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We analyse an initial-boundary value problem for the mKdV equation on a finite interval
by expressing the solution in terms of the solution of an associated matrix
Riemann-Hilbert problem in the complex -plane. This RH problem is determined by
certain spectral functions which are defined in terms of the initial-boundary values at
and . We show that the spectral functions satisfy an algebraic “global
relation” which express the implicit relation between all boundary values in terms of
spectral...
In this work we study the generalized Boussinesq equation with a dissipation term. We show that, under suitable conditions, a global solution for the initial value problem exists. In addition, we derive sufficient conditions for the blow-up of the solution to the problem. Furthermore, the instability of the stationary solutions of this equation is established.
The Hamiltonian representation for a hierarchy of Lax type equations on a dual space to the Lie algebra of integro-differential operators with matrix coefficients, extended by evolutions for eigenfunctions and adjoint eigenfunctions of the corresponding spectral problems, is obtained via some special Bäcklund transformation. The connection of this hierarchy with integrable by Lax two-dimensional Davey-Stewartson type systems is studied.
In this paper we consider the periodic Benjemin-Ono equation.We establish the invariance of the Gibbs measure associated to this equation, thus answering a question raised in Tzvetkov [28]. As an intermediate step, we also obtain a local well-posedness result in Besov-type spaces rougher than , extending the well-posedness result of Molinet [20].
We summarize the main ideas in a series of papers ([20], [21], [22], [5]) devoted to the construction of invariant measures and to the long-time behavior of solutions of the periodic Benjamin-Ono equation.
The matrix KdV equation with a negative dispersion term is considered in the right upper
quarter–plane. The evolution law is derived for the Weyl function of a corresponding
auxiliary linear system. Using the low energy asymptotics of the Weyl functions, the
unboundedness of solutions is obtained for some classes of the initial–boundary
conditions.
The aim of this note is to give a short review of our recent work (see [5]) with Miguel A. Alejo and Luis Vega, concerning the -stability, and asymptotic stability, of the -soliton of the Korteweg-de Vries (KdV) equation.
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