Singular soliton solution and bifurcation analysis of Klein-Gordon equation with power law nonlinearity

In this paper, the Klein-Gordon equation (KGE) with power law nonlinearity will be considered. The bifurcation analysis as well as the ansatz method of integration will be applied to extract soliton and other wave solutions. In particular, the second approach to integration will lead to a singular soliton solution. However, the bifurcation analysis will reveal several other solutions that are of prime importance in relativistic quantum mechanics where this equation appears.     

New explicit solutions for the Klein-Gordon equation with cubic nonlinearity

In this paper, we investigate Klein-Gordon equation with cubic nonlinearity. All explicit expressions of the bounded travelling wave solutions for the equation are obtained by using the bifurcation method and qualitative theory of dynamical systems. These solutions contain bell-shaped solitary wave solutions, kink-shaped solitary wave solutions and Jacobi elliptic function periodic solutions. Moreover, we point out the region which these periodic wave solutions lie in. We present the relation between the bounded travelling wave solution and the energy level h. We find that these periodic wave solutions tend to the corresponding solitary wave solutions as h increases or decreases. Finally, for some special selections of the energy level h, it is shown that the exact periodic solutions evolute into solitary wave solution.     

Well-posedness and asymptotic stability results for a viscoelastic plate equation with a logarithmic nonlinearity

ABSTRACT

In this paper, we consider a viscoelastic plate equation with a velocity-dependent material density and a logarithmic nonlinearity. Using the Faedo-Galaerkin approximations and the multiplier method, we establish the existence of the solutions of the problem and we prove an explicit and general decay rate result. These results extend and improve many results in the literature.     

Sharper global existence for the generalized 1D nonhomogeneous Ginzburg-Landau equation

We study the following generalized 1D Ginzburg-Landau equation on Ω=(0,∞)×(0,∞):

     

On the nonlinear stability and the existence of selective decay states of 3D quasi-geostrophic potential vorticity equation

In this article, we study the dynamics of large-scale motion in atmosphere and ocean governed by the 3D quasi-geostrophic potential vorticity (QGPV) equation with a constant stratification. It is shown that for a Kolmogorov forcing on the first energy shell, there exist a family of exact solutions that are dissipative Rossby waves. The nonlinear stability of these exact solutions are analyzed based on the assumptions on the growth rate of the forcing. In the absence of forcing, we show the existence of selective decay states for the 3D QGPV equation. The selective decay states are the 3D Rossby waves traveling horizontally at a constant speed. All these results can be regarded as the expansion of that of the 2D QGPV system and in the case of 3D QGPV system with isotropic viscosity. Finally, we present a geometric foundation for the model as a general equation for nonequilibrium reversible-irreversible coupling.     

非线性时滞差分议程的全局渐近稳定性

In this paper,a sufficient condition for the global asymptotic stability of the solutions of the following nonlinear delay difference equation is obtained,xn 1=xn xn-1xn-2 a/xmxm-1 xn-2 a,n=0,1…,where a∈(0,∞) and the initial values x-2,x-1,x0∈(0,∞).As a special case,a conjecture by Ladas is confirmed.     

A new method to solve the damped nonlinear Klein-Gordon equation

This paper discusses a damped nonlinear Klein-Gordon equation in the reproducing kernel space and provides a new method for solving the damped nonlinear Klein-Gordon equation based on the reproducing kernel space.Two numerical examples are given for illustrating the feasibility and accuracy of the method.     

On stabilization and control for the critical Klein-Gordon equation on a 3-D compact manifold

In this article, we study the internal stabilization and control of the critical nonlinear Klein-Gordon equation on 3-D compact manifolds. Under a geometric assumption slightly stronger than the classical geometric control condition, we prove exponential decay for some solutions bounded in the energy space but small in a lower norm. The proof combines profile decomposition and microlocal arguments. This profile decomposition, analogous to the one of Bahouri and Gérard (1999) [2] on R³, is performed by taking care of possible geometric effects. It uses some results of S. Ibrahim (2004) [21] on the behavior of concentrating waves on manifolds.     

