symmetric condition by Ohta‐Todorova in 2007. In the one space dimensional case, for the nondegenerate situation, we first check that the Klein‐Gordon‐Zakharov system satisfies Grillakis‐Shatah‐Strauss' assumptions on the stability and instability theorems for abstract Hamiltonian systems; see Grillakis‐Shatah‐Strauss (J. Funct. Anal. 1987). As to the degenerate case that the frequency , we follow the recent splendid work of Wu (2017) to prove the instability of the standing waves for the Klein‐Gordon‐Zakharov system, by using the modulation argument combining with the virial identity. For this purpose, we establish a modified virial identity to overcome several troublesome terms left in the traditional virial identity.     

The Gelfand–Tsetlin bases for Hodge–de Rham systems in Euclidean spaces

The main aim of this paper is to construct explicitly orthogonal bases for the spaces of k‐homogeneous polynomial solutions of the Hodge–de Rham system in the Euclidean space , which take values in the space of s‐vectors. Actually, we describe even the so‐called Gelfand–Tsetlin bases for such spaces in terms of Gegenbauer polynomials. As an application, we obtain an algorithm on how to compute an orthogonal basis of the space of homogeneous solutions for an arbitrary generalized Moisil–Théodoresco system in . Copyright © 2012 John Wiley & Sons, Ltd.     

Global well posedness of 3D‐NSE in Fourier–Lei–Lin spaces

In this paper, we prove a global well posedness of the three‐dimensional incompressible Navier–Stokes equation under an initial data, which belong to the non‐homogeneous Fourier–Lei–Lin space for σ ? ? 1 and if the norm of the initial data in the Lei–Lin space is controlled by the viscosity. 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.     

A conformal group approach to the Dirac–Kähler system on the lattice

Starting from the representation of the (n  ? 1) + n  ? dimensional Lorentz pseudo‐sphere on the projective space , we propose a method to derive a class of solutions underlying to a Dirac–Kähler type equation on the lattice. We make use of the Cayley transform to show that the resulting group representation arises from the same mathematical framework as the conformal group representation in terms of the general linear group . That allows us to describe such class of solutions as a commutative n  ? ary product, involving the quasi‐monomials ) with membership in the paravector space . Copyright © 2016 John Wiley & Sons, Ltd.     

A nonhomogeneous boundary value problem for the Kuramoto–Sivashinsky equation in a quarter plane

We study the initial boundary value problem for the one‐dimensional Kuramoto–Sivashinsky equation posed in a half line with nonhomogeneous boundary conditions. Through the analysis of the boundary integral operator, and applying the known results of the Cauchy problem of the Kuramoto–Sivashinsky equation posed on the whole line , the initial boundary value problem of the Kuramoto–Sivashinsky equation is shown to be globally well‐posed in Sobolev space for any s >?2. Copyright © 2017 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.     

Self‐organized criticality of cellular automata model; absorbtion in finite‐time of supercritical region into the critical one

In this work, it is studied the evolution and time behavior of solutions to nonlinear diffusion equation in where , d ≥ 1, and H is the Heaviside function. For d = 1,2,3, this equation describes the dynamics of self‐organizing sandpile process with critical state ρ_c. The main conclusion is that the supercritical region is absorbed in a finite‐time in the critical region . Copyright © 2013 John Wiley & Sons, Ltd.     

On the Spectral Stability of Kinks in 2D Klein–Gordon Model with Parity‐Time‐Symmetric Perturbation

In a series of recent works by Demirkaya et al., stability analysis for the static kink solutions to the one‐dimensional continuous and discrete Klein–Gordon equations with a ‐symmetric perturbation has been performed. In the present paper, we study two‐dimensional (2D) quadratic operator pencil with a small localized perturbation. Such an operator pencil is motivated by the stability problem for the static kink in 2D Klein–Gordon field taking into account spatially localized ‐symmetric perturbation, which is in the form of viscous friction. Viscous regions with positive and negative viscosity coefficient are balanced. For the considered operator pencil, we show that its essential spectrum has certain critical points generating eigenvalues under the perturbation. Our main results are sufficient conditions ensuring the existence or absence of such eigenvalues as well as the asymptotic expansions for these eigenvalues if they exist.     

Low regularity solutions,blowup, and global existence for a generalization of Camassa–Holm‐type equation

We consider a generalization of Camassa–Holm‐type equation including the Camassa–Holm equation and the Novikov equation. We mainly establish the existence of solutions in lower order Sobolev space with . Then, we present a precise blowup scenario and give a global existence result of strong solutions. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

Optimal decay rate of the bipolar Euler–Poisson system with damping in dimension three

By rewriting a bipolar Euler–Poisson equations with damping into a Euler equation with damping coupled with a Euler–Poisson equation with damping and using a new spectral analysis, we obtain the optimal decay results of the solutions in L² norm. More precisely, the velocities u₁ and u₂ decay at the L²?rate , which is faster than the normal L²‐rate for the heat equation and the Navier–Stokes equations. In addition, the decay rates of the disparities of two densities ρ₁?ρ₂ and the disparity of two velocities u₁?u₂ could reach to and in L² norm, respectively. Copyright © 2015 John Wiley & Sons, Ltd.     

Global existence and asymptotic behavior of classical solutions for a 3D two‐species chemotaxis‐Stokes system with competitive kinetics

This paper considers the 2‐species chemotaxis‐Stokes system with competitive kinetics under homogeneous Neumann boundary conditions in a 3‐dimensional bounded domain with smooth boundary. Both chemotaxis‐fluid systems and 2‐species chemotaxis systems with competitive terms were studied by many mathematicians. However, there have not been rich results on coupled 2‐species–fluid systems. Recently, global existence and asymptotic stability in the above problem with (u·∇)u in the fluid equation were established in the 2‐dimensional case. The purpose of this paper is to give results for global existence, boundedness, and stabilization of solutions to the above system in the 3‐dimensional case when is sufficiently small.     

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.     

Fourth‐order compact and energy conservative scheme for solving nonlinear Klein‐Gordon equation

In this article, a fourth‐order compact and conservative scheme is proposed for solving the nonlinear Klein‐Gordon equation. The equation is discretized using the integral method with variational limit in space and the multidimensional extended Runge‐Kutta‐Nyström (ERKN) method in time. The conservation law of the space semidiscrete energy is proved. The proposed scheme is stable in the discrete maximum norm with respect to the initial value. The optimal convergent rate is obtained at the order of in the discrete ‐norm. Numerical results show that the integral method with variational limit gives an efficient fourth‐order compact scheme and has smaller error, higher convergence order and better energy conservation for solving the nonlinear Klein‐Gordon equation compared with other methods under the same condition. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1283–1304, 2017     

New results on the existence of periodic solutions for a higher‐order Li énard type p‐Laplacian differential equation

In this paper, by using the continuation theorem of coincidence degree theory, we consider the higher‐order Li énard type p‐Laplacian differential equation as follows Some new results on the existence of periodic solutions for the previous equation are obtained, which generalize and enrich some known results to some extent from the literature. Copyright © 2011 John Wiley & Sons, Ltd.     

Existence and concentration of solutions for Schrödinger–Poisson system with steep potential well

This paper is concerned with the nonlinear Schrödinger–Poisson system where λ > 0 is a parameter. We require that V≥0 and has a bounded potential well V^?1(0). Combining this with other suitable assumptions on K and f, the existence of nontrivial solutions is obtained by using variational methods. Moreover, the concentration of solutions is also explored on the set V^?1(0) as λ→∞. Copyright © 2015 John Wiley & Sons, Ltd.