Convergence of Solutions of Nonhomogeneous Linear Difference Systems with Delays

Sufficient conditions are given for the asymptotic constancy of the solutions of a linear system of difference equations with delays. Moreover, it is shown that the limits of the solutions, as t→∞, can be computed in terms of the initial function and a special matrix solution of the corresponding adjoint equation.     

Solitary Wave Solutions and Kink Wave Solutions for a Generalized PC Equation

We present in this paper a generalised PC (GPC) equation which includes several known models. The corresponding traveling wave system is derived and we show that the homoclinic orbits of the traveling wave system correspond to the solitary waves of GPC equation, and the heteroclnic orbits correspond to the kink waves. Under some parameter conditions, the existence of above two types of orbits is demonstrated and the explicit expressions of the two solutions are worked out.     

Definite Conditions and Asymptotic Solution of One Nonlinear Partial Differential Equation in Morphogenesis

One reaction-diffusion equationhas been presented in the study of Morphogenesis.In this paper,reasonable de-finite conditions of the equation are proposed and the asymptotic form of itssolution is obtained by using perturbation method.So the existence of solution ofthis probtem is solved.     

Absence of Global Solutions of a Mixed Problem for a Schrödinger-Type Nonlinear Evolution Equation

We study the problem of the absence of global solutions of the first mixed problem for one nonlinear evolution equation of Schrödinger type. We prove that global solutions of the studied problem are absent for “sufficiently large” values of the initial data.     

An Exact Solltary Wave Solutions for a New Derivative Nonlinear Schrodinger Equation

In this letter, an exact one-soliton solution for a new derivative nonlinear Schrodinger equation is obtained.     

Infinitely Many Sign-changing Solutions for a Schroedinger Equation in R^N

We consider the existence of infinitely many sign-changing solutions for the nonlinear timeindependent schroedinger equations of the form     

Nonexistence of Global Solutions of a Mixed Problem for a Ginzburg–Landau Type Nonlinear Evolution Equation

Doklady Mathematics - The nonexistence of global solutions of the Dirichlet mixed problem for a nonlinear evolution equation of the Ginzburg–Landau type is studied. For “sufficiently...     

Asymptotic Behaviour for a Nonlinear Schrödinger Equation in Domains with Moving Boundaries

We consider a nonlinear Schrödinger equation in a time-dependent domain Q _τ of ?² given by $$u_{\tau} - i u_{\varepsilon\varepsilon} + |u|^{2} u + \gamma v=0. $$ We prove the well-posedness of the above model and analyze the behaviour of the solution as t→+∞. We consider two situations: the conservative case (γ=0) and the dissipative case (γ>0). In both situations the existence of solutions are proved using the Galerkin method and the stabilization of solutions are obtained considering multiplier techniques.     

Estimate on the Dimension of Global Attractor for Nonlinear Dissipative Kirchhoff Equation

In this paper,we estimate the dimension of the global attractor for nonlinear dissipative Kirchhoff equation in Hilbert spaces H ₀¹×L ²(Ω) and D(A)×H ₀¹(Ω). Using rescaling technology and linear variation method, we obtain the upper bound for its Hausdorff and fractal dimensions.     

Infinitely Many Sign-changing Solutions for a Schrdinger Equation in R~N

We consider the existence of infinitely many sign-changing solutions for the nonlinear time-independent schrodinger equations of the form where Vλ(x) =λa(x) 1. This problem originates from various problems in physics and mathematical physics. In constructive field theory, (1.1) is called a nonlinear Euclidean scalar field equation. In chemical dynamics, a solution of (1.1) is a stationary state of the reaction diffusion equation     

Propagation of Perturbations of Solutions for a Doubly Degenerate Nonlinear Parabolic Equation

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Multiple Solutions for a Nonlinear Schrödinger Equation with Magnetic Fields

We study a nonlinear Schrödinger equation in presence of a magnetic field and relate the number of solutions with the topology of the set where the potential attains its minimum value. In the proofs we apply variational methods, penalization techniques and Ljusternik–Schnirelmann theory.     

Oscillatory and Asymptotic Behavior of Solutions for Nonlinear Impulsive Delay Differential Equations

The oscillatory and asymptotic behavior of the solutions for third order nonlinear impulsive delay differential equations are investigated. Some novel criteria for all solutions to be oscillatory or be asymptotic are established, Three illustrative examples are proposed to demonstrate the effectiveness of the conditions.     

Solutions of the Discrete Nonlinear Schrödinger Equation with a Trap

Theoretical and Mathematical Physics - We obtain solutions of the discrete nonlinear Schrödinger equation with an impurity center in two ways. In the first of them, we construct the wave...     

Exact Solitary-wave Solutions and Periodic Wave Solutions for Generalized Modified Boussinesq Equation and the Effect of Wave Velocity on Wave Shape

By means of the undetermined assumption method, we obtain some new exact solitary-wave solutions with hyperbolic secant function fractional form and periodic wave solutions with cosine function form for the generalized modified Boussinesq equation. We also discuss the boundedness of these solutions. More over, we study the correlative characteristic of the solitary-wave solutions and the periodic wave solutions along with the travelling wave velocity＇s variation.     

Convergence of Algorithms in Optimization and Solutions of Nonlinear Equations

It is desirable that an algorithm in unconstrained optimization converges when the guessed initial position is anywhere in a large region containing a minimum point. Furthermore, it is useful to have a measure of the rate of convergence which can easily be computed at every point along a trajectory to a minimum point. The Lyapunov function method provides a powerful tool to study convergence of iterative equations for computing a minimum point of a nonlinear unconstrained function or a solution of a system of nonlinear equations. It is surprising that this popular and powerful tool in the study of dynamical systems is not used directly to analyze the convergence properties of algorithms in optimization. We describe the Lyapunov function method and demonstrate how it can be used to study convergence of algorithms in optimization and in solutions of nonlinear equations. We develop an index which can measure the rate of convergence at all points along a trajectory to a minimum point and not just at points in a small neighborhood of a minimum point. Furthermore this index can be computed when the calculations are being carried out.