Global stability in the 2D Ricker equation

We improve a previous result for the 2D Ricker equation by reducing an infinite number of topological conditions to a finite number. We also give sufficient conditions in terms of the parameters where many of these topological conditions are satisfied. We also discuss the various pathologies that occur for other parameter choices.     

Well-posedness for the massive nonlinear wave equation on asymptotically AdS spacetimes

In this paper, we prove some well-posedness theorems for the nonlinear massive wave equation (with the mass satisfying the Breitenlohner-Freedman bound) on asymptotically anti-de Sitter spaces. The solution is constructed by contraction mapping method and also as a limit of solutions to an initial boundary value problem with boundary at a finite location in spacetime by finally pushing the boundary out to infinity. The solution obtained is unique within the energy class.     

Orbital stability of standing waves for semilinear wave equations with indefinite energy

The orbital stability of standing waves for semilinear wave equations is studied in the case that the energy is indefinite and the underlying space domain is bounded or a compact manifold or whole Rn with n?2. The stability is determined by the convexity on ω of the lowest energy d(ω) of standing waves with frequency ω. The arguments rely on the conservation of energy and charge and the construction of suitable invariant manifolds of solution flows.     

Convergence of the Klein-Gordon equation to the wave map equation with magnetic field

This paper is devoted to the proof of the convergence from the modulated cubic nonlinear defocusing Klein-Gordon equation with magnetic field to the wave map equation. More precisely, we discuss the nonrelativistic-semiclassical limit of the modulated cubic nonlinear Klein-Gordon equation with magnetic field where the Planck's constant ?=ε and the speed of light c are related by c=ε−α for some α?1. When α=1 the limit wave function satisfies the wave map with one extra term coming from the magnetic field. However, α>1, the effect of the magnetic filed disappears and the limit is the typical wave map equation only.     

源项反演问题的条件稳定性

对于一维扩散方程的源项反演问题,探讨了反问题数据的相容性并应用积分恒等式方法建立了非线性源项反演的一种稳定性.     

Solution blowup for the heat equation with double nonlinearity

We consider a model initial boundary value problem for the heat equation with double nonlinearity. We use a modified Levin method to prove the solution blowup.     

A sharper decay rate for a viscoelastic wave equation with power nonlinearity

We consider a viscoelastic wave equation with power nonlinearity. First, we construct a local solution by the Faedo-Galerkin approximation scheme and contraction mapping theorem. Next, we continue the local solution to the global one by a priori estimates obtained from a decreasing energy. Finally, we discuss the decay rate of the global solution by assuming that the kernel function is convex.     

1-Soliton solution of the complex KdV equation in plasmas with power law nonlinearity and time-dependent coefficients

In this paper, the complex Korteweg-de Vries equation with power law nonlinearity is studied in presence of perturbation terms. The exact 1-soliton solution is obtained. It will be seen that the time-dependent coefficients must be simply Riemann integrable for the solitons to exist. The solitary wave ansatz is used to carry out the integration.     

Sharp threshold of global existence for the Klein-Gordon equations with critical nonlinearity

In this paper, we present a cross-constrained variational method to study the Cauchy problem of the nonlinear Klein-Gordon equations with critical nonlinearity in two space dimensions. By constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow, we establish a sharp threshold of global existence and blowup of it. Furthermore, we answer the question： How small are the initial data if the solution exists globally.     

“Destruction” of the solution of a strongly nonlinear equation of pseudoparabolic type with double nonlinearity

We consider the first initial-boundary value problem for multidimensional strongly nonlinear equations with double nonlinearity of pseudoparabolic type in a bounded domain with sufficiently smooth boundary. We prove the local solvability of this problem in the weak generalized sense. Depending on the nonlinearity and initial conditions under consideration, we prove the solvability of the equation in any finite cylinder (x, t) ∈ Ω × [0, T] or the destruction of the solution in finite time.