Abstract Forced Symmetry Breaking and Forced Frequency Locking of Modulated Waves

AbstractWe consider abstract forced symmetry breaking problems of the typeF(x, λ)=y. It is supposed that for allλthe mapsF(·, λ) are equivariant with respect to the action of a compact Lie group, thatF(x0, λ0)=0 and, hence, thatF(x, λ0)=0 for all elementsxof the group orbit O(x0) ofx0. We look for solutionsxwhich bifurcate from the solution family O(x0) asλandymove away fromλ0and zero, respectively. Especially, we describe the number of different solutionsx(for fixed control parametersλandy), their dynamic stability and their asymptotic behavior forytending to zero. Further, generalizations are given to problems of the typeF(x, λ)=y(x, λ). Finally, our results are applied to a forced frequency locking problem of the typex(t)=f(x(t), λ)−y(t). Here it is supposed that the vector fieldsf(·, λ) areS1-equivariant, that the unperturbed equationx=f(x, λ0) has an orbitally stable modulated wave solution and that the forcingy(t) is a modulated wave

Abstract forced symmetry breaking

We consider abstract forced symmetry breaking problems of the type F(x,λ) = y, x ≈ O(x0), λ ≈ λ0, y ≈ O. It is supposed that for all λ the maps F(·,λ) are equivariant with respect to representations of a given compact Lie group, that F(x0, λ0) = 0 and, hence, that F(x,λ0) = 0 for all elements x of the group orbit O(x0) of x0. We look for solutions x which bifurcate from the solution family O(x0) as λ and y move away from λ0 and zero, respectively. Especially, we describe the number of different solutions x (for fixed control parameters λ and y), their dynamic stability, their asymptotic behavior for y tending to zero and the structural stability of all these results. Further, generalizations are given to problems of the type F(x,λ) = y(x,λ), x ≈ O(x0), λ ≈ λ0, y(x,λ) ≈ 0. This work is a generalization of results of J. K. HALE, P. T'ABOAS , A. VANDERBAUWHEDE and E. DANCER to such extend that the conclusions are applicable to forced frequency locking problems for rotating and modulated wave solutions of certain S1-equivariant evolution equations which arise in laser modeling

Forced frequency locking in S1-equivariant differential equations

The aim of this paper is to present a simple analytic stategy for predicting, or engineering, two frequency locking phenomena for S1-equivariant ordinary differential equations. First we consider the forced frequency locking of a rotating wave solution of the unforced equation with a forcing of "rotating wave type", and we describe the creation of modulated wave solutions which is connected with this locking phenomenon. And second, we consider the forced frequency locking of a modulated wave solution with a forcing of "modulated wave type". Especially, we describe the sets of all control parameters and of all forcings such that frequency locking occures, the dynamic stability and the asymptotic behavior (for the forcing intensity tending to zero) of the locked solutions and the structural stability of all the phenomena. This paper is essentially founded on results from our previous work [41] concerning abstract forced symmetry breaking. The equations considered in the present paper are finite dimensional prototypes of certain infinite dimensional models describing the behavior of continuous wave operated or self-pulsating multisection DFB lasers under continuous or pulsating light injection, respectively

All-Optical Clock Recovery Using Multi-Section Distributed-Feedback Lasers

Setting In order to include other laser models, we consider the abstract equation u t = Au \Gamma\Omega \Gamma u + f(u) + fflh(!t; u; ffl) ; (3.2) where u belongs to a real Hilbert space X equipped with the scalar product h\Delta; \Deltai and norm k \Delta k. Furthermore, (ffl; !;\Omega ) 2 IR 3 are parameters, and t 0 denotes the time variable. The reader may think of u as comprising the carrier densities N and the electric field \Psi . The input power of the data stream is denoted by ffl, and the variables ! and\Omega correspond to the modulation and optical frequency, respectively. Equation (3.2) then describes the time evolution of the laser. Below, we state the hypotheses on (3.2). Most importantly, we will assume that there is a stable self-pulsating laser state, which can be observed experimentally. Before formulating this hypothesis, we assume that (3.2) is well-posed: Hypothesis (A1). A : X ! X is a closed operator with dense domain D(A) and generates a C 0 -sem..

Exponential Dichotomies for Solitary-Wave Solutions of Semilinear Elliptic Equations on Infinite Cylinders

In applications, solitary-wave solutions of semilinear elliptic equations \Deltau + g(u; ru) = 0 (x; y) 2 IR \Theta\Omega in infinite cylinders frequently arise as travelling waves of parabolic equations. As such, their bifurcations are an interesting issue. Interpreting elliptic equations on infinite cylinders as dynamical systems in x has proved very useful. Still, there are major obstacles in obtaining, for instance, bifurcation results similar to those for ordinary differential equations. In this article, persistence and continuation of exponential dichotomies for linear elliptic equations is proved. With this technique at hands, Lyapunov-Schmidt reduction near solitary waves can be applied. As an example, existence of shift dynamics near solitary waves is shown if a perturbation ¯h(x; u; ru) periodic in x is added

Numerical Computation of Solitary Waves in Infinite Cylindrical Domains

The numerical computation of solitary waves to semilinear elliptic equations in infinite cylindrical domains is investigated. Rather than solving on the infinite cylinder, the equation is approximated by a boundary-value problem on a finite cylinder. Convergence and stability results for this approach are given. It is also shown that Galerkin approximations can be used to compute solitary waves of the elliptic problem on the finite cylinder. In addition, it is demonstrated that the aforementioned procedures simplify in cases where the elliptic equation admits an additional reversibility structure. Finally, the theoretical predictions are compared with numerical computations. In particular, post buckling of an infinitely long cylindrical shell under axial compression is considered; it is shown numerically that, for a fixed spatial truncation, the error in the truncation scales with the length of the cylinder as predicted theoretically

Numerical Computation of Solitary Waves on Infinite Cylinders

The numerical computation of solitary waves to semilinear elliptic equations in infinite cylinders is investigated. Rather than solving on the infinite cylinder, the equation is approximated by a boundary-value problem on a finite cylinder. Convergence and stability results for this algorithm are given. In addition, it is shown that Galerkin approximations can be used to calculate solitary waves for the elliptic problem on the finite cylinder. The theoretical predictions are compared with numerical computations. In particular, post buckling of an infinitely long cylindrical shell under axial compression is considered; it is shown numerically that, for a fixed spatial truncation, the error in the truncation on the length of the cylinder scales in accordance with the theoretical predictions. Keywords: solitary wave, boundary-value problem, elliptic equation AMS subject classification: 65N12, 35B30, 35J55, 65N30 1 Introduction The numerical computation of solitary-wave solutions to el..