Rogue waves on the background of periodic standing waves in the derivative nonlinear Schrödinger equation.

The derivative nonlinear Schrödinger (DNLS) equation is the canonical model for the dynamics of nonlinear waves in plasma physics and optics. We study exact solutions describing rogue waves on the background of periodic standing waves in the DNLS equation. We show that the space-time localization of a rogue wave is only possible if the periodic standing wave is modulationally unstable. If the periodic standing wave is modulationally stable, the rogue wave solutions degenerate into algebraic solitons propagating along the background and interacting with the periodic standing waves. Maximal amplitudes of rogue waves are found analytically and confirmed numerically.

Rogue waves on the double-periodic background in the focusing nonlinear Schrödinger equation.

The double-periodic solutions of the focusing nonlinear Schrödinger equation have been previously obtained by the method of separation of variables. We construct these solutions by using an algebraic method with two eigenvalues. Furthermore, we characterize the Lax spectrum for the double-periodic solutions and analyze rogue waves arising on the background of such solutions. Magnification of the rogue waves is studied numerically.

Rogue periodic waves of the focusing nonlinear Schrödinger equation.

Rogue periodic waves stand for rogue waves on a periodic background. The nonlinear Schrödinger equation in the focusing case admits two families of periodic wave solutions expressed by the Jacobian elliptic functions dn and cn. Both periodic waves are modulationally unstable with respect to long-wave perturbations. Exact solutions for the rogue periodic waves are constructed by using the explicit expressions for the periodic eigenfunctions of the Zakharov-Shabat spectral problem and the Darboux transformations. These exact solutions generalize the classical rogue wave (the so-called Peregrine's breather). The magnification factor of the rogue periodic waves is computed as a function of the elliptic modulus. Rogue periodic waves constructed here are compared with the rogue wave patterns obtained numerically in recent publications.

Spectral stability of periodic waves in the generalized reduced Ostrovsky equation.

We consider stability of periodic travelling waves in the generalized reduced Ostrovsky equation with respect to co-periodic perturbations. Compared to the recent literature, we give a simple argument that proves spectral stability of all smooth periodic travelling waves independent of the nonlinearity power. The argument is based on the energy convexity and does not use coordinate transformations of the reduced Ostrovsky equations to the semi-linear equations of the Klein-Gordon type.

Energy Criterion for the Spectral Stability of Discrete Breathers.

Discrete breathers are ubiquitous structures in nonlinear anharmonic models ranging from the prototypical example of the Fermi-Pasta-Ulam model to Klein-Gordon nonlinear lattices, among many others. We propose a general criterion for the emergence of instabilities of discrete breathers analogous to the well-established Vakhitov-Kolokolov criterion for solitary waves. The criterion involves the change of monotonicity of the discrete breather's energy as a function of the breather frequency. Our analysis suggests and numerical results corroborate that breathers with increasing (decreasing) energy-frequency dependence are generically unstable in soft (hard) nonlinear potentials.

Wave systems with an infinite number of localized traveling waves.

In many wave systems, propagation of steadily traveling solitons or kinks is prohibited because of resonances with linear excitations. We show that wave systems with resonances may admit an infinite number of traveling solitons or kinks if the closest to the real axis singularities of a limiting asymptotic solution in the complex upper half plane are of the form z±=±α+iß, α≠0. This quite general statement is illustrated by examples of the fifth-order Korteweg-de Vries equation, the discrete cubic-quintic Klein-Gordon equation, and the nonlocal double sine-Gordon equations.

Paradoxical transitions to instabilities in hydromagnetic Couette-Taylor flows.

By methods of modern spectral analysis, we rigorously find distributions of eigenvalues of linearized operators associated with an ideal hydromagnetic Couette-Taylor flow. The transition to instability in the limit of a vanishing magnetic field has a discontinuous change compared to the Rayleigh stability criterion for hydrodynamical flows, which is known as the Velikhov-Chandrasekhar paradox.

Feshbach resonance management of Bose-Einstein condensates in optical lattices.

We analyze gap solitons in trapped Bose-Einstein condensates (BECs) in optical lattice potentials under Feshbach resonance management. Starting with an averaged Gross-Pitaevsky equation with a periodic potential, we employ an envelope-wave approximation to derive coupled-mode equations describing the slow BEC dynamics in the first spectral gap of the optical lattice. We construct exact analytical formulas describing gap soliton solutions and examine their spectral stability using the Chebyshev interpolation method. We show that these gap solitons are unstable far from the threshold of local bifurcation and that the instability results in the distortion of their shape. We also predict the threshold of the power of gap solitons near the local bifurcation limit.

