Research |
KP equations |
---|
NLS systems
For the last 5 years I have been working in collaboration with prof. Mark J. Ablowitz, at the Department of Applied Mathematics of CU Boulder. Our common research activity mainly focuses on the IST for continuous and discrete Nonlinear Schrodinger systems (NLS), both scalar and vector.
NLS equations have become amongst the most important nonlinear systems
studied in mathematics and physics. Such systems have been derived in such
diverse fields as deep water waves, plasma physics, nonlinear fiber optics,
magnetic spin waves, etc. In fact, it turns out that almost any dispersive,
energy preserving system gives rise, in an appropriate limit, to the scalar NLS
equation. Its vector generalization (VNLS) arises, physically, under conditions
similar to those described by the NLS when there are multiple wave-trains moving
with nearly the same group velocities. Also, VNLS models physical systems in
which the field has more than one component; for example, in optical fibers and
waveguides, the propagating electric field has two components transverse to the
direction of propagation. Both the NLS and VNLS equations admit integrable
discretizations that besides being used as the basis for constructing numerical
schemes for their continuous counterparts, also have physical applications as
discrete systems. Moreover, both the scalar and vector NLS equations and some of
their discretizations attain broad significance because they are integrable via
the inverse scattering transform (IST) and admit soliton solutions.
In spite of the fact that NLS systems have been the subject of investigation by
reaserchers for over 30 years, many important problems remain to be solved.
During the last years we have devoted our efforts to studying some of
them.
In particular, we investigated soliton interactions. It was well-known that interacting
scalar solitons affect each other only by a phase shift, that depends only on
the soliton powers and velocities, both of which are conserved quantities. Thus,
when two soliton collisions occur sequentially, the outcome of the first
collision does not affect the second collision, except for a uniform phase
shift. On the other hand, the collision of vector solitons is much richer and
more complex, due to the fact that vector and matrix solitons have internal
degrees of freedom. In fact, while the dynamics of a single soliton is
essentially governed by the underlying scalar equation, the vector nature of the
soliton manifests itself in the interaction, which can be highly nontrivial:
even though the collision is elastic, in the sense that the total energy of each
soliton is conserved, there can be a significant redistribution of energy among
the components, i.e. a polarization shift. We gave explicit formulas for the
polarization shift induced by the soliton collision, both for the continuous
VNLS equation and, for the first time, for its integrable discretization. Even
for the classical VNLS, in spite of the fact that the system had been known for
over 30 years, the nature of multisoliton collisions was not clarified. For the
first time, revisiting and casting more light on Manakov's well-known results,
we were able to prove rigorously that for the vector systems, both continuous
and discrete, the multisoliton interaction process is pair-wise and the net
result of the interaction is independent of the order in which such collisions
occur.
We analyzed the various features underlying collisions of
generic N-component VNLS solitons, including changes in amplitudes, phases and
relative separation distances. Then we established a relationship between order
independence of the pair-wise multisoliton interaction processes and the fact
that the map determining the interaction of two solitons satisfies the
Yang-Baxter relation, where the same problem was addressed for the matrix
KdV equation. In particular, we showed that the (matrix) transmission
coefficients of the VNLS equation satisfy the matrix re-factorization problem
which leads to the Yang-Baxter relation.
The interaction of vector solitons also sets forth the experimental foundations
for computation with solitons. In the energy re-distributions in the two
component case was expressed as fractional linear transformations (FLT's),
showing that the parameters controlling the energy switching between components
exhibit nontrivial information transformations. As a result, sequences of
solitons operating on other sequences of solitons that effect logic operations,
including controlled gates, can be implemented. Using this background, we
analyzed multisoliton interactions in the general N-component case and expressed
them explicitly via generalized FLT's, where a soliton is described in terms of
N-1 complex parameters. Finally, we considered the basic (three collision) gate
introduced by Steiglitz et al, where it was shown that in order to implement it,
one has to be able to adjust the values of the initial polarizations of two of
the solitons participating in the gate according to a given system of equations.
Only numerical solutions where given and the issue of when they do and do not
have solutions was left open. We proved that there are parameter regimes for the
soliton amplitudes and velocities for which a unique solution for the system
exists.
We showed that the integrable discretization of the VNLS (IDVNLS) equation
admits soliton solutions. In fact,there are both "fundamental'' soliton
solutions, that are the counterpart of the vector solitons of VNLS, and
"composite'' soliton solutions that have no counterpart in the continuous limit.
The composite soliton itself consists of a localized traveling envelope with
both temporal and spatial oscillations, as well as a complex spatial modulation.
In order to shed light on the structure of composite solitons, we considered in
more detail two special cases of the composite soliton, which we referred to as
the "orthogonal'' and the "parallel'' case. An orthogonal composite soliton is
the (weighted) sum of two fundamental-soliton-like solutions. The two
constituent sech-envelopes have mutually orthogonal polarizations. Moreover, one
of the envelopes has oscillation on the grid scale, hence, there is no continuum
limit of this solution. On the other hand, the parallel composite soliton is a
scalar function multiplied by a polarization vector. In fact, the shape of the
overall envelope is a superposition of two modulated sech-envelopes.
We studied the interaction of fundamental solitons and showed that the
order independence of the multisoliton collisions is a reflection of the fact
that, in analogy with their continuous counterparts, discrete vector soliton
interactions can be recast as Yang-Baxter maps. Moreover, we showed that, like
in the continuous case, the polarization state of a discrete vector soliton can
be described by a single complex scalar and the polarization shifts induced by
soliton collisions is conveniently represented by means of fractional linear
transformations (FLTs) on these scalars. The remarkable consequence is that
interacting discrete vector solitons exhibit nontrivial information transfer and
the value of a logical ``bit'' can be encoded in the scalar polarization state
of a vector soliton. Given such an encoding, one can identify vector-soliton
interactions with logical operations. Then, using the model proposed by
Steiglitz et al for the Manakov spatial solitons, we were able to show that the
spatially-discrete vector solitons of two-component IDVNLS are also
computationally universal in the sense of Turing equivalence.
Recently, we began to investigate the IST for NLS systems with nonvanishing
boundary conditions at space infinities. While the IST for the scalar NLS, both
with vanishing and nonvanishing boundaries, was developed many years ago, the
basic formulation of direct and inverse scattering for the vector NLS equation
in the normal dispersion regime (defocusing VNLS) has still not been given in
the case of nonvanishing boundary conditions as | x| tends to infinity. The
defocusing VNLS equation admits solitons with a nontrivial background intensity
in either one or both components, so-called dark-bright or dark-dark solitons,
which makes formulating the IST for nondecaying potentials particularly
interesting.