Research Projects 
Topological Solitons: The Skyrme Model in 2+1 dimensions (babyskyrmions) What doth thy skyrmion
out of his bed at midnight? Shall I give him
his answer? 
headon collision at γ=2.29 
Skyrmions are topological soliton solutions of a classical nonlinear field theory known as the Skyrme model. As examples of two and threedimensional solitonic structures, they are of inherent interest, but they may also have applications to nuclear physics. The Skyrme model can be obtained as a low energy limit of QCD in the large Nc limit (Nc the number of colours), with baryons being identified as quantized states of the classical soliton solutions. QCD is very difficult to solve in the strong coupling regime (low energy), even numerically. Interest turned to simpler phenomenological models for the structure of hadrons. The Skyrme model in 2+1 dimensions is know as the baby skyrme model. These nonlinear σmodels in 2+1 appear to be low dimensional analogues to fourdimensional YangMills theories. Left we can see the different stages of a headon collision of two babyskyrmions in the plane. We observe the Baryon number density profiles of each soliton. The initial data is boosted to γ=2.29; (v=0.9c). We can see that they scatter at right angles. This is very typical for solitons of wavelike equations. As a numerical relativist this model represents an ideal testbed for numerical techniques, since the numerical integration of the Skyrmion field equations is technically similar to that found with Einstein Equations. Another motivation is the possibility to use skyrmions as another type of source matter, along with scalar fields and fluids. Coupling a skyrmion matter lagrangian to Einstein equations leads to interesting bound states in curved spacetime. Another possibility is to use a skyrmion equation fo state to integrate the TolmanOppenheimmerVolkoff equations to study starlike solutions. I am interested in the dynamics of
highly boosted collisions of skyrmions, since one of the most most
interesting and challenging aspects of the Skyrme model
constitutes the numerical instabilities
reported for large velocities. It has been suggested that the hyperbolic
nature
of
the
equations is lost, turning the problem
illposed I have performed extensive 2+1 and 3+1 simulations of
skyrmion collisions at high boosts (γ ~ 10) using parallel adaptive
mesh refinement techniques (PAMR). It has been shown that causality is not
violated in the Skyrme model, so the idea is to determine if the
instabilities araise from inadequate numerical methods, or they have a
physical origin. Understanding of skyrmion scattering and dynamics is also proving to be fruitful in dense hadronic matter relevant to compact stars, a system difficult to access by other approaches. The recent discovery of holograFc baryons in gravity/gauge duality which correspond to skyrmions in the infinite tower of vector mesons provides a window to the regime of strong coupling, that QCD proper has difficulty in accessing. Interest on topological solitons is manifold. They also provide models for matter at high energies and velocities. A numerical understanding of collisions of ultrarelativistic solitons in the presence of gravity may shred light on many issues at the crossroads of spacetime singularity formation, heavyion dynamics and exotic new physics in extradimensions, where blackholes are expected to form in the next generation of TeraScale energy accelerators. 
Topological
Solitons:
The
Skyrme
Model
in
3+1
dimensions

Nontopological
Solitons:
Qballs
in
2+1
and
3+1
dimensions

It is well
known that realistic
supersymmetric theories are associated with a number of scalar fields
with various charges, which allow for baryonic and leptonic
nontopological solitons known as QBalls.
A
QBall
is a solution of a relativistic (complex) scalar field theory with a
nonlinear quadratic potential, which depends on the supersymmetry
breaking mechanism, and there is a conserved Noether charge Q carried
by the QBall lump. We can also have spinning QBalls, which carry
angular momentum. The basic properties of QBalls are well understood using analytical methods, for example existence and stability under small vibrations. I intend to continue my work on numerical simulations of QBall evolution in 2+1 (two spatial dimensions and time) and 3+1 dimensions to address strong coupling dynamics such as scattering, interactions and stability, and to perform a parameter space exploration using PAMR techniques. Is there any exotic scattering? rightangle scattering for topological solitons depends on the velocity of the impact, do nontopological solitons, such as QBalls, exhibit the same behaviour? A detailed knowledge of Qball scattering is needed to answer several questions in cosmology, where they could be responsible for both the net baryon number of the universe and its dark matter component. QBalls may be trapped in neutron star interiors. So far this questions are being addressed in Flat space time. I would like to explore the possibility to search for selfgravitating Qballs. I am using qballs as a model for matter in ultrarelativistic collisions. This regime has not been exhaustively explored and it is my intention to do so. Highly boosted matter is challenging since higher resolutio is needed and instabilities are more likely to appear. There are other types of matter that show unexpected and rich behaviour at high velocities, such as skyrmions. Other objects of interest based on a complex scalar field are boson stars. The difference is that they incorporate a quadratic potential, and naturally gravity is included in the lagrangian so the competition between the collapse and the scalar field "pressure" outwards reach an equilibrium configuration that we call a star. My collegue Bruno Mundim has studied those objects intensively. 
Topological
Solitons:
The
Skyrme
Model
in
2+1
dimensions
stabilized
by
vector
mesons

