Localised patterns emerging from a subcritical modulation instability are analysed by carrying the multiple-scales analysis beyond all orders. The model studied is the Swift-Hohenberg equation of nonlinear optics, which is equivalent to the classical Swift-Hohenberg equation with a quadratic and a cubic nonlinearity. Applying the asymptotic technique away from the Maxwell point first, it is shown how exponentially small terms determine the phase of the fast spatial oscillation with respect to their slow sechtype amplitude. In the vicinity of the Maxwell point, the beyond-all-orders calculation yields the "pinning range" of parameters where stable stationary fronts connect the homogeneous and periodic states. The full bifurcation diagram for localised patterns is then computed analytically, including snake and ladder bifurcation curves. This last step requires the matching of the periodic oscillation in the middle of a localised pattern both with an up-and a down-front. To this end, a third, super-slow spatial scale needs to be introduced, in which fronts appear as boundary layers. In addition, the location of the Maxwell point and the oscillation wave number of localised patterns are required to fourth-order accuracy in the oscillation amplitude.

The world around us, natural or man-made, is built and held together by solid materials. Understanding their behaviour is the task of solid mechanics, which is in turn applied to many areas, from earthquake mechanics to industry, construction to biomechanics. The variety of materials (metals, rocks, glasses, sand, flesh and bone) and their properties (porosity, viscosity, elasticity, plasticity) is reflected by the concepts and techniques needed to understand them: a rich mixture of mathematics, physics and experiment. These are all combined in this unique book, based on years of experience in research and teaching. Starting from the simplest situations, models of increasing sophistication are derived and applied. The emphasis is on problem-solving and building intuition, rather than a technical presentation of theory. The text is complemented by over 100 carefully-chosen exercises, making this an ideal companion for students taking advanced courses, or those undertaking research in this or related disciplines.

Synchronization due to a weak global coupling with time delay in a semiconductor laser array is investigated both in the cw and self-pulsing regimes. A generalized form of the Kuramoto phase equations is derived and discussed analytically. The time delay is shown to induce in-phase synchronization in all dynamical regimes. Another form of synchronization is found which leads to local extinction of self-pulsing in the array. PACS numbers: 42.65.Sf, 05.45.Xt, 42.55.Px, 42.60.Da Many models developed in physics, chemistry, and biology are formulated in terms of weakly coupled identical oscillators. Since signals propagate with a finite velocity, the coupling among the oscillators is, in principle, time delayed. The influence of this delay was studied recently for various models such as pulse-coupled oscillators [1] and phase models [2]. Multistability [3,4] and oscillator death [5] were found to be possible consequences of the delay.The focus of this Letter is on the nonlinear dynamics of globally coupled oscillators with time delay, illustrated by an array of semiconductor lasers (SCL). Aside from fundamental interest, this subject may also have technological applications. In arrays of semiconductor lasers, synchronizing the lasing elements in phase is of importance in order to obtain a large output power~j P j E j j 2 concentrated in a single-lobed far field pattern [6]. To this end, local coupling between the neighboring laser elements via overlapping evanescent fields was considered in [7]. More recently, local and global couplings were compared in the absence of delay [8] with the conclusion that global coupling is more efficient to achieve stationary in-phase operation. A similar result concerning nonstationary regimes has just been found numerically [9] for delayed global coupling via a feedback mirror. This is the situation we analyze in this Letter, for identical SCL.SCL are known to be extremely sensitive to optical feedback and to undergo a series of instabilities from selfpulsing to coherence collapse as the feedback strength is increased [10]. We show that in the cw regime, characterized by time independent laser intensities, the time delay induces bistability between in-phase and antiphase states by increasing the stability domain of in-phase operation. For larger coupling strength, in-phase cw operation can be destabilized in two ways. First, there is a degenerate Hopf bifurcation to antiphase self-pulsing that exists independently of the delay. Second, if the delay is at least comparable to the relaxation oscillation period of the solitary SCL, another, nondegenerate, Hopf bifurcation exists which leads to in-phase self-pulsing. To study this nonstationary regime, a set of dynamical equations that generalize the Kuramoto model [11] is derived. In these equations, each SCL is a two-frequency oscillator with one phase variable describing oscillations at the optical frequency and a second phase variable related to oscillations in the laser intensities at the relaxation frequency. Our main result, ob...

We analyze stationary fronts connecting uniform and periodic states emerging from a pattern-forming instability. The size of the resulting periodic domains cannot be predicted with weakly nonlinear methods.We show that what determine this size are exponentially small (but exponentially growing in space) terms. These can only be computed by going beyond all orders of the usual multiple-scale expansion. We apply the method to the Swift-Hohenberg equation and derive analytically a snaking bifurcation curve. At each fold of this bifurcation curve, a new pair of peaks is added to the periodic domain, which can thus be seen as a bound state of localized structures. Such scenarios have been reported with optical localized structures in nonlinear cavities and localized buckling.

We propose a new model for passive mode locking that is a set of ordinary delay differential equations. We assume a ring-cavity geometry and Lorentzian spectral filtering of the pulses but do not use small gain and loss and weak saturation approximations. By means of a continuation method, we study mode-locking solutions and their stability. We find that stable mode locking can exist even when the nonlasing state between pulses becomes unstable. © 2004 Optical Society of America OCIS codes: 140.4050, 140.5960, 190.3100. Passive mode locking (ML) of lasers is a very effective technique to generate high-quality short pulses with high repetition rates. Monolithic semiconductor lasers, passively or hybrid mode locked, are ideal for applications in high-speed telecommunications on account of their compactness, low cost, and reliability. 1The basic mechanism for passive ML is well understood since the analysis by New, 2 who showed that the differential saturation of the gain and losses in the laser cavity opens a short temporal window of net gain for pulses. A wide range of experimental, numerical, and analytical methods exist to characterize ML (for an overview, see Haus 3 and Avrutin et al. 4 ). Although numerical integrations of traveling-wave field equations coupled to material equations (distributed models) faithfully reproduce experimental observations, they offer little insight into the underlying dynamics. This is why analytical approaches based on lumped element models, mainly those introduced by New 2 and Haus and co-workers 3,5 -8 are still widely used. Inevitably, though, these approaches require certain approximations that in many cases are hardly satisf ied experimentally. New, for instance, assumed small gain and loss per cavity round trip and ignored spectral f iltering. Haus did take spectral filtering into consideration under the parabolic approximation and showed how even an infinite bandwidth alters ML stability. 5Further approximations, such as the assumption of weak saturation, had to be made during the process. Yet the agreement between analytical results and experimental data on the dye laser 7 motivated many studies of variations of Haus's model. 4 In this Letter we propose and discuss a new model for passive ML that is a set of ordinary delay differential equations (DDEs). In doing so we avoid the approximations of small gain and loss per cavity round trip and weak saturation; these do not hold well in semiconductor laser devices. On the other hand, as in most lumped element models, we neglect the spatial effects, such as spatial hole burning and self-interference of the pulse near the mirrors, inherent in a linear cavity. This amounts to considering a unidirectional lasing ring cavity. Absorbing, amplifying, and spectral filtering segments are placed in succession in the cavity. Let a͑t͒ be the field amplitude at the entrance of the absorber section [def ined such that ja͑t͒j 2 is the optical power]. The relations among the input and output fields in these segments are given by a 1 ͑t͒ e...