631 research outputs found
Impact of climate induced glacial melting on coastal marine systems in the Western Antarctic Peninsula region
IMCOAST is an international research program that features a multidisciplinary approach involving geo and biological sciences, field investigations, remote sensing and modeling and knowledge into the hydrographical and biological history of the marine coastal ecosystems of the Western Antarctic Peninsula region
On the reduction of the multidimensional Schroedinger equation to a first order equation and its relation to the pseudoanalytic function theory
Given a particular solution of a one-dimensional stationary Schroedinger
equation (SE) this equation of second order can be reduced to a first order
linear differential equation. This is done with the aid of an auxiliary Riccati
equation. We show that a similar fact is true in a multidimensional situation
also. We consider the case of two or three independent variables. One
particular solution of (SE) allows us to reduce this second order equation to a
linear first order quaternionic differential equation. As in one-dimensional
case this is done with the aid of an auxiliary Riccati equation. The resulting
first order quaternionic equation is equivalent to the static Maxwell system.
In the case of two independent variables it is the Vekua equation from theory
of generalized analytic functions. We show that even in this case it is
necessary to consider not complex valued functions only, solutions of the Vekua
equation but complete quaternionic functions. Then the first order quaternionic
equation represents two separate Vekua equations, one of which gives us
solutions of (SE) and the other can be considered as an auxiliary equation of a
simpler structure. For the auxiliary equation we always have the corresponding
Bers generating pair, the base of the Bers theory of pseudoanalytic functions,
and what is very important, the Bers derivatives of solutions of the auxiliary
equation give us solutions of the main Vekua equation and as a consequence of
(SE). We obtain an analogue of the Cauchy integral theorem for solutions of
(SE). For an ample class of potentials (which includes for instance all radial
potentials), this new approach gives us a simple procedure allowing to obtain
an infinite sequence of solutions of (SE) from one known particular solution
On a factorization of second order elliptic operators and applications
We show that given a nonvanishing particular solution of the equation
(divpgrad+q)u=0 (1) the corresponding differential operator can be factorized
into a product of two first order operators. The factorization allows us to
reduce the equation (1) to a first order equation which in a two-dimensional
case is the Vekua equation of a special form. Under quite general conditions on
the coefficients p and q we obtain an algorithm which allows us to construct in
explicit form the positive formal powers (solutions of the Vekua equation
generalizing the usual powers of the variable z). This result means that under
quite general conditions one can construct an infinite system of exact
solutions of (1) explicitly, and moreover, at least when p and q are real
valued this system will be complete in ker(divpgrad+q) in the sense that any
solution of (1) in a simply connected domain can be represented as an infinite
series of obtained exact solutions which converges uniformly on any compact
subset of . Finally we give a similar factorization of the operator
(divpgrad+q) in a multidimensional case and obtain a natural generalization of
the Vekua equation which is related to second order operators in a similar way
as its two-dimensional prototype does
On the Klein-Gordon equation and hyperbolic pseudoanalytic function theory
Elliptic pseudoanalytic function theory was considered independently by Bers
and Vekua decades ago. In this paper we develop a hyperbolic analogue of
pseudoanalytic function theory using the algebra of hyperbolic numbers. We
consider the Klein-Gordon equation with a potential. With the aid of one
particular solution we factorize the Klein-Gordon operator in terms of two
Vekua-type operators. We show that real parts of the solutions of one of these
Vekua-type operators are solutions of the considered Klein-Gordon equation.
Using hyperbolic pseudoanalytic function theory, we then obtain explicit
construction of infinite systems of solutions of the Klein-Gordon equation with
potential. Finally, we give some examples of application of the proposed
procedure
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