Bright and dark rogue internal waves: The Gardner equation approach.

We have found "bright and dark" solutions of the Gardner equation which can model internal rogue waves in three-layer fluids. We provide the first four "bright" and "dark" exact rational solutions to the Gardner equation. These are the lowest-order solutions of the corresponding hierarchies of rogue-wave solutions of this equation. They have been obtained from the rogue-wave solutions of a modified Korteweg-de Vries equation by using a Lorentz-type transformation. The maximal (and minimal) amplitudes and the background levels of these solutions for arbitrary order are deduced, based on the lowest-order examples. These solutions can be useful for explanations of extremely large amplitude internal waves in the ocean, as well as for abnormally large-amplitude waves in other areas of nonlinear physics, such as optics and dusty plasmas.

Shallow-water rogue waves: An approach based on complex solutions of the Korteweg-de Vries equation.

The formation of rogue waves in shallow water is presented in this Rapid Communication by providing the three lowest-order exact rational solutions to the Korteweg-de Vries (KdV) equation. They have been obtained from the modified KdV equation by using the complex Miura transformation. It is found that the amplitude amplification factor of such waves formed in shallow water is much larger than the amplitude amplification factor of those occurring in deep water. These solutions clearly demonstrate a potential hazard for coastal areas. They can also provide a solid mathematical basis for the existence of abnormally large-amplitude waves in other branches of nonlinear physics such as optics, unidirectional crystal growth, and in quantum mechanics.

Sasa-Satsuma hierarchy of integrable evolution equations.

We present the infinite hierarchy of Sasa-Satsuma evolution equations. The corresponding Lax pairs are given, thus proving its integrability. The lowest order member of this hierarchy is the nonlinear Schrödinger equation, while the next one is the Sasa-Satsuma equation that includes third-order terms. Up to sixth-order terms of the hierarchy are given in explicit form, while the provided recurrence relation allows one to explicitly write all higher-order terms. The whole hierarchy can be combined into a single general equation. Each term in this equation contains a real independent coefficient that provides the possibility of adapting the equation to practical needs. A few examples of exact solutions of this general equation with an infinite number of terms are also given explicitly.

Rogue wave solutions for the infinite integrable nonlinear Schrödinger equation hierarchy.

We present rogue wave solutions of the integrable nonlinear Schrödinger equation hierarchy with an infinite number of higher-order terms. The latter include higher-order dispersion and higher-order nonlinear terms. In particular, we derive the fundamental rogue wave solutions for all orders of the hierarchy, with exact expressions for velocities, phase, and "stretching factors" in the solutions. We also present several examples of exact solutions of second-order rogue waves, including rogue wave triplets.

Infinite hierarchy of nonlinear Schrödinger equations and their solutions.

We study the infinite integrable nonlinear Schrödinger equation hierarchy beyond the Lakshmanan-Porsezian-Daniel equation which is a particular (fourth-order) case of the hierarchy. In particular, we present the generalized Lax pair and generalized soliton solutions, plane wave solutions, Akhmediev breathers, Kuznetsov-Ma breathers, periodic solutions, and rogue wave solutions for this infinite-order hierarchy. We find that "even- order" equations in the set affect phase and "stretching factors" in the solutions, while "odd-order" equations affect the velocities. Hence odd-order equation solutions can be real functions, while even-order equation solutions are always complex.

Integrable equations of the infinite nonlinear Schrödinger equation hierarchy with time variable coefficients.

We present an infinite nonlinear Schrödinger equation hierarchy of integrable equations, together with the recurrence relations defining it. To demonstrate integrability, we present the Lax pairs for the whole hierarchy, specify its Darboux transformations and provide several examples of solutions. These resulting wavefunctions are given in exact analytical form. We then show that the Lax pair and Darboux transformation formalisms still apply in this scheme when the coefficients in the hierarchy depend on the propagation variable (e.g., time). This extension thus allows for the construction of complicated solutions within a greatly diversified domain of generalised nonlinear systems.

Breather-to-soliton conversions described by the quintic equation of the nonlinear Schrödinger hierarchy.

We analyze the quintic integrable equation of the nonlinear Schrödinger hierarchy that includes fifth-order dispersion with matching higher-order nonlinear terms. We show that a breather solution of this equation can be converted into a nonpulsating soliton solution on a background. We calculate the locus of the eigenvalues on the complex plane which convert breathers into solitons. This transformation does not have an analog in the standard nonlinear Schrödinger equation. We also study the interaction between the new type of solitons, as well as between breathers and these solitons.

Breather solutions of the integrable quintic nonlinear Schrödinger equation and their interactions.

We present breather solutions of the quintic integrable equation of the Schrödinger hierarchy. This equation has terms describing fifth-order dispersion and matching nonlinear terms. Using a Darboux transformation, we derive first-order and second-order breather solutions. These include first- and second-order rogue-wave solutions. To some extent, these solutions are analogous with the corresponding nonlinear Schrödinger equation (NLSE) solutions. However, the presence of a free parameter in the equation results in specific solutions that have no analogues in the NLSE case. We analyze new features of these solutions.

Soliton solutions of an integrable nonlinear Schrödinger equation with quintic terms.

We present the fifth-order equation of the nonlinear Schrödinger hierarchy. This integrable partial differential equation contains fifth-order dispersion and nonlinear terms related to it. We present the Lax pair and use Darboux transformations to derive exact expressions for the most representative soliton solutions. This set includes two-soliton collisions and the degenerate case of the two-soliton solution, as well as beating structures composed of two or three solitons. Ultimately, the new quintic operator and the terms it adds to the standard nonlinear Schrödinger equation (NLSE) are found to primarily affect the velocity of solutions, with complicated flow-on effects. Furthermore, we present a new structure, composed of coincident equal-amplitude solitons, which cannot exist for the standard NLSE.

