The method of ultradiscrete limit is applied to a series of discrete systems derived from Hamiltonian systems parametrized with corresponding lattice polygons. For every ultradiscrete system, general solution is obtained from the polar set of each lattice polygon.

We propose a differential-geometric classification of the four-component hyperbolic systems of conservation laws which satisfy the following properties: (a) they do not possess Riemann invariants; (b) they are linearly degenerate; (c) their rarefaction curves are rectilinear; (d) the cross-ratio of the four characteristic speeds is harmonic. This turns out to provide a classification of projective congruences in ${\mathbb P}^5$ whose developable surfaces are planar pencils of lines, each of these lines cutting the focal variety at points forming a harmonic quadruplet. Symmetry properties and the connection of these congruences to Cartan's isoparametric hypersurfaces are discussed.

The goal of this paper is to present a solution of the cellular automaton associated with the discrete KdV equation, using an algebro-geometric solution of the discrete KP equation over a finite field out of a hyperelliptic curve.

In this paper we prove the existence of a solution for a system of nonlinear parabolic partial differential equations arising from thermoelectric modelling of metallurgical electrodes undergoing a phase change. The model consists of an electromagnetic problem for eddy current computation coupled with a Stefan problem for temperature. The proof uses a regularized problem obtained by truncating the source term in temperature equation. Passing to the limit requires fine a priori estimates leading to compactness.

We present some basic properties of two distinguished discretizations of elliptic operators: the self-adjoint 5-point and 7-point schemes on a two dimensional lattice. We first show that they allow us to solve Dirichlet boundary value problems; then we present their Moutard transformations (distinguished examples of transformation of Darboux type in two dimensions). Finally we construct their Lelieuvre formulae and we show that, at the level of the normal vector and in full analogy with their continuous counterparts, the self-adjoint 5-point scheme characterizes a two dimensional quadrilateral lattice (a lattice whose elementary quadrilaterals are planar), while the self-adjoint 7-point scheme characterizes a generic 2D lattice.

The trilinear form of the discrete Tzitzeica equation by Schief is found to be a discrete Toda molecule equation with a special boundary condition. Based on this fact, a higher order discrete Tzitzeica equation and an ultradiscrete Tzitzeica equation are obtained.

A number theoretical aspect of the fundamental cycle of a periodic box-ball system is investigated. Using the formulae for the fundamental cycle of a class of initial states, we point out that the asymptotic behaviour of the fundamental cycle is closely related to the celebrated Riemann hypothesis.

We extend a solution method used for the one-dimensional Toda lattice in [1], [2] to the two-dimensional Toda lattice. The idea is

1. to study the lattice not with values in $\mathbb{C}$ but in the Banach algebra ${\cal L}$ of bounded operators and

2. to derive solutions of the original lattice ($\mathbb{C}$-solutions) by applying a functional $\tau$ to the ${\cal L}$-solutions constructed in 1.

In the second part we focus on applications. We start by rederiving line-solitons and briefly discuss discrete resonance phenomena. Moreover, we are able to find conditions under which it is possible to superpose even countably many line-solitons.

Two themes are considered in this paper. First of all in the Introduction a comment is made about the Tzitzéica equation which occurred in the author's work as reported in [1]. Much recent work has referred to this equation and a representative bibliography is given in this paper with brief comments. The second theme of this paper is concerned with work surrounding Quantum Solitons and the author's talk at the ISLAND 2 meeting under the title ‘Quantum solitons’ is briefly summarised. An extended review of this subject matter has now appeared in [2] and as there the quantum soliton of the quantum attractive NLS model is seen as a ‘qubit’ for quantum information purposes. It is hoped that this summary and/or the reference [2] can help to stimulate further interest in this ‘quantum’ aspect of our subject in the solitons community. The actual observation of a quantum soliton is also reported and a proper theoretical description of it given. Reference is made to $q$-boson lattices first as a simple example of the quantum inverse method in the Introduction and then as a subject matter in its own right at the end of the paper.

Extending the so called coupled KP hierarchy to negative time flows one obtains coupled Toda-type equations defined on a two-dimensional lattice. These equations allow for reductions to 1+1 dimensional integrable systems that are defined on a finite part of this lattice. A system of coupled Hirota bilinear equations, obtained from such a reduction and defined on only 5 points of the lattice, will be shown to correspond to a coupling of a Tzitzeica equation to two linear equations. The Lax representation of this system is also presented.

The hierarchy structure of a derivative nonlinear Schrödinger equation is investigated in terms of the Sato-Segal-Wilson formulation. Special solutions are constructed as ratios of Wronski determinants. Relations to the Painlevé IV and the discrete Painlevé I are discussed by applying a similarity reduction.

