1,700 research outputs found
On integrability of the differential constraints arising from the singularity analysis
Integrability of the differential constraints arising from the singularity
analysis of two (1+1)-dimensional second-order evolution equations is studied.
Two nonlinear ordinary differential equations are obtained in this way, which
are integrable by quadratures in spite of very complicated branching of their
solutions.Comment: arxiv version is already offcia
A Bi-Hamiltonian Structure for the Integrable, Discrete Non-Linear Schrodinger System
This paper shows that the Ablowitz-Ladik hierarchy of equations (a well-known
integrable discretization of the Non-linear Schrodinger system) can be
explicitly viewed as a hierarchy of commuting flows which: (a) are Hamiltonian
with respect to both a standard, local Poisson operator J and a new non-local,
skew, almost Poisson operator K, on the appropriate space; (b) can be
recursively generated from a recursion operator R (obtained by composing K and
the inverse of J.) In addition, the proof of these facts relies upon two new
pivotal resolvent identities which suggest a general method for uncovering
bi-Hamiltonian structures for other families of discrete, integrable equations.Comment: 33 page
Sharp bounds on enstrophy growth in the viscous Burgers equation
We use the Cole--Hopf transformation and the Laplace method for the heat
equation to justify the numerical results on enstrophy growth in the viscous
Burgers equation on the unit circle. We show that the maximum enstrophy
achieved in the time evolution is scaled as , where
is the large initial enstrophy, whereas the time needed for
reaching the maximal enstrophy is scaled as . These bounds
are sharp for sufficiently smooth initial conditions.Comment: 12 page
Collisions of solitons and vortex rings in cylindrical Bose-Einstein condensates
Interactions of solitary waves in a cylindrically confined Bose-Einstein
condensate are investigated by simulating their head-on collisions. Slow vortex
rings and fast solitons are found to collide elastically contrary to the
situation in the three-dimensional homogeneous Bose gas. Strongly inelastic
collisions are absent for low density condensates but occur at higher densities
for intermediate velocities. The scattering behaviour is rationalised by use of
dispersion diagrams. During inelastic collisions, spherical shell-like
structures of low density are formed and they eventually decay into depletion
droplets with solitary wave features. The relation to similar shells observed
in a recent experiment [Ginsberg et al. Phys Rev. Lett. 94, 040403 (2005)] is
discussed
Ablowitz-Ladik system with discrete potential. I. Extended resolvent
Ablowitz-Ladik linear system with range of potential equal to {0,1} is
considered. The extended resolvent operator of this system is constructed and
the singularities of this operator are analyzed in detail.Comment: To be published in Theor. Math. Phy
Energy transmission in the forbidden bandgap of a nonlinear chain
A nonlinear chain driven by one end may propagate energy in the forbidden
band gap by means of nonlinear modes. For harmonic driving at a given
frequency, the process ocurs at a threshold amplitude by sudden large energy
flow, that we call nonlinear supratransmission. The bifurcation of energy
transmission is demonstrated numerically and experimentally on the chain of
coupled pendula (sine-Gordon and nonlinear Klein-Gordon equations) and
sustained by an extremely simple theory.Comment: LaTex file, 6 figures, published in Phys Rev Lett 89 (2002) 13410
Spectral decomposition for the Dirac system associated to the DSII equation
A new (scalar) spectral decomposition is found for the Dirac system in two
dimensions associated to the focusing Davey--Stewartson II (DSII) equation.
Discrete spectrum in the spectral problem corresponds to eigenvalues embedded
into a two-dimensional essential spectrum. We show that these embedded
eigenvalues are structurally unstable under small variations of the initial
data. This instability leads to the decay of localized initial data into
continuous wave packets prescribed by the nonlinear dynamics of the DSII
equation
Exact solution of Riemann--Hilbert problem for a correlation function of the XY spin chain
A correlation function of the XY spin chain is studied at zero temperature.
This is called the Emptiness Formation Probability (EFP) and is expressed by
the Fredholm determinant in the thermodynamic limit. We formulate the
associated Riemann--Hilbert problem and solve it exactly. The EFP is shown to
decay in Gaussian.Comment: 7 pages, to be published in J. Phys. Soc. Jp
Yang-Baxter and reflection maps from vector solitons with a boundary
Based on recent results obtained by the authors on the inverse scattering
method of the vector nonlinear Schr\"odinger equation with integrable boundary
conditions, we discuss the factorization of the interactions of N-soliton
solutions on the half-line. Using dressing transformations combined with a
mirror image technique, factorization of soliton-soliton and soliton-boundary
interactions is proved. We discover a new object, which we call reflection map,
that satisfies a set-theoretical reflection equation which we also introduce.
Two classes of solutions for the reflection map are constructed. Finally, basic
aspects of the theory of set-theoretical reflection equations are introduced.Comment: 29 pages. Featured article in Nonlinearit
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