7,257 research outputs found
Regular and Singular Pulse and Front Solutions and Possible Isochronous Behavior in the Short-Pulse Equation: Phase-Plane, Multi-Infinite Series and Variational Approaches
In this paper we employ three recent analytical approaches to investigate the
possible classes of traveling wave solutions of some members of a family of
so-called short-pulse equations (SPE). A recent, novel application of
phase-plane analysis is first employed to show the existence of breaking kink
wave solutions in certain parameter regimes. Secondly, smooth traveling waves
are derived using a recent technique to derive convergent multi-infinite series
solutions for the homoclinic (heteroclinic) orbits of the traveling-wave
equations for the SPE equation, as well as for its generalized version with
arbitrary coefficients. These correspond to pulse (kink or shock) solutions
respectively of the original PDEs.
Unlike the majority of unaccelerated convergent series, high accuracy is
attained with relatively few terms. And finally, variational methods are
employed to generate families of both regular and embedded solitary wave
solutions for the SPE PDE. The technique for obtaining the embedded solitons
incorporates several recent generalizations of the usual variational technique
and it is thus topical in itself. One unusual feature of the solitary waves
derived here is that we are able to obtain them in analytical form (within the
assumed ansatz for the trial functions). Thus, a direct error analysis is
performed, showing the accuracy of the resulting solitary waves. Given the
importance of solitary wave solutions in wave dynamics and information
propagation in nonlinear PDEs, as well as the fact that not much is known about
solutions of the family of generalized SPE equations considered here, the
results obtained are both new and timely.Comment: accepted for publication in Communications in Nonlinear Science and
Numerical Simulatio
The laminar generalized Stokes layer and turbulent drag reduction
This paper considers plane channel flow modified by waves of spanwise
velocity applied at the wall and travelling along the streamwise direction.
Laminar and turbulent regimes for the streamwise flow are both studied.
When the streamwise flow is laminar, it is unaffected by the spanwise flow
induced by the waves. This flow is a thin, unsteady and streamwise-modulated
boundary layer that can be expressed in terms of the Airy function of the first
kind. We name it the generalized Stokes layer because it reduces to the
classical oscillating Stokes layer in the limit of infinite wave speed.
When the streamwise flow is turbulent, the laminar generalized Stokes layer
solution describes well the space-averaged turbulent spanwise flow, provided
that the phase speed of the waves is sufficiently different from the turbulent
convection velocity, and that the time scale of the forcing is smaller than the
life time of the near-wall turbulent structures. Under these conditions, the
drag reduction is found to scale with the Stokes layer thickness, which renders
the laminar solution instrumental for the analysis of the turbulent flow.
A classification of the turbulent flow regimes induced by the waves is
presented by comparing parameters related to the forcing conditions with the
space and time scales of the turbulent flow.Comment: Accepted for publication on J. Fluid Mec
Particle-Hole Asymmetry and Brightening of Solitons in A Strongly Repulsive BEC
We study solitary wave propagation in the condensate of a system of hard-core
bosons with nearest-neighbor interactions. For this strongly repulsive system,
the evolution equation for the condensate order parameter of the system,
obtained using spin coherent state averages is different from the usual
Gross-Pitaevskii equation (GPE). The system is found to support two kinds of
solitons when there is a particle-hole imbalance: a dark soliton that dies out
as the velocity approaches the sound velocity, and a new type of soliton which
brightens and persists all the way up to the sound velocity, transforming into
a periodic wave train at supersonic speed. Analogous to the GPE soliton, the
energy-momentum dispersion for both solitons is characterized by Lieb II modes.Comment: Accepted for publication in PRL, Nov 12, 200
A novel implementation of Petrov-Galerkin method to shallow water solitary wave pattern and superperiodic traveling wave and its multistability: generalized Korteweg-de Vries equation
This work deals with the constitute of numerical solutions of the generalized Korteweg-de Vries (GKdV) equation with Petrov-Galerkin finite element approach utilising a cubic B-spline function as the trial function and a quadratic function as the test function. Accurateness and effectiveness of the submitted methods are shown by employing propagation of single solitary wave. The L2, L∞error norms and I1, I2and I3invariants are used to validate the applicability and durability of our numerical algorithm. Implementing the Von-Neumann theory, it is manifested that the suggested method is marginally stable. Furthermore, supernonlinear traveling wave solution of the GKdV equation is presented using phase plots. It is seen that the GKdV equation supports superperiodic traveling wave solution only and it is significantly affected by velocity and nonlinear parameters. Also, considering a superficial periodic forcing multistability of traveling waves of perturbed GKdV equation is presented. It is found that the perturbed GKdV equation
supports coexisting chaotic and various quasiperiodic features with same parametric values at different initial condition
Relaxed micromorphic broadband scattering for finite-size meta-structures -- a detailed development
The conception of new metamaterials showing unorthodox behaviors with respect
to elastic wavepropagation has become possible in recent years thanks to
powerful dynamical homogenization techniques. Such methods effectively allow to
describe the behavior of an infinite medium generated by periodically
architectured base materials. Nevertheless, when it comes to the study of the
scattering properties of finite-sized structures, dealing with the correct
boundary conditions at the macroscopicscale becomes challenging. In this paper,
we show how finite-domain boundary value problems canbe set-up in the framework
of enriched continuum mechanics (relaxed micromorphic model) by imposing
continuity of macroscopic displacement and of generalized traction when
non-local effects areneglected.The case of a metamaterial slab of finite width
is presented, its scattering properties are studied viaa semi-analytical
solution of the relaxed micromorphic model and compared to numerical
simulationsencoding all details of the selected microstructure. The reflection
coefficient obtained via the twomethods is presented as a function of the
frequency and of the direction of propagation of the incidentwave. We find
excellent agreement for a large range of frequencies going from the long-wave
limitto frequencies beyond the first band-gap and for angles of incidence
ranging from normal to nearparallel incidence. The case of a semi-infinite
metamaterial is also presented and is seen to be areliable measure of the
average behavior of the finite metastructure. A tremendous gain in termsof
computational time is obtained when using the relaxed micromorphic model for
the study of theconsidered metastructure
Hybrid analytical/numerical coupled-mode modeling of guided-wave devices
A general version of coupled-mode-theory for frequency domain scattering problems in integrated optics is proposed. As a prerequisite a physically reasonable field template is required, that typically combines modes of the optical channels in the structure with coefficient functions of in principle arbitrary coordinates. Upon 1-D discretizations of these amplitude functions into finite elements, a Galerkin procedure reduces the problem to a system of linear equations in the element coefficients, where given input amplitudes are included. Smooth approximate solutions are obtained by solving the system in a least squares sense. The versatility of the approach is illustrated by means of a series of 2-D examples, including a perpendicular crossing of waveguides, and a grating-assisted rectangular resonator. As an appendix, we show that alternatively a similar procedure can be derived by variational means, i.e. by restricting a suitable functional representation of the full 2-D/3-D vectorial scattering problem (with transparent influx boundary conditions for inhomogeneous exterior) to the respective field templates.\u
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