Incomplete collisions in strongly dispersion-managed return-to-zero communication systems

Incomplete collisions in wavelength-division-multiplexed return-to-zero transmission systems are analyzed by asymptotic methods. Formulas for frequency and timing shifts are obtained. The results agree with direct numerical calculations.     

Long Internal Waves in Fluids of Great Depth

An equation is derived that governs the evolution in two spatial dimensions of long internal waves in fluids of great depth. The equation is a natural generalization of Benjamin's (1967) one-dimensional equation, and relates to it in the same way that the equation of Kadomtsev and Petviashvili relates to the Kortewegde-Vries equation. The stability of one-dimensional solitons with respect to long transverse disturbances is studied in the context of this equation. Solitons are found to be unstable with respect to such perturbations in any system in which the phase speed is a minimum (rather than a maximum) for the longest waves. Internal waves do not have this property, and are not unstable with respect to such perturbations.     

Unified Orbital Description of the Envelope Dynamics in Two‐Dimensional Simple Periodic Lattices

The propagation of wave envelopes in two‐dimensional (2‐D) simple periodic lattices is studied. A discrete approximation, known as the tight‐binding (TB) approximation, is employed to find the equations governing a class of nonlinear discrete envelopes in simple 2‐D periodic lattices. Instead of using Wannier function analysis, the orbital approximation of Bloch modes that has been widely used in the physical literature, is employed. With this approximation the Bloch envelope dynamics associated with both simple and degenerate bands are readily studied. The governing equations are found to be discrete nonlinear Schrödinger (NLS)‐type equations or coupled NLS‐type systems. The coefficients of the linear part of the equations are related to the linear dispersion relation. When the envelopes vary slowly, the continuous limit of the general discrete NLS equations are effective NLS equations in moving frames. These continuous NLS equations (from discrete to continuous) also agree with those derived via a direct multiscale expansion. Rectangular and triangular lattices are examples.     

Initial Time Layers and Kadomtsev–Petviashvili-Type Equations

The Kadomtsev–Petviashvili (KP) equation and generalizations (GKP) have temporal discontinuities at the initial instant of time. Motivated by the study of water waves, a generalized Boussinesq equation that contains the GKP equations as an "outer" limit is introduced. Within the context of matched asymptotic expansions the discontinuities are resolved. The linear system is analyzed in more detail and the limit process is rigorously established.     

A Multidimensional Inverse-Scattering Method

A formal solution of the inverse scattering problem for the n-dimensional; time-dependent and time-independent Schrödinger equations is given. Equations are found for reconstructing the potential from scattering data purely by quadratures. The solution also helps elucidate the problem of characterizing admissible scattering data.     

Soliton strings and interactions in mode-locked lasers

Soliton strings in mode-locked lasers are obtained using a variant of the nonlinear Schrödinger equation, appropriately modified to model power (intensity) and energy saturation. This equation goes beyond the well-known master equation often used to model these systems. It admits mode-locking and soliton strings in both the constant dispersion and dispersion-managed systems in the (net) anomalous and normal regimes; the master equation is contained as a limiting case. Analysis of soliton interactions show that soliton strings can form when pulses are a certain distance apart relative to their width. Anti-symmetric bi-soliton states are also obtained. Initial states mode-lock to these states under evolution. In the anomalous regime individual soliton pulses are well approximated by the solutions of the unperturbed nonlinear Schrödinger equation, while in the normal regime the pulses are much wider and strongly chirped.     

Piston dispersive shock wave problem

The piston shock problem is a classical result of shock wave theory. In this work, the analogous dispersive shock wave (DSW) problem for a fluid described by the nonlinear Schr?dinger equation is analyzed. Asymptotic solutions are calculated for a piston (step potential) moving with uniform speed into a dispersive fluid at rest. In contrast to the classical case, there is a bifurcation of shock behavior where, for large enough piston velocities, the DSW develops a periodic wave train in its wake with vacuum points and a maximum density that remains fixed as the piston velocity is increased further. These results have application to Bose-Einstein condensates and nonlinear optics.     

Interactions of dispersive shock waves

Collisions and interactions of dispersive shock waves in defocusing (repulsive) nonlinear Schrödinger type systems are investigated analytically and numerically. Two canonical cases are considered. In one case, two counterpropagating dispersive shock waves experience a head-on collision, interact and eventually exit the interaction region with larger amplitudes and altered speeds. In the other case, a fast dispersive shock overtakes a slower one, giving rise to an interaction. Eventually the two merge into a single dispersive shock wave. In both cases, the interaction region is described by a modulated, quasi-periodic two-phase solution of the nonlinear Schrödinger equation. The boundaries between the background density, dispersive shock waves and their interaction region are calculated by solving the Whitham modulation equations. These asymptotic results are in excellent agreement with full numerical simulations. It is further shown that the interactions of two dispersive shock waves have some qualitative similarities to the interactions of two classical shock waves.