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Mark J Ablowitz

Publications and source records attributed to Mark J Ablowitz.

9 recordsLinked to original sources
PubMed2005

Spectral renormalization method for computing self-localized solutions to nonlinear systems.

A new numerical scheme for computing self-localized states--or solitons--of nonlinear waveguides is proposed. The idea behind the method is to transform the underlying equation governing the soliton, such as a nonlinear Schrödinger-type equation, into Fourier space and determine a nonlinear nonlocal integral equation coupled to an algebraic equation. The coupling prevents the numerical scheme from diverging. The nonlinear guided mode is then determined from a convergent fixed point iteration scheme. This spectral renormalization method can find wide applications in nonlinear optics and related fields such as Bose-Einstein condensation and fluid mechanics.

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PubMed2005

Wave dynamics in optically modulated waveguide arrays.

A model describing wave propagation in optically modulated waveguide arrays is proposed. In the weakly guided regime, a two-dimensional semidiscrete nonlinear Schrödinger equation with the addition of a bulk diffraction term and an external "optical trap" is derived from first principles, i.e., Maxwell equations. When the nonlinearity is of the defocusing type, a family of unstaggered localized modes are numerically constructed. It is shown that the equation with an induced potential is well-posed and gives rise to localized dynamically stable nonlinear modes. The derived model is of the Gross-Pitaevskii type, a nonlinear Schrödinger equation with a linear optical potential, which also models Bose-Einstein condensates in a magnetic trap.

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PubMed2004

Reduction of collision-induced timing shifts in dispersion-managed quasi-linear systems with periodic-group-delay dispersion compensation.

Periodic-group-delay (PGD) dispersion-compensation modules were recently proposed as mechanisms to alleviate collision-induced timing shifts in dispersion-managed (DM) systems. Frequency and timing shifts in quasi-linear DM systems with PGDs were obtained, and it is shown that significant reductions are achieved when even a small fraction of the total dispersion is compensated for by PGDs.

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PubMed2004

Carrier-envelope phase slip of ultrashort dispersion-managed solitons.

The carrier-envelope phase slip of an ultrashort pulse circulating in a mode-locked Ti:sapphire laser is analyzed. The laser cavity is modeled by a dispersion- and nonlinearity-managed nonlinear Schrödinger equation. The combined contributions to the phase slip induced by nonlinear phase and nonlinear dispersion are found to approach zero for strong dispersion maps. The dependence of the slip on third-order dispersion is found as well. The analytical results are verified using numerical simulations.

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PubMed2004

Dispersion management for randomly varying optical fibers.

An asymptotic theory for optical pulse propagation in a dispersion-managed (DM) fiber with random dispersion is presented. The validity of the theory is verified with direct numerical simulation. The equations that describe the slow evolution of initial pulses have special solutions that, for fibers with moderate noise in the dispersion profile, perform much better than ideal DM solitons optimized for the unperturbed fiber.

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PubMed2003

Dark and gray strong dispersion-managed solitons.

Dark and gray solitons in communication systems with strong dispersion management (DM) are obtained. These new modes are characterized by a decaying oscillatory background. Unlike the bright DM solitons in which the oscillations are observed only on a logarithmic scale, here the oscillations are dominant on a linear scale and become very strong for moderate map strength.

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PubMed2002

Discrete vector spatial solitons in a nonlinear waveguide array.

A vector discrete diffraction managed soliton system is introduced. The vector model describes propagation of two polarization modes interacting in a nonlinear waveguide array with varying diffraction via the cross-phase modulation coupling. In the limit of strong diffraction we derive averaged equations governing the slow dynamics of the beam's amplitudes, and their stationary (in the form of bright-bright vector bound state) and traveling wave solutions are found. Through an extensive series of direct numerical simulations, interactions between diffraction-managed solitons for different values of velocities, diffraction, and cross-phase modulation coefficient are studied. We compare each collision case with its classical counterpart (constant diffraction) and find that in both the scalar and vector diffraction management cases, the interaction picture involves beam shaping, fusion, fission, nearly elastic collisions, and, in some cases, multihump structures. The collision scenario is found, in both the scalar and vector diffraction managed cases, to be rather different from the classical case.

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PubMed2002

Methods for discrete solitons in nonlinear lattices.

A method to find discrete solitons in nonlinear lattices is introduced. Using nonlinear optical waveguide arrays as a prototype application, both stationary and traveling-wave solitons are investigated. In the limit of small wave velocity, a fully discrete perturbative analysis yields formulas for the mode shapes and velocity.

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