168 research outputs found
Spectral theory of soliton and breather gases for the focusing nonlinear Schrödinger equation
Solitons and breathers are localized solutions of integrable systems that can be viewed as “particles” of complex statistical objects called soliton and breather gases. In view of the growing evidence of their ubiquity in fluids and nonlinear optical media, these “integrable” gases present a fundamental interest for nonlinear physics. We develop an analytical theory of breather and soliton gases by considering a special, thermodynamic-type limit of the wave-number–frequency relations for multiphase (finite-gap) solutions of the focusing nonlinear Schrödinger equation. This limit is defined by the locus and the critical scaling of the band spectrum of the associated Zakharov-Shabat operator, and it yields the nonlinear dispersion relations for a spatially homogeneous breather or soliton gas, depending on the presence or absence of the “background” Stokes mode. The key quantity of interest is the density of states defining, in principle, all spectral and statistical properties of a soliton (breather) gas. The balance of terms in the nonlinear dispersion relations determines the nature of the gas: from an ideal gas of well separated, noninteracting breathers (solitons) to a special limiting state, which we term a breather (soliton) condensate, and whose properties are entirely determined by the pairwise interactions between breathers (solitons). For a nonhomogeneous breather gas, we derive a full set of kinetic equations describing the slow evolution of the density of states and of its carrier wave counterpart. The kinetic equation for soliton gas is recovered by collapsing the Stokes spectral band. A number of concrete examples of breather and soliton gases are considered, demonstrating the efficacy of the developed general theory with broad implications for nonlinear optics, superfluids, and oceanography. In particular, our work provides the theoretical underpinning for the recently observed remarkable connection of the soliton gas dynamics with the long-term evolution of spontaneous modulational instability
Semiclassical dynamics of quasi-one-dimensional, attractive Bose-Einstein condensates
The strongly interacting regime for attractive Bose-Einstein condensates
(BECs) tightly confined in an extended cylindrical trap is studied. For
appropriately prepared, non-collapsing BECs, the ensuing dynamics are found to
be governed by the one-dimensional focusing Nonlinear Schr\"odinger equation
(NLS) in the semiclassical (small dispersion) regime. In spite of the
modulational instability of this regime, some mathematically rigorous results
on the strong asymptotics of the semiclassical limiting solutions were obtained
recently. Using these results, "implosion-like" and "explosion-like" events are
predicted whereby an initial hump focuses into a sharp spike which then expands
into rapid oscillations. Seemingly related behavior has been observed in
three-dimensional experiments and models, where a BEC with a sufficient number
of atoms undergoes collapse. The dynamical regimes studied here, however, are
not predicted to undergo collapse. Instead, distinct, ordered structures,
appearing after the "implosion", yield interesting new observables that may be
experimentally accessible.Comment: 9 pages, 3 figure
Universality in the profile of the semiclassical limit solutions to the focusing Nonlinear Schroedinger equation at the first breaking curve
We consider the semiclassical (zero-dispersion) limit of the one-dimensional
focusing Nonlinear Schroedinger equation (NLS) with decaying potentials. If a
potential is a simple rapidly oscillating wave (the period has the order of the
semiclassical parameter epsilon) with modulated amplitude and phase, the
space-time plane subdivides into regions of qualitatively different behavior,
with the boundary between them consisting typically of collection of piecewise
smooth arcs (breaking curve(s)). In the first region the evolution of the
potential is ruled by modulation equations (Whitham equations), but for every
value of the space variable x there is a moment of transition (breaking), where
the solution develops fast, quasi-periodic behavior, i.e., the amplitude
becomes also fastly oscillating at scales of order epsilon. The very first
point of such transition is called the point of gradient catastrophe. We study
the detailed asymptotic behavior of the left and right edges of the interface
between these two regions at any time after the gradient catastrophe. The main
finding is that the first oscillations in the amplitude are of nonzero
asymptotic size even as epsilon tends to zero, and they display two separate
natural scales; of order epsilon in the parallel direction to the breaking
curve in the (x,t)-plane, and of order epsilon ln(epsilon) in a transversal
direction. The study is based upon the inverse-scattering method and the
nonlinear steepest descent method.Comment: 40 pages, 10 figure
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