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It is commonly held that a necessary condition for the existence of solitons in nonlinear-wave systems is that the soliton's frequency (spatial or temporal) must not fall into the continuous spectrum of radiation modes. However, this is not always true. We present a new class of codimension-one solitons (i.e., those existing at isolated frequency values) that are embedded into the continuous spectrum. This is possible if the spectrum of the linearized system has (at least) two branches, one corresponding to exponentially localized solutions, and the other to radiation modes. An embedded soliton (ES) is obtained when the latter component exactly vanishes in the solitary-wave's tail. The paper contains both a survey of recent results obtained by the authors and some new results, the aim being to draw together several different mechanism underlying the existence of ESs. We also consider the distinctive property of semi-stability of ES, and moving ESs. Results are presented for four different physical models, including an extended 5th-order KdV equation describing surface waves in inviscid fluids, and three models from nonlinear optics. One of them pertains to a resonant Bragg grating in an optical fiber with a cubic nonlinearity, while two others describe second-harmonic generation (SHG) in the temporal or spatial domain (i.e., respectively, propagating pulses in nonlinear optical fibers, or stationary patterns in nonlinear planar waveguides). Special attention is paid to the SHG model in the temporal domain for a case of competing quadratic and cubic nonlinearities. An essential new result is that ES is, virtually, fully stable in the latter model in the case when both harmonics have anomalous dispersion.

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The stability of multiple-pulse solutions to the discrete nonlinear Schrödinger equation is considered. A bound state of widely separated single pulses is rigorously shown to be unstable, unless the phase shift Delta phi between adjacent pulses satisfies Delta phi=pi. This instability is accounted for by positive real eigenvalues in the linearized system. The analysis leading to the instability result does not, however, determine the linear stability of those multiple pulses for which Delta phi=pi between adjacent pulses. A direct variational approach for a two-pulse predicts that it is linearly stable if Delta phi=pi, and if the separation between the individual pulses satisfies a certain condition. The variational approach can easily be generalized to study the stability of N pulses for any N>or=3. The analysis is supplemented with a detailed numerical stability analysis.

The interaction between a pair of Bloch fronts forming a traveling domain in a bistable medium is studied. A parameter range beyond the nonequilibrium Ising-Bloch bifurcation is found where traveling domains collapse. Only beyond a second threshold the repulsive front interactions become strong enough to balance attractive interactions and asymmetries in front speeds, and form stable traveling pulses. The analysis is carried out for the forced complex Ginzburg-Landau equation. Similar qualitative behavior is found in the bistable FitzHugh-Nagumo model. PACS number(s):Traveling waves far from equilibrium are often formed when a uniform state is destabilized by a Hopf bifurcation occurring at a finite wavenumber [1]. Traveling waves or pulses also form from parity breaking bifurcations of stationary patterns [2]. A related mechanism that has not received adequate attention involves a parity breaking front bifurcation in which a stationary front solution loses stability to a pair of counter-propagating front solutions [3][4][5][6]. This bifurcation, sometimes referred to as the nonequilibrium Ising-Bloch (NIB) bifurcation, has been found in chemical reactions [7,8] and in liquid crystals [9,10]. Bistable systems, which do not necessarily support stationary patterns, may exhibit traveling pulses and waves beyond the NIB bifurcation. Activator-inhibitor systems with non-diffusing inhibitors provide a good example. For fast inhibitor kinetics initial domain patterns always coarse grain and converge to a uniform state. For sufficiently slow kinetics, and beyond the NIB bifurcation, traveling pulses, periodic wavetrains, and spiral waves appear.Numerical studies of systems with a NIB bifurcation indicate that traveling pulses do not appear immediately at the front bifurcation point. Instead, there is an intermediate parameter range where initial domains may travel but eventually collapse. Only past a second threshold parameter value do initial domains converge to stable traveling pulses [5]. In this paper we study the interactions between a pair of traveling fronts in this intermediate parameter range. We find that the balance of repulsive front interactions with attractive interactions and an asymmetry between leading and trailing fronts gives this threshold parameter value.We choose to study the parametrically forced complex Ginzburg-Landau (CGL) equationwhere A(x, t) is a complex field and ν, c 1 , c 3 and γ are real parameters. The parameter α can be a complex number, but since the final results we present here do not depend on its imaginary part we assume α is also real [11]. The CGL equation (γ = α = 0) is often obtained as an envelope equation for an extended system undergoing a Hopf bifurcation at zero wavenumber [12]. Then, the variable A(x, t) describes weak modulations of the homogeneous oscillations. The terms α and γA * in (1) represent, respectively, the effect of parametric forcing with equal and twice the system's natural oscillation frequency [13,14]. Equation (1) has been introduced recently in the con...

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We study the stability and interactions of chirped solitary pulses in a system of nonlinearly coupled cubic Ginzburg-Landau (CGL) equations with a group-velocity mismatch between them, where each CGL equation is stabilized by linearly coupling it to an additional linear dissipative equation. In the context of nonlinear fiber optics, the model describes transmission and collisions of pulses at different wavelengths in a dual-core fiber, in which the active core is furnished with bandwidth-limited gain, while the other, passive (lossy) one is necessary for stabilization of the solitary pulses. Complete and incomplete collisions of pulses in two channels in the cases of anomalous and normal dispersion in the active core are analyzed by means of perturbation theory and direct numerical simulations. It is demonstrated that the model may readily support fully stable pulses whose collisions are quasielastic, provided that the group-velocity difference between the two channels exceeds a critical value. In the case of quasielastic collisions, the temporal shift of pulses, predicted by the analytical approach, is in semiquantitative agreement with direct numerical results in the case of anomalous dispersion (in the opposite case, the perturbation theory does not apply). We also consider a simultaneous collision between pulses in three channels, concluding that this collision remains quasielastic, and the pulses remain completely stable. Thus, the model may be a starting point for the design of a stabilized wavelength-division-multiplexed transmission system.

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