135 research outputs found
Supernova Hosts for Gamma-Ray Burst Jets: Dynamical Constraints
I constrain a possible supernova origin for gamma-ray bursts by modeling the
dynamical interaction between a relativistic jet and a stellar envelope
surrounding it. The delay in observer's time introduced by the jet traversing
the envelope should not be long compared to the duration of gamma-ray emission;
also, the jet should not be swallowed by a spherical explosion it powers. The
only stellar progenitors that comfortably satisfy these constraints, if one
assumes that jets move ballistically within their host stars, are compact
carbon-oxygen or helium post-Wolf-Rayet stars (type Ic or Ib supernovae); type
II supernovae are ruled out. Notably, very massive stars do not appear capable
of producing the observed bursts at any redshift unless the stellar envelope is
stripped prior to collapse. The presence of a dense stellar wind places an
upper limit on the Lorentz factor of the jet in the internal shock model;
however, this constraint may be evaded if the wind is swept forward by a photon
precursor. Shock breakout and cocoon blowout are considered individually;
neither presents a likely source of precursors for cosmological GRBs.
These envelope constraints could conceivably be circumvented if jets are
laterally pressure-confined while traversing the outer stellar envelope. If so,
jets responsible for observed GRBs must either have been launched from a region
several hundred kilometers wide, or have mixed with envelope material as they
travel. A phase of pressure confinement and mixing would imprint correlations
among jets that may explain observed GRB variability-luminosity and
lag-luminosity correlations.Comment: 17 pages, MNRAS, accepted. Contains new analysis of pressure-confined
jets, of jets that experience oblique shocks or mix with their cocoons, and
of cocoons after breakou
Algebro-geometric approach in the theory of integrable hydrodynamic type systems
The algebro-geometric approach for integrability of semi-Hamiltonian
hydrodynamic type systems is presented. This method is significantly simplified
for so-called symmetric hydrodynamic type systems. Plenty interesting and
physically motivated examples are investigated
Nonlocalized modulation of periodic reaction diffusion waves: The Whitham equation
In a companion paper, we established nonlinear stability with detailed
diffusive rates of decay of spectrally stable periodic traveling-wave solutions
of reaction diffusion systems under small perturbations consisting of a
nonlocalized modulation plus a localized perturbation. Here, we determine
time-asymptotic behavior under such perturbations, showing that solutions
consist to leading order of a modulation whose parameter evolution is governed
by an associated Whitham averaged equation
Modulational Instability in Equations of KdV Type
It is a matter of experience that nonlinear waves in dispersive media,
propagating primarily in one direction, may appear periodic in small space and
time scales, but their characteristics --- amplitude, phase, wave number, etc.
--- slowly vary in large space and time scales. In the 1970's, Whitham
developed an asymptotic (WKB) method to study the effects of small
"modulations" on nonlinear periodic wave trains. Since then, there has been a
great deal of work aiming at rigorously justifying the predictions from
Whitham's formal theory. We discuss recent advances in the mathematical
understanding of the dynamics, in particular, the instability of slowly
modulated wave trains for nonlinear dispersive equations of KdV type.Comment: 40 pages. To appear in upcoming title in Lecture Notes in Physic
Modeling water waves beyond perturbations
In this chapter, we illustrate the advantage of variational principles for
modeling water waves from an elementary practical viewpoint. The method is
based on a `relaxed' variational principle, i.e., on a Lagrangian involving as
many variables as possible, and imposing some suitable subordinate constraints.
