72 research outputs found
Initial value problems in linear integral operator equations
For some general linear integral operator equations, we investigate consequent initial value problems by using the theory of reproducing kernels. A new method is proposed which -- in particular -- generates a new field among initial value problems, linear integral operators, eigenfunctions and values, integral transforms and reproducing kernels. In particular, examples are worked out for the integral equations of Lalesco-Picard, Dixon and Tricomi types
Functional separable solutions for two classes of nonlinear equations of mathematical physics
This study describes a new modification of the method of functional separation of variables for nonlinear equations of mathematical physics. Solutions are sought in an implicit form that involves several free functions (specific expressions for these functions are determined by analyzing the arising functional differential equations). The effectiveness of the method is illustrated by examples of nonlinear reaction–diffusion equations and Klein–Gordon type equations with variable coefficients that depend on one or more arbitrary functions. A number of new exact functional separable solutions and generalized traveling-wave solutions are obtained
Group classification of variable coefficient KdV-like equations
The exhaustive group classification of the class of KdV-like equations with
time-dependent coefficients is carried out using
equivalence based approach. A simple way for the construction of exact
solutions of KdV-like equations using equivalence transformations is described.Comment: 8 pages; minor misprints are corrected. arXiv admin note: substantial
text overlap with arXiv:1104.198
On the complete analytic structure of the massive gravitino propagator in four-dimensional de Sitter space
With the help of the general theory of the Heun equation, this paper
completes previous work by the authors and other groups on the explicit
representation of the massive gravitino propagator in four-dimensional de
Sitter space. As a result of our original contribution, all weight functions
which multiply the geometric invariants in the gravitino propagator are
expressed through Heun functions, and the resulting plots are displayed and
discussed after resorting to a suitable truncation in the series expansion of
the Heun function. It turns out that there exist two ranges of values of the
independent variable in which the weight functions can be divided into
dominating and sub-dominating family.Comment: 21 pages, 9 figures. The presentation has been further improve
Solvable Systems of Linear Differential Equations
The asymptotic iteration method (AIM) is an iterative technique used to find
exact and approximate solutions to second-order linear differential equations.
In this work, we employed AIM to solve systems of two first-order linear
differential equations. The termination criteria of AIM will be re-examined and
the whole theory is re-worked in order to fit this new application. As a result
of our investigation, an interesting connection between the solution of linear
systems and the solution of Riccati equations is established. Further, new
classes of exactly solvable systems of linear differential equations with
variable coefficients are obtained. The method discussed allow to construct
many solvable classes through a simple procedure.Comment: 13 page
Resonant Absorption as Mode Conversion?
Resonant absorption and mode conversion are both extensively studied
mechanisms for wave "absorption" in solar magnetohydrodynamics (MHD). But are
they really distinct? We re-examine a well-known simple resonant absorption
model in a cold MHD plasma that places the resonance inside an evanescent
region. The normal mode solutions display the standard singular resonant
features. However, these same normal modes may be used to construct a ray
bundle which very clearly undergoes mode conversion to an Alfv\'en wave with no
singularities. We therefore conclude that resonant absorption and mode
conversion are in fact the same thing, at least for this model problem. The
prime distinguishing characteristic that determines which of the two
descriptions is most natural in a given circumstance is whether the converted
wave can provide a net escape of energy from the conversion/absorption region
of physical space. If it cannot, it is forced to run away in wavenumber space
instead, thereby generating the arbitrarily small scales in situ that we
recognize as fundamental to resonant absorption and phase mixing. On the other
hand, if the converted wave takes net energy way, singularities do not develop,
though phase mixing may still develop with distance as the wave recedes.Comment: 23 pages, 8 figures, 2 tables; accepted by Solar Phys (July 9 2010
Quantum evolution across singularities: the case of geometrical resolutions
We continue the study of time-dependent Hamiltonians with an isolated
singularity in their time dependence, describing propagation on singular
space-times. In previous work, two of us have proposed a "minimal subtraction"
prescription for the simplest class of such systems, involving Hamiltonians
with only one singular term. On the other hand, Hamiltonians corresponding to
geometrical resolutions of space-time tend to involve multiple operator
structures (multiple types of dependence on the canonical variables) in an
essential way.