Kinks, Domain Walls and branes 
The video in the right
shows the collision of a kink and an antikink of a φ^4 field theory in
1+1 dimensions, at low velocity (v=0.3c). This relatively simple setup
can be used to model the 3+1 boundary (a brane) of a 4+1
bulk, given certain symmetry assumptions. Fermions become localized since
they tend to stick to the kink defect rather than escaping as radiation to the bulk. The kink in the scalar field acts as an impurity that traps the fermion on the brane, similarly to some condensed matter systems where a magnetic field provides the
"impurity". This trapping is analogous to the requirement that fermions must live on the brane. The particle number (fermion content) on moving kinks changes
after a kinkantikink collision, or when these 1+1 "domain walls"
collide with the spacetime boundaries, and we can compute this particle transfer calculating Bogoliubov
(transmissionreflection) coefficients. Pionnering work in this direction has been performed by Gibbons, Maeda and Takamizu, and subsecuently by Saffin and Tranberg . These toy models for fermions on colliding branes represent a very
instructive effort to understand more realistic braneworld
scenarios. My idea is to extend the kink collisions code I have to
incorporate mesh refinement. It may also be interesting to study highly boosted
kinkantikink
collisions in this framework. I would like to test convergence and conserved quantities
and extend all this interesting work, which constitutes one more example of the very healthy interplay between condensed matter, high energy theory and numerical analysis. 
Constrained Evolution and Constraint Damping 
ParticleParticle and Particle Mesh Methods with PAMR 
Left: initial configuration;
Right: end configuration with the analytic velocity field
independent of the radius.
The domain is a square with the
left lower corner at (1, 1) with its
center at (0, 0). We can evaluate the velocity field at each point of
the domain analitically , i.e., we know the functions vx (x, y) and vy
(x, y). I use a second order RungeKutta scheme to advance
each particle position from time level n to time level n + 1 by Δt,
The next step is to migrate particles that leave a processor's subdomain into the corresponding new one. The way implemented here is that each processor broadcasts the positions of particles that left its subdomain. Each processor then selects the particles belonging to its subdomain. Note that in general the domain is not filled with particles uniformly, so that processors must handle different number of them at a given time. The bbhutils offer an easy way for reading parameters from files, Especially useful is the ivec type. The ivec type allows for easy and compact notation of specifying an index vector. This is useful for output control. The next snippet of code shows how to use the variable out ivec to control the output frequency. The parameter file is passed as the first command line argument. The data can be viewed by DV by selecting the item Particles from the options menu of the DV local view window. The following is an example of a C routine that saves a 2D particle data: within a desired domain. Source Code
Grid
Velocity
Field
I
set the analytic velocity field so the particles go in circles; 1=N*dt
where N is the number of time steps and dt is the timelength of the
time step; the local error or a RungeKutta scheme of second order is
supossed to be proportional to dt^3, and the global error proportional
to dt^2; I was not sure about this and I decided to write a program
that would evolve a complete orbit of a single particle located at
radius=1 using this scheme; then i calculated the error of the radius
after the orbit and the relative error dr/r for different values of dt
and N time steps such that N*dt=1; the results are shown in this
file and the following graph. Starting from N=27 and dt=0.037, the
relative error is ~1%, so if we use a dt=0.01 and 100 or so
time steps, we are safely well below 1 percent of relative error in the
radius after one orbit.
We can appreciate theese
results in the following loglog plot:

Preprints of Interest 