This approach allows the construction of approximations without necessarily
relying on a small parameter. This is illustrated via simple examples, namely
the Serre equations in shallow water, a generalization of the Klein-Gordon
equation in deep water and how to unify these equations in arbitrary depth. The
chapter ends with a discussion and caution on how this approach should be used
in practice.Comment: 15 pages, 1 figure, 39 references. This document is a contributed
chapter to an upcoming volume to be published by Springer in Lecture Notes in
Physics Series. Other author's papers can be downloaded at
http://www.denys-dutykh.com
Alfv\'en Reflection and Reverberation in the Solar Atmosphere
Magneto-atmospheres with Alfv\'en speed [a] that increases monotonically with
height are often used to model the solar atmosphere, at least out to several
solar radii. A common example involves uniform vertical or inclined magnetic
field in an isothermal atmosphere, for which the Alfv\'en speed is exponential.
We address the issue of internal reflection in such atmospheres, both for
time-harmonic and for transient waves. It is found that a mathematical boundary
condition may be devised that corresponds to perfect absorption at infinity,
and, using this, that many atmospheres where a(x) is analytic and unbounded
present no internal reflection of harmonic Alfv\'en waves. However, except for
certain special cases, such solutions are accompanied by a wake, which may be
thought of as a kind of reflection. For the initial-value problem where a
harmonic source is suddenly switched on (and optionally off), there is also an
associated transient that normally decays with time as O(t-1) or O(t-1 ln t),
depending on the phase of the driver. Unlike the steady-state harmonic
solutions, the transient does reflect weakly. Alfv\'en waves in the solar
corona driven by a finite-duration train of p-modes are expected to leave such
transients.Comment: Accepted by Solar Physic
Multimode solutions of first-order elliptic quasilinear systems obtained from Riemann invariants
Two new approaches to solving first-order quasilinear elliptic systems of
PDEs in many dimensions are proposed. The first method is based on an analysis
of multimode solutions expressible in terms of Riemann invariants, based on
links between two techniques, that of the symmetry reduction method and of the
generalized method of characteristics. A variant of the conditional symmetry
method for constructing this type of solution is proposed. A specific feature
of that approach is an algebraic-geometric point of view, which allows the
introduction of specific first-order side conditions consistent with the
original system of PDEs, leading to a generalization of the Riemann invariant
method for solving elliptic homogeneous systems of PDEs. A further
generalization of the Riemann invariants method to the case of inhomogeneous
systems, based on the introduction of specific rotation matrices, enables us to
weaken the integrability condition. It allows us to establish a connection
between the structure of the set of integral elements and the possibility of
constructing specific classes of simple mode solutions. These theoretical
considerations are illustrated by the examples of an ideal plastic flow in its
elliptic region and a system describing a nonlinear interaction of waves and
particles. Several new classes of solutions are obtained in explicit form,
including the general integral for the latter system of equations
On the fourth-order accurate compact ADI scheme for solving the unsteady Nonlinear Coupled Burgers' Equations
The two-dimensional unsteady coupled Burgers' equations with moderate to
severe gradients, are solved numerically using higher-order accurate finite
difference schemes; namely the fourth-order accurate compact ADI scheme, and
the fourth-order accurate Du Fort Frankel scheme. The question of numerical
stability and convergence are presented. Comparisons are made between the
present schemes in terms of accuracy and computational efficiency for solving
problems with severe internal and boundary gradients. The present study shows
that the fourth-order compact ADI scheme is stable and efficient
New Lump-like Structures in Scalar-field Models
In this work we investigate lump-like solutions in models described by a
single real scalar field. We start considering non-topological solutions with
the usual lump-like form, and then we study other models, where the bell-shape
profile may have varying amplitude and width, or develop a flat plateau at its
top, or even induce a lump on top of another lump. We suggest possible
applications where these exotic solutions might be used in several distinct
branches of physics.Comment: REvTex4, twocolumn, 10 pages, 9 figures; new reference added, to
appear in EPJ
k-Essence, superluminal propagation, causality and emergent geometry
The k-essence theories admit in general the superluminal propagation of the
perturbations on classical backgrounds. We show that in spite of the
superluminal propagation the causal paradoxes do not arise in these theories
and in this respect they are not less safe than General Relativity.Comment: 34 pages, 5 figure
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