Solutions of the higher-order Manakov-type continuous and discrete equations.

We derive exact and approximate localized solutions for the Manakov-type continuous and discrete equations. We establish the correspondence between the solutions of the coupled Ablowitz-Ladik equations and the solutions of the coupled higher-order Manakov equations.

Velocity of heat dissipative solitons in optical fibers.

In the fiber fuse, a pulse of high temperature travels toward the input end of the fiber, where high-power laser light is launched into the fiber. At any point along the fiber, the soliton can be ignited. The fiber core is damaged in the process so that light cannot propagate beyond the hot spot. This phenomenon is an example of a dissipative soliton that can exist only in the presence of an external energy supply and internal loss. We analyze this phenomenon, derive an expression for the velocity of the soliton, and determine its width as functions of the physical parameters of the laser and the fiber material.

Pulsating, creeping, and erupting solitons in dissipative systems.

We present three novel pulsating solutions of the cubic-quintic complex Ginzburg-Landau equation. They describe some complicated pulsating behavior of solitons in dissipative systems. We study their main features and the regions of parameter space where they exist.

Birefringence control in plasma-enhanced chemical vapor deposition planar waveguides by ultraviolet irradiation.

Complete birefringence compensation is demonstrated in plasma-enhanced chemical vapor deposition waveguides by 193-nm postexposure. A single relaxation process dominates the decay in stress anisotropy, indicating that compressive stress from the substrate leads to an elastic stress anisotropy at the core.

Partially coherent solitons of variable shape in a slow Kerr-like medium: exact solutions.

We carry out a theoretical investigation of the properties of partially coherent solitons for media which have a slow Kerr-like nonlinearity. We find exact solutions of the Nth-order Manakov equations in a general form. These describe partially coherent solitons (PCSs) and their collisions. In fact, the exact solutions allow us to analyze important properties of PCSs such as stationary profiles of the spatial beams and effects resulting from their collisions. In particular, we find, analytically, the number of parameters that control the soliton shape. We present profiles which are symmetric as well as those which are asymmetric. We also find that collisions allow the profiles to remain stationary but cause their shapes to change.

Hamiltonian-versus-energy diagrams in soliton theory.

Parametric curves featuring Hamiltonian versus energy are useful in the theory of solitons in conservative nonintegrable systems with local nonlinearities. These curves can be constructed in various ways. We show here that it is possible to find the Hamiltonian (H) and energy (Q) for solitons of non-Kerr-law media with local nonlinearities without specific knowledge of the functional form of the soliton itself. More importantly, we show that the stability criterion for solitons can be formulated in terms of H and Q only. This allows us to derive all the essential properties of solitons based only on the concavity of the curve H vs Q. We give examples of these curves for various nonlinearity laws and show that they confirm the general principle. We also show that solitons of an unstable branch can transform into solitons of a stable branch by emitting small amplitude waves. As a result, we show that simple dynamics like the transformation of a soliton of an unstable branch into a soliton of a stable branch can also be predicted from the H-Q diagram.

Ultrashort pulses generated by mode-locked lasers with either a slow or a fast saturable-absorber response.

We present a new exact solution for ultrashort pulses generated by passively mode-locked lasers, taking into account the slow and the fast parts of the semiconductor saturable-absorber response in the nonsaturated limit.

Generalized LP(1m) Modes in an Optical Fiber with Constant Radius of Curvature.

Accurate forms for the generalized LP(1m) mode in a uniform circular-core curved fiber are given. We show that each generalized LP(1m) mode is composed of four linearly polarized partial fields. We also show that, when the propagation constants of HE(2m), TM(0m), and TE(0m) modes are degenerate, there are four linearly polarized modes for each generalized LP(1m) mode.

Berry's Phase Analysis of Polarization Rotation in Helicoidal Fibers.

We relate Berry's topological phase to the polarization rotation of linearly polarized light in helicoidal single-mode ideal fibers where the pitch length and coil radius are allowed to change adiabatically. First we present an alternative derivation for this phase using the Serret-Frenet coordinate system and show that this phase can be derived and interpreted in terms of both solid and planar angles. The results obtained are then applied to various helicoidal fiber structures, and from this we show that the total change in the polarization rotation angle can be tailored through a judicious choice of the fiber geometry. Finally, we propose that certain helicoidal fiber configurations can be used as fiber sensors.

Optical conveyor belts: a new scheme for fiber communications.

A scheme is proposed for high-bandwidth, robust data transmission in silica fibers in the normal-dispersion regime. The scheme uses a uniform periodic train of dark solitons to eliminate completely the dispersion of a data stream encoded in linear pulses of small but otherwise arbitrary amplitude in the orthogonal polarization. Data pulses carried by dark solitons of differing contrast do not interfere with one another when the solitons collide. Thus many channels, each associated with a different contrast, can be multiplexed on the same fiber.

Accurate expressions for the LP modes in a uniform circular-core curved fiber.

Accurate forms for the LP(nm) modes (n = 0 and n ≧ 2) in a uniform circular-core curved fiber are given. We show that the LP(nm) modes (n ≧ 2) are composed of two spatially orthogonal components and that, to the zeroth order, there is no special polarization axis for the LP modes in a uniform circular-core curved fiber.