Global Dynamics Above the Ground State for the Nonlinear Klein–Gordon Equation Without a Radial Assumption

We extend our result Nakanishi and Schlag in J. Differ. Equ. 250(5):2299–2333, 2011) to the non-radial case, giving a complete classification of global dynamics of all solutions with energy that is at most slightly above that of the ground state for the nonlinear Klein–Gordon equation with the focusing cubic nonlinearity in three space dimensions.     

Vortex Dynamics for the Nonlinear Maxwell–Klein–Gordon Equation

We derive the vortex dynamics for the nonlinear Maxwell–Klein–Gordon equation with the Ginzburg–Landau type potential. In particular, we consider the case when the external electric fields are of order \({O( | \log \epsilon |^{\frac{1}{2}})}\). We study the convergence of the space–time Jacobian \({\partial_t \psi \cdot i \nabla \psi}\) as an interaction term between the vortices and electric fields. An explicit form of the limiting vector measure is shown.     

Global Attractor of a Dissipative Fractional Klein Gordon Schrödinger System

Journal of Dynamics and Differential Equations - In this paper we study the local and global well posedness of a fractional dissipative Klein–Gordon–Schrödinger type system in...     

Spatial Analyticity on the Global Attractor for the Kuramoto–Sivashinsky Equation

For the Kuramoto–Sivashinsky equation with L-periodic boundary conditions we show that the radius of space analyticity on the global attractor is lower-semicontinuous function at the stationary solutions, and thereby deduce the existence of a neighborhood in the global attractor of the set of all stationary solutions in which the radius of analyticity is independent of the bifurcation parameter L. As an application of the result, we prove that the number of rapid spatial oscillations of functions belonging to this neighborhood is, up to a logarithmic correction, at most linear in L.     

Singular Limits of the Klein–Gordon Equation

We establish the singular limits, including semiclassical, nonrelativistic and nonrelativistic-semiclassical limits, of the Cauchy problem for the modulated defocusing nonlinear Klein–Gordon equation. For the semiclassical limit, ${\hbar\to 0}$ , we show that the limit wave function of the modulated defocusing cubic nonlinear Klein–Gordon equation solves the relativistic wave map and the associated phase function satisfies a linear relativistic wave equation. The nonrelativistic limit, c → ∞, of the modulated defocusing nonlinear Klein–Gordon equation is the defocusing nonlinear Schrödinger equation. The nonrelativistic-semiclassical limit, ${\hbar\to 0, c=\hbar^{-\alpha}\to \infty}$ for some α > 0, of the modulated defocusing cubic nonlinear Klein–Gordon equation is the classical wave map for the limit wave function and a typical linear wave equation for the associated phase function.     

A Higdon-like non-reflecting boundary condition for the Klein–Gordon equation with evanescent waves

The non-reflecting boundary condition developed by Higdon and automated by Givoli and Neta is highly effective at absorbing propagating waves in a finite difference setting, but it does not absorb evanescent waves. In this paper, we augment the Higdon scheme with additional terms to absorb these evanescent waves in the context of the two-dimensional Klein–Gordon equation. Numerical examples illustrate the performance of this technique.     

A hybrid computational approach for Klein–Gordon equations on Cantor sets

Nonlinear Dynamics - In this letter, we present a hybrid computational approach established on local fractional Sumudu transform method and homotopy perturbation technique to procure the solution...     

A high-order nonlinear Schrödinger equation as a variational problem for the averaged Lagrangian of the nonlinear Klein–Gordon equation

Nonlinear Dynamics - We use Whitham’s averaged Lagrangian method extended with the multiple-scale formalism to derive a sixth-order nonlinear Schrödinger equation for the complex...     

Global Solutions for a Coupled Compressible Navier–Stokes/Allen–Cahn System in 1D

In this paper, we investigate a coupled compressible Navier–Stokes/Allen–Cahn system which describes the motion of a mixture of two viscous compressible fluids. We prove the existence and uniqueness of global classical solution, the existence of weak solutions and the existence of unique strong solution of the Navier–Stokes/Allen–Cahn system in 1D for initial data ρ ₀ without vacuum states.     

The Visous Cahn–Hilliard Equation as a Limit of the Phase Field Model: Lower Semicontinuity of the Attractor

We consider the one-dimensional viscous Cahn–Hilliard equation with Dirichlet boundary conditions as the limit of a corresponding Dirichlet boundary value problem for the phase field model and we prove the convergence of the attractor. No assumption on the hyperbolicity of the stationary solutions is made.     

Numerical solution of the Klein–Gordon equation via He’s variational iteration method

In this paper, we present the solution of the Klein--Gordon equation. Klein--Gordon equation is the relativistic version of the Schrödinger equation, which is used to describe spinless particles. The He’s variational iteration method (VIM) is implemented to give approximate and analytical solutions for this equation. The variational iteration method is based on the incorporation of a general Lagrange multiplier in the construction of correction functional for the equation. Application of variational iteration technique to this problem shows rapid convergence of the sequence constructed by this method to the exact solution. Moreover, this technique reduces the volume of calculations by avoiding discretization of the variables, linearization or small perturbations.     

