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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
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 Zakharov-Shabat spectral problem on the semi-line: Hilbert formulation and applications
The inverse spectral transform for the Zakharov-Shabat equation on the
semi-line is reconsidered as a Hilbert problem. The boundary data induce an
essential singularity at large k to one of the basic solutions. Then solving
the inverse problem means solving a Hilbert problem with particular prescribed
behavior. It is demonstrated that the direct and inverse problems are solved in
a consistent way as soon as the spectral transform vanishes with 1/k at
infinity in the whole upper half plane (where it may possess single poles) and
is continuous and bounded on the real k-axis. The method is applied to
stimulated Raman scattering and sine-Gordon (light cone) for which it is
demonstrated that time evolution conserves the properties of the spectral
transform.Comment: LaTex file, 1 figure, submitted to J. Phys.
A transform method for linear evolution PDEs on a finite interval
We study initial boundary value problems for linear scalar partial
differential equations with constant coefficients, with spatial derivatives of
{\em arbitrary order}, posed on the domain . We first show that
by analysing the so-called {\em global relation}, which is an algebraic
relation defined in the complex -plane coupling all boundary values of the
solution, it is possible to identify how many boundary conditions must be
prescribed at each end of the space interval in order for the problem to be
well posed. We then show that the solution can be expressed as an integral in
the complex -plane. This integral is defined in terms of an -transform of
the initial condition and a -transform of the boundary conditions. For
particular cases, such as the case of periodic boundary conditions, or the case
of boundary value problems for {\em second} order PDEs, the integral can be
rewritten as an infinite series. However, there exist initial boundary value
problems for which the only representation is an integral which {\em cannot} be
written as an infinite series. An example of such a problem is provided by the
linearised version of the KdV equation. Thus, contrary to common belief, the
solution of many linear initial boundary value problems on a finite interval
{\em cannot} be expressed in terms of an infinite series