Computation of conservation laws for nonlinear lattices

An algorithm to compute polynomial conserved densities of polynomial nonlinear lattices is presented. The algorithm is implemented in Mathematica and can be used as an automated integrability test. With the code diffdens.m, conserved densities are obtained for several well-known lattice equations. For systems with parameters, the code allows one to determine the conditions on these parameters so that a sequence of conservation laws exist.Comment: To appear in Physica D, 17 pages, Latex, uses the style files elsart.sty and elsart12.st

Computation of conserved densities for systems of nonlinear differential-difference equations

A new method for the computation of conserved densities of nonlinear differential-difference equations is applied to Toda lattices and discretizations of the Korteweg-de Vries and nonlinear Schrodinger equations. The algorithm, which can be implemented in computer algebra languages such as Mathematica, can be used as an indicator of integrability.Comment: submitted to Phys. Lett A, 10 pages, late

Nonlinear Schrodinger Equations and N=1 Superconformal Algebra

By using AKNS scheme and soliton connection taking values in N=1 superconformal algebra we obtain new coupled super Nonlinear Schrodinger equations.Comment: 12 pages, no figures, LaTex fil

Unidirectional Propagation of an Ultra-Short Electromagnetic Pulse in a Resonant Medium with High Frequency Stark Shift

We consider in the unidirectional approximation the propagation of an ultra short electromagnetic pulse in a resonant medium consisting of molecules characterized by a transition operator with both diagonal and non-diagonal matrix elements. We find the zero-curvature representation of the reduced Maxwell-Bloch equations in the sharp line limit. This can be used to develop the inverse scattering transform method to solve these equations. Finally we obtain two types of exact traveling pulse solutions, one with the usual exponential decay and another with an algebraic decay.Comment: Latex, 10 pages, no figure

On the dynamics of rational solutions for 1-D generalized Volterra system

The Hirota bilinear formalism and soliton solutions for a generalized Volterra system is presented. Also, starting from the soliton solutions, we obtain a class of nonsingular rational solutions using the "long wave" limit procedure of Ablowitz and Satsuma, and appropriate "gauge" transformations. Their properties are also discussed and it is shown that these solutions interact elastically with no phase shift.Comment: 9 pages, Late

Passive mode-locking under higher order effects

The response of a passive mode-locking mechanism, where gain and spectral filtering are saturated with the energy and loss saturated with the power, is examined under the presence of higher order effects. These include third order dispersion, self-steepening and Raman gain. The locking mechanism is maintained even with these terms; mode-locking occurs for both the anomalous and normal regimes. In the anomalous regime, these perturbations are found to affect the speed but not the structure of the (locked) pulses. In fact, these pulses behave like solitons of a classical nonlinear Schrodinger equation and as such a soliton perturbation theory is used to verify the numerical observations. In the normal regime, the effect of the perturbations is small, in line with recent experimental observations. The results in the normal regime are verified mathematically using a WKB type asymptotic theory. Finally, bi-solitons are found to behave as dark solitons on top of a stable background and are significantly affected by these perturbations

Strong transmission and reflection of edge modes in bounded photonic graphene

The propagation of linear and nonlinear edge modes in bounded photonic honeycomb lattices formed by an array of rapidly varying helical waveguides is studied. These edge modes are found to exhibit strong transmission (reflection) around sharp corners when the dispersion relation is topologically nontrivial (trivial), and can also remain stationary. An asymptotic theory is developed that establishes the presence (absence) of edge states on all four sides, including in particular armchair edge states, in the topologically nontrivial (trivial) case. In the presence of topological protection, nonlinear edge solitons can persist over very long distances.Comment: 5 pages, 4 figures. Minor updates on the presentation and interpretation of results. The movies showing transmission and reflection of linear edge modes are available at https://www.youtube.com/watch?v=XhaZZlkMadQ and https://www.youtube.com/watch?v=R8NOw0NvRu

Integrable systems and modular forms of level 2

A set of nonlinear differential equations associated with the Eisenstein series of the congruent subgroup $\Gamma_0(2)$ of the modular group $SL_2(\mathbb{Z})$ is constructed. These nonlinear equations are analogues of the well known Ramanujan equations, as well as the Chazy and Darboux-Halphen equations associated with the modular group. The general solutions of these equations can be realized in terms of the Schwarz trianle function $S(0,0,1/2; z)$.Comment: PACS numbers: 02.30.Ik, 02.30.Hq, 02.10.De, 02.30.G

On an integrable discretization of the modified Korteweg-de Vries equation

We find time discretizations for the two ''second flows'' of the Ablowitz-Ladik hierachy. These discretizations are described by local equations of motion, as opposed to the previously known ones, due to Taha and Ablowitz. Certain superpositions of our maps allow a one-field reduction and serve therefore as valid space-time discretizations of the modified Korteweg-de Vries equation. We expect the performance of these discretizations to be much better then that of the Taha-Ablowitz scheme. The way of finding interpolating Hamiltonians for our maps is also indicated, as well as the solution of an initial value problem in terms of matrix factorizations.Comment: 23 pages, LaTe