879 research outputs found
The Unified Method: III Non-Linearizable Problems on the Interval
Boundary value problems for integrable nonlinear evolution PDEs formulated on
the finite interval 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 six scalar functions of , called
spectral functions. Two of these functions depend on the initial data, whereas
the other four depend on all boundary values. The most difficult step of the
new method is the characterization of the latter four spectral functions in
terms of the given initial and boundary data, i.e. the elimination of the
unknown boundary values. Here, we present an effective characterization of the
spectral functions in terms of the given initial and boundary data. We present
two different characterizations of this problem. One 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. The other is based on the analysis of the global relation and on the
introduction of the so-called Gelfand-Levitan-Marchenko representations of the
eigenfunctions defining the relevant spectral functions. We also show that
these two different characterizations are equivalent and that in the limit when
the length of the interval tends to infinity, the relevant formulas reduce to
the analogous formulas obtained recently for the case of boundary value
problems formulated on the half-line.Comment: 22 page
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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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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