We study spatial and temporal solitons in the PT symmetric coupler with gain in one waveguide and loss in the other. Stability properties of the high-and low-frequency solitons are found to be completely determined by a single combination of the soliton's amplitude and the gain/loss coefficient of the waveguides. The unstable perturbations of the high-frequency soliton break the symmetry between its active and lossy components which results in a blowup of the soliton or a formation of a long-lived breather state. The unstable perturbations of the low-frequency soliton separate its two components in space blocking the power drainage of the active component and cutting the power supply to the lossy one. Eventually this also leads to the blowup or breathing.

The parametrically driven, damped NLS equation is shown to describe the dynamics of small-amplitude breathers of the easy-plane ferromagnet and the long Josephson junction under the influence of the parametric pumping and dissipation. The soliton solutions are found exactly and stability problem for the dissipative case is reduced rigorously to the one for the undamped soliton.

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Stability of optical gap solitons is analyzed within a coupled-mode theory. Lower intensity solitons are shown to always possess a vibration mode responsible for their long-lived oscillations. As the intensity of the soliton is increased, the vibration mode falls into resonance with two branches of the long-wavelength radiation producing a cascade of oscillatory instabilities of higher intensity solitons.[S0031-9007 (98)06265-6] PACS numbers: 42.65.Tg, 03.40.Kf, 05.45.+b, 75.30.Ds In the late 1970s and early 1980s, the theory of elementary particles [1] and condensed matter physics (in particular, the Su-Schrieffer-Heeger polyacetylene model [2]) stimulated a wide interest in particlelike solutions of classical spinor field equations. Recently there has been a remarkable upsurge of the interest; the localized solutions of spinorlike systems have made a comeback under the new name of gap solitons.Thanks to the gap in the linear spectrum, solitons in spinorlike systems can propagate without losing their energy to resonantly excited radiation waves [3,4]. An example of the gap-soliton bearing system is given by an optical fiber with periodically varying refractive index [3]; here the gap is produced by the Bragg reflection and resonance of the waves along the grating. Another class of gap solitons arises in two-wave resonant optical materials with a x ͑2͒ susceptibility and diatomic crystal lattices (see [4] and references therein). Finally, in the already mentioned polyacetylene model [2], the gap in the electron spectrum is due to the electron-phonon interaction and effective period doubling of the lattice.The aim of this Letter is to analyze the stability of gap solitons. Previous analytical studies of the spinor soliton stability faced serious obstacles (cf.[5]), while results of computer simulations were contradictory (cf. [6,7]). As a result, no stability or instability criterion is available to date. The main difficulty of the previous analyses was that they were all based on a postulate that stable solutions must render the energy minimum. In the actual fact, however, the minimality of energy is not necessary for stability in systems with indefinite metrics [8]. As far as optical gap solitons are concerned, they have been commonly deemed stable following recent computer simulations carried out for certain particular parameter values [9]. In this Letter we demonstrate that the gap solitons can be unstable, elucidate the mechanism of instability, and demarcate the stability/instability regions on the plane of their parameters.In nonlinear optics the gap solitons are usually analyzed within the coupled-mode theory [3] which reduces to a system of coupled equations for the amplitudes of the forward-and backward-propagating waves, i͑u t 1 u x ͒ 1 y 1 ͑jyj 2 1 rjuj 2 ͒u 0 , i͑y t 2 y x ͒ 1 u 1 ͑juj 2 1 rjyj 2 ͒y 0 .In the periodic Kerr medium one typically has r 1͞2 [3]; in other problems of the fiber optics r may range up to infinity [10]. In the case r 0 Eqs. (1) yield the massive Thirring model of the field the...

For most discretizations of the phi4 theory, the stationary kink can only be centered either on a lattice site or midway between two adjacent sites. We search for exceptional discretizations that allow stationary kinks to be centered anywhere between the sites. We show that this translational invariance of the kink implies the existence of an underlying one-dimensional map phi(n+1) =F (phi(n)) . A simple algorithm based on this observation generates three families of exceptional discretizations.