We consider some of the general properties of such (near-)singular
Hamiltonian systems, and further specialize to the case of a free scalar field
on a two-parameter generalization of the null-brane space-time. We find that
the singular limit of free scalar field evolution exists for a discrete subset
of the possible values of the two parameters. The coordinates we introduce
reveal a peculiar reflection property of scalar field propagation on the
generalized (as well as the original) null-brane. We further present a simple
family of pp-wave geometries whose singular limit is a light-like hyperplane
(discontinuously) reflecting the positions of particles as they pass through
it.Comment: 25 pages, 1 figur
Local time and the pricing of time-dependent barrier options
A time-dependent double-barrier option is a derivative security that delivers
the terminal value at expiry if neither of the continuous
time-dependent barriers b_\pm:[0,T]\to \RR_+ have been hit during the time
interval . Using a probabilistic approach we obtain a decomposition of
the barrier option price into the corresponding European option price minus the
barrier premium for a wide class of payoff functions , barrier functions
and linear diffusions . We show that the barrier
premium can be expressed as a sum of integrals along the barriers of
the option's deltas \Delta_\pm:[0,T]\to\RR at the barriers and that the pair
of functions solves a system of Volterra integral
equations of the first kind. We find a semi-analytic solution for this system
in the case of constant double barriers and briefly discus a numerical
algorithm for the time-dependent case.Comment: 32 pages, to appear in Finance and Stochastic
Conformally flat spacetimes and Weyl frames
We discuss the concepts of Weyl and Riemann frames in the context of metric
theories of gravity and state the fact that they are completely equivalent as
far as geodesic motion is concerned. We apply this result to conformally flat
spacetimes and show that a new picture arises when a Riemannian spacetime is
taken by means of geometrical gauge transformations into a Minkowskian flat
spacetime. We find out that in the Weyl frame gravity is described by a scalar
field. We give some examples of how conformally flat spacetime configurations
look when viewed from the standpoint of a Weyl frame. We show that in the
non-relativistic and weak field regime the Weyl scalar field may be identified
with the Newtonian gravitational potential. We suggest an equation for the
scalar field by varying the Einstein-Hilbert action restricted to the class of
conformally-flat spacetimes. We revisit Einstein and Fokker's interpretation of
Nordstr\"om scalar gravity theory and draw an analogy between this approach and
the Weyl gauge formalism. We briefly take a look at two-dimensional gravity as
viewed in the Weyl frame and address the question of quantizing a conformally
flat spacetime by going to the Weyl frame.Comment: LATEX - 18 page
Dynamics with Infinitely Many Derivatives: The Initial Value Problem
Differential equations of infinite order are an increasingly important class
of equations in theoretical physics. Such equations are ubiquitous in string
field theory and have recently attracted considerable interest also from
cosmologists. Though these equations have been studied in the classical
mathematical literature, it appears that the physics community is largely
unaware of the relevant formalism. Of particular importance is the fate of the
initial value problem. Under what circumstances do infinite order differential
equations possess a well-defined initial value problem and how many initial
data are required? In this paper we study the initial value problem for
infinite order differential equations in the mathematical framework of the
formal operator calculus, with analytic initial data. This formalism allows us
to handle simultaneously a wide array of different nonlocal equations within a
single framework and also admits a transparent physical interpretation. We show
that differential equations of infinite order do not generically admit
infinitely many initial data. Rather, each pole of the propagator contributes
two initial data to the final solution. Though it is possible to find
differential equations of infinite order which admit well-defined initial value
problem with only two initial data, neither the dynamical equations of p-adic
string theory nor string field theory seem to belong to this class. However,
both theories can be rendered ghost-free by suitable definition of the action
of the formal pseudo-differential operator. This prescription restricts the
theory to frequencies within some contour in the complex plane and hence may be
thought of as a sort of ultra-violet cut-off.Comment: 40 pages, no figures. Added comments concerning fractional operators
and the implications of restricting the contour of integration. Typos
correcte
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