Solitons in nonintegrable systems.

We introduce the concept of this special focus issue on solitons in nonintegrable systems. A brief overview of some recent developments is provided, and the various contributions are described. The topics covered in this focus issue are the modulation of solitons, bores, and shocks, the dynamical evolution of solitary waves, and existence and stability of solitary waves and embedded solitons.

Stability analysis of embedded solitons in the generalized third-order nonlinear Schrodinger equation.

We study the generalized third-order nonlinear Schrodinger (NLS) equation which admits a one-parameter family of single-hump embedded solitons. Analyzing the spectrum of the linearization operator near the embedded soliton, we show that there exists a resonance pole in the left half-plane of the spectral parameter, which explains linear stability, rather than nonlinear semistability, of embedded solitons. Using exponentially weighted spaces, we approximate the resonance pole both analytically and numerically. We confirm in a near-integrable asymptotic limit that the resonance pole gives precisely the linear decay rate of parameters of the embedded soliton. Using conserved quantities, we qualitatively characterize the stable dynamics of embedded solitons.

Translationally invariant discrete kinks from one-dimensional maps.

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.

Oscillations of dark solitons in trapped Bose-Einstein condensates.

We consider a one-dimensional defocusing Gross-Pitaevskii equation with a parabolic potential. Dark solitons oscillate near a center of the potential trap and their amplitude decays due to radiative losses (sound emission). We develop a systematic asymptotic multiscale expansion method in the limit when the potential trap is flat. The first-order approximation predicts a uniform frequency of oscillations for the dark soliton of arbitrary amplitude. The second-order approximation predicts the nonlinear growth rate of the oscillation amplitude, which results in decay of the dark soliton. The results are compared with previous publications and numerical computations.

Bifurcations and stability of gap solitons in periodic potentials.

We analyze the existence, stability, and internal modes of gap solitons in nonlinear periodic systems described by the nonlinear Schrödinger equation with a sinusoidal potential, such as photonic crystals, waveguide arrays, optically-induced photonic lattices, and Bose-Einstein condensates loaded onto an optical lattice. We study bifurcations of gap solitons from the band edges of the Floquet-Bloch spectrum, and show that gap solitons can appear near all lower or upper band edges of the spectrum, for focusing or defocusing nonlinearity, respectively. We show that, in general, two types of gap solitons can bifurcate from each band edge, and one of those two is always unstable. A gap soliton corresponding to a given band edge is shown to possess a number of internal modes that bifurcate from all band edges of the same polarity. We demonstrate that stability of gap solitons is determined by location of the internal modes with respect to the spectral bands of the inverted spectrum and, when they overlap, complex eigenvalues give rise to oscillatory instabilities of gap solitons.

Stable vortex and dipole vector solitons in a saturable nonlinear medium.

We study both analytically and numerically the existence, uniqueness, and stability of vortex and dipole vector solitons in a saturable nonlinear medium in (2+1) dimensions. We construct perturbation series expansions for the vortex and dipole vector solitons near the bifurcation point, where the vortex and dipole components are small. We show that both solutions uniquely bifurcate from the same bifurcation point. We also prove that both vortex and dipole vector solitons are linearly stable in the neighborhood of the bifurcation point. Far from the bifurcation point, the family of vortex solitons becomes linearly unstable via oscillatory instabilities, while the family of dipole solitons remains stable in the entire domain of existence. In addition, we show that an unstable vortex soliton breaks up either into a rotating dipole soliton or into two rotating fundamental solitons.

Persistent oscillations of scalar and vector dispersion-managed solitons.

We show that both orthogonal and parallel internal modes exist on the background of a dispersion-managed (DM) soliton in randomly birefringent fibers. The orthogonal modes exist for arbitrarily small values of the dispersion map strength, while the parallel modes exist only when the map strength exceeds a certain threshold value. We demonstrate that initial perturbations of a DM soliton's profile that consist of one or more internal modes, exhibit nearly stable oscillations over very long propagation distances, before decaying into radiation. (c) 2000 American Institute of Physics.