10 research outputs found
Recommended from our members
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
Skyrmion States In Chiral Liquid Crystals
Within the framework of Oseen-Frank theory, we analyse the static
configurations for chiral liquid crystals. In particular, we find numerical
solutions for localised axisymmetric states in confined chiral liquid crystals
with weak homeotropic anchoring at the boundaries. These solutions describe the
distortions of two-dimensional skyrmions, known as either \textit{spherulites}
or \textit{cholesteric bubbles}, which have been observed experimentally in
these systems. Relations with nonlinear integrable equations have been outlined
and are used to study asymptotic behaviors of the solutions. By using
analytical methods, we build approximated solutions of the equilibrium
equations and we analyse the generation and stabilization of these states in
relation to the material parameters, the external fields and the anchoring
boundary conditions.Comment: 13 pages, 13 figures, Conference: PMNP 2017: 50 years of IST,
Gallipoli (LE)- Italy June 17-24, 201
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