32 research outputs found
Complex plane representations and stationary states in cubic and quintic resonant systems
Weakly nonlinear energy transfer between normal modes of strongly resonant
PDEs is captured by the corresponding effective resonant systems. In a previous
article, we have constructed a large class of such resonant systems (with
specific representatives related to the physics of Bose-Einstein condensates
and Anti-de Sitter spacetime) that admit special analytic solutions and an
extra conserved quantity. Here, we develop and explore a complex plane
representation for these systems modelled on the related cubic Szego and LLL
equations. To demonstrate the power of this representation, we use it to give
simple closed form expressions for families of stationary states bifurcating
from all individual modes. The conservation laws, the complex plane
representation and the stationary states admit furthermore a natural
generalization from cubic to quintic nonlinearity. We demonstrate how two
concrete quintic PDEs of mathematical physics fit into this framework, and thus
directly benefit from the analytic structures we present: the quintic nonlinear
Schroedinger equation in a one-dimensional harmonic trap, studied in relation
to Bose-Einstein condensates, and the quintic conformally invariant wave
equation on a two-sphere, which is of interest for AdS/CFT-correspondence.Comment: v2: version accepted for publicatio
A nonrelativistic limit for AdS perturbations
The familiar nonrelativistic limit converts the
Klein-Gordon equation in Minkowski spacetime to the free Schroedinger equation,
and the Einstein-massive-scalar system without a cosmological constant to the
Schroedinger-Newton (SN) equation. In this paper, motivated by the problem of
stability of Anti-de Sitter (AdS) spacetime, we examine how this limit is
affected by the presence of a negative cosmological constant .
Assuming for consistency that the product tends to a negative
constant as , we show that the corresponding
nonrelativistic limit is given by the SN system with an external harmonic
potential which we call the Schrodinger-Newton-Hooke (SNH) system. We then
derive the resonant approximation which captures the dynamics of small
amplitude spherically symmetric solutions of the SNH system. This resonant
system turns out to be much simpler than its general-relativistic version,
which makes it amenable to analytic methods. Specifically, in four spatial
dimensions, we show that the resonant system possesses a three-dimensional
invariant subspace on which the dynamics is completely integrable and hence can
be solved analytically. The evolution of the two-lowest-mode initial data (an
extensively studied case for the original general-relativistic system), in
particular, is described by this family of solutions.Comment: v3: slightly expanded published versio
Solvable cubic resonant systems
Weakly nonlinear analysis of resonant PDEs in recent literature has generated
a number of resonant systems for slow evolution of the normal mode amplitudes
that possess remarkable properties. Despite being infinite-dimensional
Hamiltonian systems with cubic nonlinearities in the equations of motion, these
resonant systems admit special analytic solutions, which furthermore display
periodic perfect energy returns to the initial configurations. Here, we
construct a very large class of resonant systems that shares these properties
that have so far been seen in specific examples emerging from a few standard
equations of mathematical physics (the Gross-Pitaevskii equation, nonlinear
wave equations in Anti-de Sitter spacetime). Our analysis provides an
additional conserved quantity for all of these systems, which has been
previously known for the resonant system of the two-dimensional
Gross-Pitaevskii equation, but not for any other cases.Comment: v2: 23 pages, 1 figure, minor corrections, published versio
Exact lowest-Landau-level solutions for vortex precession in Bose-Einstein condensates
The Lowest Landau Level (LLL) equation emerges as an accurate approximation
for a class of dynamical regimes of Bose-Einstein Condensates (BEC) in
two-dimensional isotropic harmonic traps in the limit of weak interactions.
Building on recent developments in the field of spatially confined extended
Hamiltonian systems, we find a fully nonlinear solution of this equation
representing periodically modulated precession of a single vortex. Motions of
this type have been previously seen in numerical simulations and experiments at
moderately weak coupling. Our work provides the first controlled analytic
prediction for trajectories of a single vortex, suggests new targets for
experiments, and opens up the prospect of finding analytic multi-vortex
solutions.Comment: v2: minor improvements, published in PR
Formation of singularities for equivariant 2+1 dimensional wave maps into the two-sphere
In this paper we report on numerical studies of the Cauchy problem for
equivariant wave maps from 2+1 dimensional Minkowski spacetime into the
two-sphere. Our results provide strong evidence for the conjecture that large
energy initial data develop singularities in finite time and that singularity
formation has the universal form of adiabatic shrinking of the degree-one
harmonic map from into .Comment: 14 pages, 5 figures, final version to be published in Nonlinearit
Dispersion and collapse of wave maps
We study numerically the Cauchy problem for equivariant wave maps from 3+1
Minkowski spacetime into the 3-sphere. On the basis of numerical evidence
combined with stability analysis of self-similar solutions we formulate two
conjectures. The first conjecture states that singularities which are produced
in the evolution of sufficiently large initial data are approached in a
universal manner given by the profile of a stable self-similar solution. The
second conjecture states that the codimension-one stable manifold of a
self-similar solution with exactly one instability determines the threshold of
singularity formation for a large class of initial data. Our results can be
considered as a toy-model for some aspects of the critical behavior in
formation of black holes.Comment: 14 pages, Latex, 9 eps figures included, typos correcte
Two infinite families of resonant solutions for the Gross-Pitaevskii equation
We consider the two-dimensional Gross-Pitaevskii equation describing a
Bose-Einstein condensate in an isotropic harmonic trap. In the small coupling
regime, this equation is accurately approximated over long times by the
corresponding nonlinear resonant system whose structure is determined by the
fully resonant spectrum of the linearized problem. We focus on two types of
consistent truncations of this resonant system: first, to sets of modes of
fixed angular momentum, and second, to excited Landau levels. Each of these
truncations admits a set of explicit analytic solutions with initial conditions
parametrized by three complex numbers. Viewed in position space, the fixed
angular momentum solutions describe modulated oscillations of dark rings, while
the excited Landau level solutions describe modulated precession of small
arrays of vortices and antivortices. We place our findings in the context of
similar results for other spatially confined nonlinear Hamiltonian systems in
recent literature.Comment: v2: published version (commentary added
Ground state in the energy super-critical Gross-Pitaevskii equation with a harmonic potential
The energy super-critical Gross--Pitaevskii equation with a harmonic
potential is revisited in the particular case of cubic focusing nonlinearity
and dimension d > 4. In order to prove the existence of a ground state (a
positive, radially symmetric solution in the energy space), we develop the
shooting method and deal with a one-parameter family of classical solutions to
an initial-value problem for the stationary equation. We prove that the
solution curve (the graph of the eigenvalue parameter versus the supremum) is
oscillatory for d = 13. Compared to the existing
literature, rigorous asymptotics are derived by constructing three families of
solutions to the stationary equation with functional-analytic rather than
geometric methods.Comment: 42 page
Saddle point solutions in Yang-Mills-dilaton theory
The coupling of a dilaton to the -Yang-Mills field leads to
interesting non-perturbative static spherically symmetric solutions which are
studied by mixed analitical and numerical methods. In the abelian sector of the
theory there are finite-energy magnetic and electric monopole solutions which
saturate the Bogomol'nyi bound. In the nonabelian sector there exist a
countable family of globally regular solutions which are purely magnetic but
have zero Yang-Mills magnetic charge. Their discrete spectrum of energies is
bounded from above by the energy of the abelian magnetic monopole with unit
magnetic charge. The stability analysis demonstrates that the solutions are
saddle points of the energy functional with increasing number of unstable
modes. The existence and instability of these solutions are "explained" by the
Morse-theory argument recently proposed by Sudarsky and Wald.Comment: 11 page