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We study the zero-dispersion limit for certain initial boundary value problems for the defocusing nonlinear Schrödinger (NLS) equation and for the Korteweg-de Vries (KdV) equation with dominant surface tension. These problems are formulated on the half-line and they involve linearisable boundary conditions.
We treat some recent results concerning sampling expansions of Kramer type. The link of the sampling theorem of Whittaker-Shannon-Kotelnikov with the Kramer sampling theorem is considered and the connection of these theorems with boundary value problems is specified. Essentially, this paper surveys certain results in the field of sampling theories and linear, ordinary, first-, and second-order boundary value problems that generate Kramer analytic kernels. The investigation of the first-order problems is tackled in a joint work with Everitt. For the second-order problems, we refer to the work of Everitt and Nasri-Roudsari in their survey paper in 1999. All these problems are represented by unbounded selfadjoint differential operators on Hilbert function spaces, with a discrete spectrum which allows the introduction of the associated Kramer analytic kernel. However, for the first-order problems, the analysis of this paper is restricted to the specification of conditions under which the associated operators have a discrete spectrum.
Let be a real, locally convex, locally solid vector lattice of (AM)-type. First, we prove an approximation theorem of Bishop's type for a vector subspace of such a lattice. Second, using this theorem, we obtain a generalization of Nachbin's density theorem for weighted spaces.
We deal with the general initial-boundary value problem for a second-order nonlinear nonstationary evolution equation. The associated operator equation is studied by the Fredholm and Nemitskii operator theory. Under local Hölder conditions for the nonlinear member, we observe quantitative and qualitative properties of the set of solutions of the given problem. These results can be applied to different mechanical and natural science models.
We establish sufficient conditions for the existence of solutions for semilinear differential inclusions, with nonlocal conditions. We rely on a fixed-point theorem for contraction multivalued maps due to Covitz and Nadler and on the Schaefer's fixed-point theorem combined with lower semicontinuous multivalued operators with decomposable values.
Several inverse spectral problems are solved by a method which is based on exact solutions of the semi-infinite Toda lattice. In fact, starting with a well-known and appropriate probability measure , the solution , of the Toda lattice is exactly determined and by taking , the solution , of the inverse spectral problem is obtained. The solutions of the Toda lattice which are found in this way are finite for every and can also be obtained from the solutions of a simple differential equation. Many other exact solutions obtained from this differential equation show that there exist initial conditions and such that the semi-infinite Toda lattice is not integrable in the sense that the functions and are not finite for every .