Finite Dimensional Global Attractor for a Fractional Schrödinger Type Equation with Mixed Anisotropic Dispersion

Journal of Dynamics and Differential Equations - We study the Cauchy problem for a class of nonlinear damped fractional Schrödinger type equation in a two dimensional unbounded domain. Then,...     

The Response of a Rayleigh–Liénard Oscillator to a Fundamental Resonance

A Rayleigh–Liénard oscillator excited by a fundamentalresonance is investigated by using an asymptotic perturbation method based on Fourier expansion and time rescaling. Two first-order nonlinear ordinarydifferential equations governing the modulation of the amplitude andthe phase of solutions are derived. These equations are used todetermine steady-state responses and their stability. Excitationamplitude-response and frequency-response curves are shown and checkedby numerical integration. Dulac's criterion, the Poincaré–Bendixsontheorem, and energy considerations are used in order to study the existenceand characteristics of limit cycles of the two modulation equations. Alimit cycle corresponds to a modulated motion for the Rayleigh–Liénardoscillator. For small excitation amplitude, the analytical results arein excellent agreement with the numerical solutions. In certain caseswhen the excitation amplitude is very low, an approximate analyticsolution corresponding to a modulated motion can be obtained andnumerically checked. Moreover, if the excitation amplitude is increased,an infinite-period bifurcation occurs because the modulation periodlengthens and becomes infinite, while the modulation amplitude remainsfinite and suddenly the attractor settles down into a periodic motion.     

The effects of horizontal singular straight line in a generalized nonlinear Klein–Gordon model equation

In this paper, we investigate bounded traveling waves of the generalized nonlinear Klein–Gordon model equations by using bifurcation theory of planar dynamical systems to study the effects of horizontal singular straight lines in nonlinear wave equations. Besides the well-known smooth traveling wave solutions and the non-smooth ones, four kinds of new bounded singular traveling wave solution are found for the first time. These singular traveling wave solutions are characterized by discontinuous second-order derivatives at some points, even though their first-order derivatives are continuous. Obviously, they are different from the singular traveling wave solutions such as compactons, cuspons, peakons. Their implicit expressions are also studied in this paper. These new interesting singular solutions, which are firstly founded, enrich the results on the traveling wave solutions of nonlinear equations. It is worth mentioning that the nonlinear equations with horizontal singular straight lines may have abundant and interesting new kinds of traveling wave solution.     

Nonlinear Wave-Wave Interaction and Stability Criterion for Parametrically Coupled Nonlinear Schrödinger Equations

A rigorous mathematical reduction of the procedure widely usedfor studying a class of the nonlinear problems with perturbations,namely the method of the multiple scales, is used. A profound analysis,which provides an approach for deriving a coupled nonlinearSchrödinger equations. The investigation has been achieved byperturbing the nonlinear dynamical system about the linear dynamicalproblem. Modulated wavetrains are described to all orders ofapproximation. Moreover, we extend our approach to deal with equationshaving periodic terms. Two types of simultaneous nonlinearSchrödinger equations are derived. One type is valid at thenon-parametric system and the second type represents a modification forthe first type which is governed the non-resonance case. Two parametriccoupled nonlinear Schrödeinger equations are derived to govern thesecond-sub-harmonic resonance. In addition other two coupled equationsare found for the third-sub-harmonic resonance case. These systems ofequations control the stability behavior at the parametric resonancecases. The stability criteria for the several types of coupled nonlinearSchrödinger equations are studied. These criteria are achieved by atemporal periodic perturbation.     

Finite Dimensional Global Attractor for 3D MHD-α Models: A Comparison

We consider two magnetohydrodynamic-α (MHDα) models with kinematic viscosity and magnetic diffusivity for an incompressible fluid in a three-dimensional periodic box (torus). More precisely, we consider the Navier–Stokes-α-MHD and the Modified Leray-α-MHD models. Similar models are useful to study the turbulent behavior of fluids in presence of a magnetic field because of the current impossibility to handle non-regularized systems neither analytically nor via numerical simulations. In both cases, the global existence of the solution and of a global attractor can be shown. We provide an upper bound for the Hausdorff and the fractal dimension of the attractor. This bound can be interpreted in terms of degrees of freedom of the long-time dynamics of the involved system and gives information about the numerical stability of the model. We get the same bound that holds for the Simplified Bardina-MHD model, considered in a previous paper (this result provides, in some sense, an intermediate bound between the number of degrees of freedom for the Simplified Bardina model and the Navier–Stokes-α equation in the nonmagnetic case). However, the Navier–Stokes-α-MHD system is preferable since, in the ideal case, it conserves more quadratic invariants derived from the standard MHD model.     

Topological and non-topological solitons of the Klein–Gordon equations in 1+2 dimensions

This paper obtains the topological and non-topological 1-soliton solution of the Klein–Gordon equation in 1+2 dimensions. There are five various forms of this equation that will be studied. The solitary wave ansatz will be used to carry out the integration.     

Global Existence for the Generalized Two-Component Hunter–Saxton System

We study the global existence of solutions to a two-component generalized Hunter–Saxton system in the periodic setting. We first prove a persistence result for the solutions. Then for some particular choices of the parameters (α, κ), we show the precise blow-up scenarios and the existence of global solutions to the generalized Hunter–Saxton system under proper assumptions on the initial data. This significantly improves recent results.