Exponential dichotomy and smooth invariant center manifolds for semilinear hyperbolic systems

Es wird gezeigt, dass ein Satz über die Abbildung spektraler Lücken, welcher exponentielle Dichotomie charakterisiert, für eine allgemeine Klasse (SH) von semilinearen hyperbolischen Systemen von partiellen Differentialgleichungen in einem Banach-Raum X von stetigen Funktionen gilt. Dies beantwortet ein Schlüsselproblem für die Existenz und Glattheit invarianter Mannigfaltigkeiten semilinearer hyperbolischer Systeme. Unter natürlichen Annahmen an die Nichtlinearitäten wird gezeigt, dass schwache Lösungen von (SH) einen glatten Halbfluß im Raum X bilden. Für Linearisierungen werden hochfrequente Abschätzungen für Spektren sowie Resolventen unter Verwendung von reduzierten (block)diagonal Systemen hergestellt. Darauf aufbauend wird der Abbildungssatz für spektrale Lücken im kleinen Raum X bewiesen: Eine offene spektrale Lücke des Generators wird exponentiell auf eine offene spektrale Lücke der Halbruppe abgebildet und umgekehrt. Es folgt, dass ein Phänomen wie im Gegenbeispiel von Renardy nicht auftreten kann. Unter Verwendung der allgemeinen Theorie implizieren die Ergebnisse die Existenz von glatten Zentrumsmannigfaltigkeiten für (SH). Die Ergebnisse werden auf traveling wave Modelle für die Dynamik von Halbleiter Lasern angewandt. Für diese werden Moden Approximationen (Systeme von gewöhnlichen Differentialgleichungen, welche die Dynamik auf gewissen Zentrumsmannigfaltigkeiten approximativ beschreiben) hergeleitet und gerechtfertigt, die generische Bifurkation von modulierten Wellen aus rotierenden Wellen wird gezeigt. Globale Existenz und glatte Abhängigkeit von nichtautonomen traveling wave Modellen werden betrachtet, außerdem werden Moden Approximationen für solche nichtautonomen Modelle rigoros hergeleitet. Insbesondere arbeitet die Theorie für die Stabilitäts- und Bifurkationsanalyse von Turing Modellen mit korellierter Zufallsbewegung. Ferner beinhaltet die Klasse (SH) neutrale und retardierte funktionale Differentialgleichungen.A spectral gap mapping theorem, which characterizes exponential dichotomy, is proven for a general class of semilinear hyperbolic systems of PDEs in a Banach space X of continuous functions. This resolves a key problem on existence and smoothness of invariant manifolds for semilinear hyperbolic systems. It is shown that weak solutions to (SH) form a smooth semiflow in X under natural conditions on the nonlinearities. For linearizations high frequency estimates of spectra and resolvents in terms of reduced diagonal and blockdiagonal systems are given. Using these estimates a spectral gap mapping theorem in the small Banach space X is proven: An open spectral gap of the generator is mapped exponentially to an open spectral gap of the semigroup and vice versa. Hence, a phenomenon like in Renardy''s counterexample cannot appear for linearizations of (SH). By the general theory the results imply existence of smooth center manifolds for (SH). Moreoever, the results are applied to traveling wave models of semiconductor laser dynamics. For such models mode approximations (ODE systems which approximately describe the dynamics on center manifolds) are derived and justified, and generic bifurcations of modulated waves from rotating waves are shown. Global existence and smooth dependence of nonautonomous traveling wave models with more general solutions, which possess jumps, are considered, and mode approximations are derived for such nonautonomous models. In particular the theory applies to stability and bifurcation analysis for Turing models with correlated random walk. Moreover, the class (SH) includes neutral and retarded functional differential equations

Parallel simulation of high power semiconductor lasers

High power tapered semiconductor lasers are characterized by a huge amount of structural and geometrical design parameters and are subject to time-space instabilities like pulsations, self-focussing, filamentation and thermal lensing which yield restrictions to output power, beam quality and wavelength stability. Numerical simulations are an important tool for finding optimal design parameters, understanding the complicated dynamical behavior and for predicting new laser designs. We present fast dynamic high performance parallel simulation results based on traveling wave equations which are suitable for model calibration and parameter scanning of the long time dynamics in reasonable time. Simulation results are compared to experimental data

Between cohesion and division: reconciling the faultines of Europe’s past

Despite movement towards integration in the form of a shared currency and political institutions over the last 20 years, Europe shows signs of slipping back into populism and rancour. But do the faultlines of Europe’s past make full reconciliation impossible? Giacomo Lichtner, Mark Seymour, Maartja Abbenhuis explore this possibility, arguing that doing so is a necessity if the continent’s functional cohesion, anduniquely social-democratic vision, are to be sustained into the future

Variation of constants formula for hyperbolic systems

A smooth variation of constants formula for semilinear hyperbolic systems is established using a suitable Banach space $X$ of continuous functions together with its sun dual space $X^{\odot \ast}$. It is shown that mild solutions of this variation of constants formula generate a smooth semiflow in $X$. This proves that the stability of stationary states for the nonlinear flow is determined by the stability of the linearized semigroup

