12 research outputs found
The Generalized Dirichlet to Neumann map for the KdV equation on the half-line
For the two versions of the KdV equation on the positive half-line an
initial-boundary value problem is well posed if one prescribes an initial
condition plus either one boundary condition if and have the
same sign (KdVI) or two boundary conditions if and have
opposite sign (KdVII). Constructing the generalized Dirichlet to Neumann map
for the above problems means characterizing the unknown boundary values in
terms of the given initial and boundary conditions. For example, if
and are given for the KdVI
and KdVII equations, respectively, then one must construct the unknown boundary
values and , respectively. We
show that this can be achieved without solving for by analysing a
certain ``global relation'' which couples the given initial and boundary
conditions with the unknown boundary values, as well as with the function
, where satisifies the -part of the associated
Lax pair evaluated at . Indeed, by employing a Gelfand--Levitan--Marchenko
triangular representation for , the global relation can be solved
\emph{explicitly} for the unknown boundary values in terms of the given initial
and boundary conditions and the function . This yields the unknown
boundary values in terms of a nonlinear Volterra integral equation.Comment: 21 pages, 3 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
The solution of the global relation for the derivative nonlinear Schr\"odinger equation on the half-line
We consider initial-boundary value problems for the derivative nonlinear
Schr\"odinger (DNLS) equation on the half-line . In a previous work, we
showed that the solution can be expressed in terms of the solution of
a Riemann-Hilbert problem with jump condition specified by the initial and
boundary values of . However, for a well-posed problem, only part of
the boundary values can be prescribed; the remaining boundary data cannot be
independently specified, but are determined by the so-called global relation.
In general, an effective solution of the problem therefore requires solving the
global relation. Here, we present the solution of the global relation in terms
of the solution of a system of nonlinear integral equations. This also provides
a construction of the Dirichlet-to-Neumann map for the DNLS equation on the
half-line.Comment: 20 pages, 2 figures, minor corrections mad
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
The Unified Method: II NLS on the Half-Line with -Periodic Boundary Conditions
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 first 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 and on the introduction of the so-called
Gelfand-Levitan-Marchenko representations of the eigenfunctions defining the
spectral functions. We then concentrate on the physically significant case of
-periodic Dirichlet boundary data. After presenting certain heuristic
arguments which suggest that the Neumann boundary values become periodic as
, we show that for the case of the NLS with a sine-wave as
Dirichlet data, the asymptotics of the Neumann boundary values can be computed
explicitly at least up to third order in a perturbative expansion and indeed at
least up to this order are asymptotically periodic.Comment: 29 page
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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