Research
A key element in the study of Applied Mathematics is to understand and describe physical phenomena by employing detailed mathematical models. Frequently such models lead to large amplitude or nonlinear systems. Remarkably, in many cases certain prototypical equations are found to be the fundamental underlying systems that can be used to approximate the physical problem. The main feature of my research is to understand by exact and numerical methods, or by approximation, solutions to these underlying equations and their properties.
Below I list some of the topics I have worked or I am currently working on.
Rogue Waves
It is common for mid-ocean storm waves to reach seven meters in height, and in extreme conditions such waves can reach heights of fifteen meters. However, for centuries maritime folklore told of the existence of vastly more massive waves that could appear without warning in mid-ocean, against the prevailing current and wave direction, and often in perfectly clear weather. Such waves were said to consist of an almost vertical wall of water preceded by a trough so deep that it was referred to as a hole in the sea. These waves are called rogue waves (also known as freak waves, monster waves, killer waves, extreme waves, and abnormal waves). A rogue wave is a highly localized phenomenon both in space and duration, most frequently occurring far out at sea. Historically oceanographers have discounted these reports as tall tales, i.e. the embellished stories of mariners with too much time at sea. But in the last years scientists have discovered strong evidence indicating that such massive rogue waves do exist. The phenomenon has become the subject of recent scientific study, but their origin still remains a mystery of the
deep.
Our goal is to study the generation of these waves at a fundamental level and, in some sense, challenge common believes on the subject. Starting from the Euler equations which model fluid motion under certain criteria we will derive the appropriate set of equations that describe these waves. The main difference with the already known literature, is that we will start from a complete two dimensional system and posing certain conditions we will reduce the 2D system in a set of coupled 1D equations. This set of equations results from the different crossing sea states with an additional condition: these states will have nearly the same group velocities. The breakthrough is that this posses the condition for rogue wave generation. Namely, when different sea states cross rogue waves will be generated when these states have the same, or almost the same, group velocities. Since is is truly a rare oceanic state this also explains the rarity of rogue wave occurrence.
Solitons in Liquid Crystals (Nematicons)
Liquid crystals are a state of matter that has properties between those of conventional liquid and those of solid crystal. For instance, a liquid crystal may flow like a liquid, but its molecules may be oriented in a crystal-like way. There are many different types of liquid-crystal phases, which can be distinguished by their different optical properties. One of the most common phases is the nematic. The word nematic comes from the Greek νήμα (nema), which means thread. This term originates from the thread-like topological defects observed in nematics, which are formally called disclinations. In a nematic phase, the calamitic or rod-shaped organic molecules have no positional order, but they self-align to have long-range directional order with their long axes roughly parallel. Thus, the molecules are free to flow and their center of mass positions are randomly distributed as in a liquid, but still maintain their long-range directional order.
The equations that describe coherent structures (or solitons) in liquid crystals are nonlocal in (2+1) dimensions (2 space, 1 time) which makes their analysis very demanding. We have developed an asymptotic theory, based on multiple scales, that reduces this system into the well known and much studied Kadomtsev-Petviashvilli (KP) equation. As such we are able to describe the solutions of the original system by those of the KP equation. This reduction is also confirmed by direct numerical simulations.
Perturbed Dark Solitons
Dark solitons are intensity dips on a constant background with a phase jump across their intensity minimum. Since their discovery, dark solitons have attracted considerable attention, especially in the fields of nonlinear optics and Bose-Einstein condensates. We have developed a direct perturbation method for approximating these solitons, of the nonlinear Schrodinger equation, under the influence of perturbations. Unlike their bright counterparts, i.e., solitons that decay at infinity, dark solitons -under the influence of perturbations- develop a shelf that propagates with speed determined by the background intensity. The dynamical properties of the soliton and the shelf are obtained from integral relations arising from the conservation laws of the NLS equation. The analysis is developed for both constant and slowly evolving backgrounds.
Ultra-short Pulses in Mode-Locked Lasers
In our research investigations we derived, based on the asymptotic procedure of multiple scales, a nonlinear integro-differential equation which governs the dynamics of dispersion managed pulse propagation. This governing equation is referred to as the dispersion managed nonlinear Schrodinger (DMNLS) equation. Since modeling ultra-short pulse mode-locked lasers should take into account not only dispersion management, but also amplification and loss we recently introduced a modified system which takes into account these effects; the gain medium is modeled by a saturated energy term, whereas the Kerr-lens effect is modeled by a saturated with power term. This equation was employed to model the propagation of Ti:sapphire laser pulses in both the anomalous and normal dispersion regimes. This system naturally describes the locking and evolution of pulses in mode-locked lasers that are operating in the soliton regime.
The Short Pulse and related Equations
The standard model for describing propagation of a pulse-shaped complex field envelope in nonlinear dispersive media is the nonlinear Schrodinger (NLS) equation. In the context of nonlinear optics, the main assumption made when deriving the NLS equation from Maxwell's equations is that the pulse-width is large as compared to the period of the carrier frequency. When this assumption is no longer valid, i.e., for pulse duration of the order of a few cycles of the carrier, the evolution of such short pulses' is better described by the so-called short-pulse equation (SPE). Unlike the NLS, the SPE does not exhibit smooth, well behaved travelling wave soulutions. Instead, one has to add higher order terms to recover these solutions. The integrability properties of the SPE and the related equations is the subject of our investigations.
Solitons in Bose-Einstein Condensates
From a theoretical standpoint, and for sufficiently low temperatures, the static and dynamical properties of a BEC can be described by means of an effective mean-field equation known as the Gross–Pitaevskii (GP) equation. This is a variant of the NLS equation, incorporating an external potential used to confine the condensate. In realistic situations, finite temperature effects dramatically alter the dynamics of BECs as they induce dissipation. We intend to analyze this problem and investigate, in particular, the effect of temperature on the static and dynamical properties of dark solitons and vortices. We will also investigate if temperature-induced dissipation may be compensated by gain mechanisms (relevant to the BEC context), similarly to the situation occurring in nonlinear optics.
Self-Fourier Functions
In recent years researchers have developed an interest in functions which are their
own Fourier transform (FT). These functions are called self-Fourier functions (SFFs). Applications of SFFs can be found in such diverse areas as optics, image processing, quantum mechanics and even analytic number theory. Recent developments focus on generalizations of known SFFs and their applications to coherent laser design. Another direction is the prominent role Fourier analysis plays in the study of boundary value problems (BVPs). To date, no algorithm has been provided to systematically generate such functions without having to evaluate some type of integral transform. We recently introduced a formalism that allows us to show the existence of these families of functions without actually evaluating any Fourier, or other transform-type integrals, a task often challenging, and frequently not even possible. More importantly, our method associates SFFs with BVPs of differential operators in such a way as to suggest that this may be the natural framework to use when dealing with this problem.
