246 research outputs found
Spectrum of a non-self-adjoint operator associated with the periodic heat equation
We study the spectrum of the linear operator subject to the
periodic boundary conditions on . We prove that the
operator is closed in with the domain in for , its spectrum consists of an infinite
sequence of isolated eigenvalues and the set of corresponding eigenfunctions is
complete. By using numerical approximations of eigenvalues and eigenfunctions,
we show that all eigenvalues are simple, located on the imaginary axis and the
angle between two subsequent eigenfunctions tends to zero for larger
eigenvalues. As a result, the complete set of linearly independent
eigenfunctions does not form a basis in .Comment: 22 pages, 10 figure
Rigorous justification of the short-pulse equation
We prove that the short-pulse equation, which is derived from Maxwell
equations with formal asymptotic methods, can be rigorously justified. The
justification procedure applies to small-norm solutions of the short-pulse
equation. Although the small-norm solutions exist for infinite times and
include modulated pulses and their elastic interactions, the error bound for
arbitrary initial data can only be controlled over finite time intervals.Comment: 15 pages, no figure
Orbital stability in the cubic defocusing NLS equation: I. Cnoidal periodic waves
Periodic waves of the one-dimensional cubic defocusing NLS equation are
considered. Using tools from integrability theory, these waves have been shown
in [Bottman, Deconinck, and Nivala, 2011] to be linearly stable and the
Floquet-Bloch spectrum of the linearized operator has been explicitly computed.
We combine here the first four conserved quantities of the NLS equation to give
a direct proof that cnoidal periodic waves are orbitally stable with respect to
subharmonic perturbations, with period equal to an integer multiple of the
period of the wave. Our result is not restricted to the periodic waves of small
amplitudes.Comment: 28 pages, 3 figures. Main result strengthened by removing a smallness
condition. Limiting case of the black soliton now postponed to a companion
pape
Surface gap solitons at a nonlinearity interface
We demonstrate existence of waves localized at the interface of two nonlinear
periodic media with different coefficients of the cubic nonlinearity via the
one-dimensional Gross--Pitaevsky equation. We call these waves the surface gap
solitons (SGS). In the case of smooth symmetric periodic potentials, we study
analytically bifurcations of SGS's from standard gap solitons and determine
numerically the maximal jump of the nonlinearity coefficient allowing for the
SGS existence. We show that the maximal jump vanishes near the thresholds of
bifurcations of gap solitons. In the case of continuous potentials with a jump
in the first derivative at the interface, we develop a homotopy method of
continuation of SGS families from the solution obtained via gluing of parts of
the standard gap solitons and study existence of SGS's in the photonic band
gaps. We explain the termination of the SGS families in the interior points of
the band gaps from the bifurcation of linear bound states in the continuous
non-smooth potentials.Comment: 23 pages, 6 figures, 3 tables corrections in v.2: sign error in the
energy functional on p.3; discussion of the symmetries of Bloch functions on
p. 5-6 corrected; derivative symbol missing in (3.5) and in the formula for
\mu below (3.6
Global existence of small-norm solutions in the reduced Ostrovsky equation
We use a novel transformation of the reduced Ostrovsky equation to the
integrable Tzitz\'eica equation and prove global existence of small-norm
solutions in Sobolev space . This scenario is an alternative to
finite-time wave breaking of large-norm solutions of the reduced Ostrovsky
equation. We also discuss a sharp sufficient condition for the finite-time wave
breaking.Comment: 11 pages; 1 figur
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