18,205 research outputs found
Oscillatory spatially periodic weakly nonlinear gravity waves on deep water
A weakly nonlinear Hamiltonian model is derived from the exact water wave equations to study the time evolution of spatially periodic wavetrains. The model assumes that the spatial spectrum of the wavetrain is formed by only three free waves, i.e. a carrier and two side bands. The model has the same symmetries and invariances as the exact equations. As a result, it is found that not only the permanent form travelling waves and their stability are important in describing the time evolution of the waves, but also a new kind of family of solutions which has two basic frequencies plays a crucial role in the dynamics of the waves. It is also shown that three is the minimum number of free waves which is necessary to have chaotic behaviour of water waves
Decay of travelling waves in dissipative Poisson systems
In many finite and infinite dimensional systemslow-dimensional behaviour is often observed. That is to say, the dynamics, observed experimentally or numerically, looks as if it can be described (approximately) with only a few essential parameters. Choosing the correct set of such ldquorobust observablesrdquo is an essential ingredient of a successful low dimensional description. This paper reports on a specific example of a more general approach that aims at describing certain (low dimensional) phenomena in (high dimensional) damped/driven equations with parameters that are essentially determined from the underlying conservative part of the equation. In particular, a Hamiltonian or a Poisson structure of the conservative part is exploited to find (characterize) families of exact solutions. These solutions are then used as the ldquobase functionsrdquo with the aid of which the solutions of the disturbed system are approximated. This approximation is accomplished using the parameters that characterize the family as variables that depend on time. In this paper, this procedure is applied to a class of systems which admit travelling waves when dissipation is ignored
Conservation laws, exact travelling waves and modulation instability for an extended nonlinear Schr\"odinger equation
We study various properties of solutions of an extended nonlinear
Schr\"{o}dinger (ENLS) equation, which arises in the context of geometric
evolution problems -- including vortex filament dynamics -- and governs
propagation of short pulses in optical fibers and nonlinear metamaterials. For
the periodic initial-boundary value problem, we derive conservation laws
satisfied by local in time, weak (distributional) solutions, and
establish global existence of such weak solutions. The derivation is obtained
by a regularization scheme under a balance condition on the coefficients of the
linear and nonlinear terms -- namely, the Hirota limit of the considered ENLS
model. Next, we investigate conditions for the existence of traveling wave
solutions, focusing on the case of bright and dark solitons. The balance
condition on the coefficients is found to be essential for the existence of
exact analytical soliton solutions; furthermore, we obtain conditions which
define parameter regimes for the existence of traveling solitons for various
linear dispersion strengths. Finally, we study the modulational instability of
plane waves of the ENLS equation, and identify important differences between
the ENLS case and the corresponding NLS counterpart. The analytical results are
corroborated by numerical simulations, which reveal notable differences between
the bright and the dark soliton propagation dynamics, and are in excellent
agreement with the analytical predictions of the modulation instability
analysis.Comment: 27 pages, 5 figures. To be published in Journal of Physics A:
Mathematical and Theoretica
Falling liquid films with blowing and suction
Flow of a thin viscous film down a flat inclined plane becomes unstable to
long wave interfacial fluctuations when the Reynolds number based on the mean
film thickness becomes larger than a critical value (this value decreases as
the angle of inclination with the horizontal increases, and in particular
becomes zero when the plate is vertical). Control of these interfacial
instabilities is relevant to a wide range of industrial applications including
coating processes and heat or mass transfer systems. This study considers the
effect of blowing and suction through the substrate in order to construct from
first principles physically realistic models that can be used for detailed
passive and active control studies of direct relevance to possible experiments.
Two different long-wave, thin-film equations are derived to describe this
system; these include the imposed blowing/suction as well as inertia, surface
tension, gravity and viscosity. The case of spatially periodic blowing and
suction is considered in detail and the bifurcation structure of forced steady
states is explored numerically to predict that steady states cease to exist for
sufficiently large suction speeds since the film locally thins to zero
thickness giving way to dry patches on the substrate. The linear stability of
the resulting nonuniform steady states is investigated for perturbations of
arbitrary wavelengths, and any instabilities are followed into the fully
nonlinear regime using time-dependent computations. The case of small amplitude
blowing/suction is studied analytically both for steady states and their
stability. Finally, the transition between travelling waves and non-uniform
steady states is explored as the suction amplitude increases
A KdV-like advection-dispersion equation with some remarkable properties
We discuss a new non-linear PDE, u_t + (2 u_xx/u) u_x = epsilon u_xxx,
invariant under scaling of dependent variable and referred to here as SIdV. It
is one of the simplest such translation and space-time reflection-symmetric
first order advection-dispersion equations. This PDE (with dispersion
coefficient unity) was discovered in a genetic programming search for equations
sharing the KdV solitary wave solution. It provides a bridge between non-linear
advection, diffusion and dispersion. Special cases include the mKdV and linear
dispersive equations. We identify two conservation laws, though initial
investigations indicate that SIdV does not follow from a polynomial Lagrangian
of the KdV sort. Nevertheless, it possesses solitary and periodic travelling
waves. Moreover, numerical simulations reveal recurrence properties usually
associated with integrable systems. KdV and SIdV are the simplest in an
infinite dimensional family of equations sharing the KdV solitary wave. SIdV
and its generalizations may serve as a testing ground for numerical and
analytical techniques and be a rich source for further explorations.Comment: 15 pages, 4 figures, corrected sign typo in KdV Lagrangian above
equation 3
Standing and travelling waves in cylindrical Rayleigh-Benard convection
The Boussinesq equations for Rayleigh-Benard convection are simulated for a
cylindrical container with an aspect ratio near 1.5. The transition from an
axisymmetric stationary flow to time-dependent flows is studied using nonlinear
simulations, linear stability analysis and bifurcation theory. At a Rayleigh
number near 25,000, the axisymmetric flow becomes unstable to standing or
travelling azimuthal waves. The standing waves are slightly unstable to
travelling waves. This scenario is identified as a Hopf bifurcation in a system
with O(2) symmetry
Nonlinear waves in Newton's cradle and the discrete p-Schroedinger equation
We study nonlinear waves in Newton's cradle, a classical mechanical system
consisting of a chain of beads attached to linear pendula and interacting
nonlinearly via Hertz's contact forces. We formally derive a spatially discrete
modulation equation, for small amplitude nonlinear waves consisting of slow
modulations of time-periodic linear oscillations. The fully-nonlinear and
unilateral interactions between beads yield a nonstandard modulation equation
that we call the discrete p-Schroedinger (DpS) equation. It consists of a
spatial discretization of a generalized Schroedinger equation with p-Laplacian,
with fractional p>2 depending on the exponent of Hertz's contact force. We show
that the DpS equation admits explicit periodic travelling wave solutions, and
numerically find a plethora of standing wave solutions given by the orbits of a
discrete map, in particular spatially localized breather solutions. Using a
modified Lyapunov-Schmidt technique, we prove the existence of exact periodic
travelling waves in the chain of beads, close to the small amplitude modulated
waves given by the DpS equation. Using numerical simulations, we show that the
DpS equation captures several other important features of the dynamics in the
weakly nonlinear regime, namely modulational instabilities, the existence of
static and travelling breathers, and repulsive or attractive interactions of
these localized structures
- …