96 research outputs found
Reformulating the Schrodinger equation as a Shabat-Zakharov system
We reformulate the second-order Schrodinger equation as a set of two coupled
first order differential equations, a so-called "Shabat-Zakharov system",
(sometimes called a "Zakharov-Shabat" system). There is considerable
flexibility in this approach, and we emphasise the utility of introducing an
"auxiliary condition" or "gauge condition" that is used to cut down the degrees
of freedom. Using this formalism, we derive the explicit (but formal) general
solution to the Schrodinger equation. The general solution depends on three
arbitrarily chosen functions, and a path-ordered exponential matrix. If one
considers path ordering to be an "elementary" process, then this represents
complete quadrature, albeit formal, of the second-order linear ODE.Comment: 18 pages, plain LaTe
Long-Time Asymptotics for the Korteweg-de Vries Equation via Nonlinear Steepest Descent
We apply the method of nonlinear steepest descent to compute the long-time
asymptotics of the Korteweg-de Vries equation for decaying initial data in the
soliton and similarity region. This paper can be viewed as an expository
introduction to this method.Comment: 31 page
Existence and stability of hole solutions to complex Ginzburg-Landau equations
We consider the existence and stability of the hole, or dark soliton,
solution to a Ginzburg-Landau perturbation of the defocusing nonlinear
Schroedinger equation (NLS), and to the nearly real complex Ginzburg-Landau
equation (CGL). By using dynamical systems techniques, it is shown that the
dark soliton can persist as either a regular perturbation or a singular
perturbation of that which exists for the NLS. When considering the stability
of the soliton, a major difficulty which must be overcome is that eigenvalues
may bifurcate out of the continuous spectrum, i.e., an edge bifurcation may
occur. Since the continuous spectrum for the NLS covers the imaginary axis, and
since for the CGL it touches the origin, such a bifurcation may lead to an
unstable wave. An additional important consideration is that an edge
bifurcation can happen even if there are no eigenvalues embedded in the
continuous spectrum. Building on and refining ideas first presented in Kapitula
and Sandstede (Physica D, 1998) and Kapitula (SIAM J. Math. Anal., 1999), we
show that when the wave persists as a regular perturbation, at most three
eigenvalues will bifurcate out of the continuous spectrum. Furthermore, we
precisely track these bifurcating eigenvalues, and thus are able to give
conditions for which the perturbed wave will be stable. For the NLS the results
are an improvement and refinement of previous work, while the results for the
CGL are new. The techniques presented are very general and are therefore
applicable to a much larger class of problems than those considered here.Comment: 41 pages, 4 figures, submitte
Parametric Forcing of Waves with Non-Monotonic Dispersion Relation: Domain Structures in Ferrofluids?
Surface waves on ferrofluids exposed to a dc-magnetic field exhibit a
non-monotonic dispersion relation. The effect of a parametric driving on such
waves is studied within suitable coupled Ginzburg-Landau equations. Due to the
non-monotonicity the neutral curve for the excitation of standing waves can
have up to three minima. The stability of the waves with respect to long-wave
perturbations is determined a phase-diffusion equation. It shows that the
band of stable wave numbers can split up into two or three sub-bands. The
resulting competition between the wave numbers corresponding to the respective
sub-bands leads quite naturally to patterns consisting of multiple domains of
standing waves which differ in their wave number. The coarsening dynamics of
such domain structures is addressed.Comment: 23 pages, 6 postscript figures, composed using RevTeX. Submitted to
PR
On the Cauchy Problem for the Korteweg-de Vries Equation with Steplike Finite-Gap Initial Data I. Schwartz-Type Perturbations
We solve the Cauchy problem for the Korteweg-de Vries equation with initial
conditions which are steplike Schwartz-type perturbations of finite-gap
potentials under the assumption that the respective spectral bands either
coincide or are disjoint.Comment: 29 page
Boundary Limitation of Wavenumbers in Taylor-Vortex Flow
We report experimental results for a boundary-mediated wavenumber-adjustment
mechanism and for a boundary-limited wavenumber-band of Taylor-vortex flow
(TVF). The system consists of fluid contained between two concentric cylinders
with the inner one rotating at an angular frequency . As observed
previously, the Eckhaus instability (a bulk instability) is observed and limits
the stable wavenumber band when the system is terminated axially by two rigid,
non-rotating plates. The band width is then of order at small
() and agrees well with
calculations based on the equations of motion over a wide -range.
When the cylinder axis is vertical and the upper liquid surface is free (i.e.
an air-liquid interface), vortices can be generated or expelled at the free
surface because there the phase of the structure is only weakly pinned. The
band of wavenumbers over which Taylor-vortex flow exists is then more narrow
than the stable band limited by the Eckhaus instability. At small
the boundary-mediated band-width is linear in . These results are
qualitatively consistent with theoretical predictions, but to our knowledge a
quantitative calculation for TVF with a free surface does not exist.Comment: 8 pages incl. 9 eps figures bitmap version of Fig
Instability of small-amplitude convective flows in a rotating layer with stress-free boundaries
We consider stability of steady convective flows in a horizontal layer with
stress-free boundaries, heated below and rotating about the vertical axis, in
the Boussinesq approximation (the Rayleigh-Benard convection). The flows under
consideration are convective rolls or square cells, the latter being
asymptotically equal to the sum of two orthogonal rolls of the same wave number
k. We assume, that the Rayleigh number R is close to the critical one, R_c(k),
for the onset of convective flows of this wave number: R=R_c(k)+epsilon^2; the
amplitude of the flows is of the order of epsilon. We show that the flows are
always unstable to perturbations, which are a sum of a large-scale mode not
involving small scales, and two large-scale modes, modulated by the original
rolls rotated by equal small angles in the opposite directions. The maximal
growth rate of the instability is of the order of max(epsilon^{8/5},(k-k_c)^2),
where k_c is the critical wave number for the onset of convection.Comment: Latex, 12 pp., 15 refs. An improved version of the manuscript
submitted to "Mechanics of fluid and gas", 2006 (in Russian; English
translation "Fluid Dynamics"
Asymmetric gap soliton modes in diatomic lattices with cubic and quartic nonlinearity
Nonlinear localized excitations in one-dimensional diatomic lattices with
cubic and quartic nonlinearity are considered analytically by a
quasi-discreteness approach. The criteria for the occurence of asymmetric gap
solitons (with vibrating frequency lying in the gap of phonon bands) and
small-amplitude, asymmetric intrinsic localized modes (with the vibrating
frequency being above all the phonon bands) are obtained explicitly based on
the modulational instabilities of corresponding linear lattice plane waves. The
expressions of particle displacement for all these nonlinear localized
excitations are also given. The result is applied to standard two-body
potentials of the Toda, Born-Mayer-Coulomb, Lennard-Jones, and Morse type. The
comparison with previous numerical study of the anharmonic gap modes in
diatomic lattices for the standard two-body potentials is made and good
agreement is found.Comment: 24 pages in Revtex, 2 PS figure
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