Spectral mapping theorem for linear hyperbolic systems

We prove spectral mapping theorem for linear hyperbolic systems of PDEs. The system is of the following form: For $0 0$ $${\rm{(H)}} \quad \left \{ \begin{array}{l} \displaystyle {\partial \over {\partial t}} \begin{pmatrix} u(t,x) \\ v(t,x) \end{pmatrix} + K(x) {\partial \over {\partial x}} \begin{pmatrix} u(t,x) \\ v(t,x) \end{pmatrix} + C(x) \begin{pmatrix} u(t,x) \\ v(t,x) \end{pmatrix} = 0, \\ \displaystyle {d \over {dt}} \left [ v(t,l) - D u(t,l) \right ] = F u(t,\cdot) + G v(t,\cdot) , \\ \displaystyle u(t,0) = E v(t,0), \end{array} \right .$$ where u(t,x) \in \C^{n_1}, v(t,x) \in \C^{n_2}, $K(x) = \mathrm{diag} \, \left( k_i(x) \right )_{1 \le i \le n}$ is a diagonal matrix of functions $k_i \in C^1\left( [0,l], \R \right)$, $k_i(x) > 0$ for $i = 1, \dots, n_1$ and $k_i(x) < 0$ for $i = n_1+1, \dots, n=n_1+n_2$, and $D$,$E$ are matrices. We show high frequency estimates of spectra and resolvents in terms of reduced (block)diagonal systems. Let $A$ denote the infinitesimal generator for $\mathrm{(H)}$ which generates $C_0$ semigroup $e^{At}$ on L^2 \times \C^{n_2}. Our main result is the following spectral mapping theorem $$\sigma(e^{At}) \setminus \{ 0 \} = \overline{e^{\sigma(A)t}} \setminus \{ 0 \}.$

Principle of linearized stability and smooth center manifold theorem for semilinear hyperbolic systems

We prove principle of linearized stability and smooth center manifold theorem for a general class of semilinear hyperbolic systems $\mathrm{(SH)}$ in one space dimension, which are of the following form: For $0 0$ $$\mathrm{(SH)} \left \{ \begin{array}{l} {\partial \over {\partial t}} \begin{pmatrix} u(t,x) \\ v(t,x) \end{pmatrix} + K(x) {\partial \over {\partial x}} \begin{pmatrix} u(t,x) \\ v(t,x) \end{pmatrix} + H(x, u(t,x), v(t,x)) = 0, \\ {d \over {dt}} \left [ v(t,l) - D u(t,l) \right ] = F(u(t,\cdot),v(t,\cdot)), \\ u(t,0) = E \, v(t,0), \\ u(0,x) = u_0(x), \; v(0,x) = v_0(x), \end{array} \right .$$ where $u(t,x) \in \R^{n_1}$, $v(t,x) \in \R^{n_2}$, $K(x) = \mathrm{diag} \, \left( k_i(x) \right )_{1 \le i \le n}$ is a diagonal matrix of functions $k_i \in C^1\left( [0,l], \R \right)$, $k_i(x) > 0$ for $i = 1, \dots, n_1$ and $k_i(x) < 0$ for $i = n_1+1, \dots, n=n_1+n_2$, and $D$,$E$ are matrices. First we prove that weak solutions to $\mathrm{(SH)}$ form a smooth semiflow in a Banach space $X$ of continuous functions under natural conditions on the nonlinearities $H$ and $F$. Then we show a spectral gap mapping theorem for linearizations of $\mathrm{(SH)}$ in the complexification of $X$, which implies that growth and spectral bound coincide. Consequently we obtain principle of linearized stability for $\mathrm{(SH)}$. Moreover, the spectral gap mapping theorem characterizes exponential dichotomy in terms of a spectral gap of the infinitesimal generator for linearized hyperbolic systems. This resolves a key problem in applying invariant manifold theory to prove smooth center manifold theorem for $\mathrm{(SH)}$

Improvement of output beam quality in broad area lasers with off-axis feedback

We report a method to improve the beam quality of broad area lasers by using a V-shaped external cavity formed by two off-axis feedback mirrors that allow to select a single transverse mode with the intensity modulated in the transverse direction. We find that in the case when one of the two feedback mirrors is absent a spontaneous formation of self-induced transverse population grating leading to a reduction of the lasing threshold is observed. Most favorable conditions for stabilization of single transverse supermode and creation of a high power and high brightness plane wave traveling in the extended cavity are obtained for equal re ectivities of the two external reflectors

Improving the stability of distributed-feedback tapered master-oscillator power-amplifiers

We report theoretical results on the wavelength stabilization in distributed-feedback master-oscillator power-amplifiers which are compact semiconductor laser devices capable of emitting a high brilliance beam at an optical power of several Watts. Based on a traveling wave equation model we calculate emitted optical power and spectral maps in dependence on the pump of the power amplifier. We show that a proper choice of the Bragg grating type and coupling coefficient allows to optimize the laser operation, such that for a wide range of injection currents the laser emits a high intensity continuous wave beam

The spectrum of delay differential equations with large delay

We show that the spectrum of linear delay differential equations with large delay splits into two different parts. One part, called the strong spectrum, converges to isolated points when the delay parameter tends to infinity. The other part, called the pseudocontinuous spectrum, accumulates near criticality and converges after rescaling to a set of spectral curves, called the asymptotic continuous spectrum. We show that the spectral curves and strong spectral points provide a complete description of the spectrum for sufficiently large delay and can be comparatively easily calculated by approximating expressions

Mode transitions in distributed-feedback tapered master-oscillator power-amplifier

Theoretical and experimental investigations have been carried out to study the spectral and spatial behavior of monolithically integrated distributed-feedback tapered master-oscillators power-amplifiers emitting around 973 nm. Introduction of self and cross heating effects and the analysis of longitudinal optical modes allows us to explain experimental results. The results show a good qualitative agreement between measured and calculated characteristics