253 research outputs found
Nonlocal and multipoint boundary value problems for linear evolution equations
We derive the solution representation for a large class of nonlocal boundary
value problems for linear evolution PDEs with constant coefficients in one
space variable. The prototypical such PDE is the heat equation, for which
problems of this form model physical phenomena in chemistry and for which we
formulate and prove a full result. We also consider the third order case, which
is much less studied and has been shown by the authors to have very different
structural properties in general.
The nonlocal conditions we consider can be reformulated as \emph{multipoint
conditions}, and then an explicit representation for the solution of the
problem is obtained by an application of the Fokas transform method. The
analysis is carried out under the assumption that the problem being solved is
well posed, i.e.\ that it admits a unique solution. For the second order case,
we also give criteria that guarantee well-posedness.Comment: 28 pages, 4 figure
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The Dirichlet-to-Neumann map for the elliptic sine Gordon
We analyse the Dirichlet problem for the elliptic sine Gordon equation in the upper half plane. We express the solution in terms of a Riemann-Hilbert problem whose jump matrix is uniquely defined by a certain function b(\la), \la\in\R, explicitly expressed in terms of the given Dirichlet data and the unknown Neumann boundary value , where and are related via the global relation \{b(\la)=0, \la\geq 0\}. Furthermore, we show that the latter relation can be used to characterise the Dirichlet to Neumann map, i.e. to express in terms of . It appears that this provides the first case that such a map is explicitly characterised for a nonlinear integrable {\em elliptic} PDE, as opposed to an {\em evolution} PDE
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Two-point boundary value problems for linear evolution equations
We study boundary value problems for a linear evolution equation with spatial derivatives of arbitrary order, on the domain 0 < x < L, 0 < t < T, with L and T positive nite constants. We present a general method for identifying well-posed problems, as well as for constructing an explicit representation of the solution of such problems. This representation has explicit x and t dependence, and it consists of an integral in the k-complex plane and of a discrete sum. As illustrative examples we solve some two-point boundary value problems for the equations iqt + qxx = 0 and qt + qxxx = 0
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Boundary value problems for the elliptic sine-Gordon equation in a semi-strip
We study boundary value problems posed in a semistrip for the elliptic sine-Gordon equation, which is the paradigm of an elliptic integrable PDE in two variables. We use the method introduced by one of the authors, which provides a substantial generalization of the inverse scattering transform and can be used for the analysis of boundary as opposed to initial-value problems. We first express the solution in terms of a 2 by 2 matrix Riemann-Hilbert problem
whose \jump matrix" depends on both the Dirichlet and the Neumann boundary values. For
a well posed problem one of these boundary values is an unknown function. This unknown
function is characterised in terms of the so-called global relation, but in general this characterisation is nonlinear. We then concentrate on the case that the prescribed boundary conditions
are zero along the unbounded sides of a semistrip and constant along the bounded side. This
corresponds to a case of the so-called linearisable boundary conditions, however a major difficulty for this problem is the existence of non-integrable singularities of the function q_y at
the two corners of the semistrip; these singularities are generated by the discontinuities of the
boundary condition at these corners. Motivated by the recent solution of the analogous problem for the modified Helmholtz equation, we introduce an appropriate regularisation which
overcomes this difficulty. Furthermore, by mapping the basic Riemann-Hilbert problem to an
equivalent modified Riemann-Hilbert problem, we show that the solution can be expressed in
terms of a 2 by 2 matrix Riemann-Hilbert problem whose jump matrix depends explicitly on
the width of the semistrip L, on the constant value d of the solution along the bounded side,
and on the residues at the given poles of a certain spectral function denoted by h. The
determination of the function h remains open
Evolution equations on time-dependent intervals
We study initial boundary value problems for linear evolution partial
differential equations (PDEs) posed on a time-dependent interval
, , where and are given, real,
differentiable functions, and is an arbitrary constant. For such problems,
we show how to characterise the unknown boundary values in terms of the given
initial and boundary conditions. As illustrative examples we consider the heat
equation and the linear Schr\"{o}dinger equation. In the first case, the
unknown Neumann boundary values are expressed in terms of the Dirichlet
boundary values and of the initial value through the unique solution of a
system of two linear integral equations with explicit kernels. In the second
case, a similar result can be proved but only for a more restrictive class of
boundary curves.
Four-loop large-n_f contributions to the non-singlet structure functions F_2 and F_L
We have calculated the n_f^2 and n_f^3 contributions to the flavour
non-singlet structure functions F_2 and F_L in inclusive deep-inelastic
scattering at the fourth order in the strong coupling alpha_s. The coefficient
functions have been obtained by computing a very large number of Mellin-N
moments using the method of differential equations, and then determining the
analytic forms in N and Bjorken-x from these. Our new n_f^2 terms are
numerically much larger than the n_f^3 leading large-nf parts which were
already known; they agree with predictions of the threshold and high-energy
resummations. Furthermore our calculation confirms the earlier determination of
the four-loop n_f^2 part of the corresponding anomalous dimension. Via the
no-pi^2 conjecture/theorem for Euclidean physical quantities, we predict the z4
n_f^3 part of the fifth-order anomalous dimension for the evolution of
non-singlet quark distributions.Comment: 48 pages, LaTex, 5 eps-based figures. Analytical results in ancillary
FORM file
The Unified Method: I Non-Linearizable Problems on the Half-Line
Boundary value problems for integrable nonlinear evolution PDEs formulated on
the half-line can be analyzed by the unified method introduced by one of the
authors and used extensively in the literature. The implementation of this
general method to this particular class of problems yields the solution in
terms of the unique solution of a matrix Riemann-Hilbert problem formulated in
the complex -plane (the Fourier plane), which has a jump matrix with
explicit -dependence involving four scalar functions of , called
spectral functions. Two of these functions depend on the initial data, whereas
the other two depend on all boundary values. The most difficult step of the new
method is the characterization of the latter two spectral functions in terms of
the given initial and boundary data, i.e. the elimination of the unknown
boundary values. For certain boundary conditions, called linearizable, this can
be achieved simply using algebraic manipulations. Here, we present an effective
characterization of the spectral functions in terms of the given initial and
boundary data for the general case of non-linearizable boundary conditions.
This characterization is based on the analysis of the so-called global
relation, on the analysis of the equations obtained from the global relation
via certain transformations leaving the dispersion relation of the associated
linearized PDE invariant, and on the computation of the large asymptotics
of the eigenfunctions defining the relevant spectral functions.Comment: 39 page
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