We consider the problem of modelling the flow of a slightly compressible fluid in a periodic fractured medium assuming that the fissures are thin with respect to the block size. As a starting point we used a formulation applied to a system comprising a fractured porous medium made of blocks and fractures separated by a thin layer which is considered as an interface. The inter-relationship between these three characteristics comprise the triple porosity model. The microscopic model consists of the usual equation describing Darcy flow with the permeability being highly discontinuous. Over the matrix domain, the permeability is scaled by $(\varepsilon \delta)^2$, where $\varepsilon$ is the size of a typical porous block, with $\delta$ representing the relative size of the fracture. We then consider a model with Robin type transmission conditions: a jump of the density across the interface block-fracture is taken into account and proportional to the flux by the mean of a function $(\varepsilon\delta)^{-\gamma}$, where $\gamma$ is a parameter. Using two-scale convergence, we get homogenized models which govern the global behaviour of the flow as $\varepsilon$ and $\delta$ tend to zero. The resulting homogenized problem is a dual-porosity type model that contains a term representing memory effects for $\gamma\le 1$, and it is a single porosity model with effective coefficients for $\gamma >1$.

we prove in this work the trace and trace lifting theorems for the sobolev spaces associated with a system of hörmander’s vectors fields of order 2. the case of non-degenerate characteristic points for left invariant vector fields on the heisenberg group is also studied.

The aim of this paper is to provide a survey on the recent development in level set methods in inverse problems and optimal design. We give introductions on the general features of such problems involving geometries and on the general framework of the level set method. In subsequent parts we discuss shape sensitivity analysis and its relation to level set methods, various approaches on constructing optimization algorithms based on the level set approach, and special tools needed for the application of level set based optimization methods to ill-posed problems. Furthermore, we provide a review on numerical methods important in this context, and give an overview of applications treated with level set methods. Finally, we provide a discussion of the most challenging and interesting open problems in this field, that might be of interest for scientists who plan to start future research in this field.

Let D1 ⊂ R3 be a non-empty simply connected open bounded set and Γ = ∂D1 be its boundary. We assume Γ to be a C2 surface and let (Γn)n∈N be a sequence of surfaces, such that Γn is closed, , and the surface measure of Γ\Γn is positive for n ∈ N. The set D1 is the cavity associated with (Γn)n∈N. Let HΓn, n ∈ N, HΓ be the self-adjoint operators on L2 associated with the minus Laplace operator with Dirichlet boundary condition on Γn, n ∈ N and on Γ, respectively. Note that, roughly speaking, HΓ is the operator associated with minus the Laplacian on R3\Γ with the Dirichlet boundary condition on Γ. A similar statement holds for HΓn, n ∈ N. We show that the spectral measure, dEΓn (λ), λ ∈ R, associated with the operator HΓn, n ∈ N, when n → +∞ converges to the spectral measure dEΓ (λ), λ ∈ R, associated with the operator HΓ. That is in a sense made precise later we prove that Σ (HΓn) → Σ (HΓ) when n → +∞, where Σ (HΓ) and Σ (HΓn), n ∈ N, are the spectra of HΓ and HΓn, n ∈ N, respectively. Moreover, we show that the point spectrum of HΓ is made of an infinite number of positive eigenvalues of finite multiplicity and that the point spectrum of HΓn, n ∈ N, is empty and that the essential spectrum of HΓn, n ∈ N, and of HΓ is continuous and is contained in [0, +∞). Under the extra hypothesis that Γ is a C4 surface and that the Gaussian curvature of Γ is positive at every point of Γ, we prove that the essential spectrum of HΓ is a continuous spectrum and is given by [0, +∞). So that the convergence of dEΓn (λ), λ ∈ R to dEΓ (λ), λ ∈ R, when n → +∞ can be interpreted as a spectral concentration phenomenon that consists in the fact that the limit of the spectrum of operators with purely continuous spectrum is the spectrum of an operator that is made of eigenvalues embedded in a continuous spectrum.

We call a quasi-adequate semi-group whose set of idempotents forms a left [right] quasi-normal band a left [right] semi-perfect abundant semi-group. After obtaining some characterization theorems of such quasi-adequate semi-groups, we establish a structure for left [right] semi-perfect abundant semi-groups of type W. Our results generalize and strengthen the results of El-Qallali and Fountain on quasi-adequate semi-groups.

In this paper we consider Beurling-type distributions in the Hankel setting. The Hankel transform and Hankel convolution are studied on Beurling-type distributions. We also introduce a class of ultra-differential operators that allows us to show a Hankel version of the second structure theorem of Komatsu and Braun. Necessary and sufficient conditions are established in order that a Beurling distribution generates a surjective Hankel convolution operator.

We obtain spatial and temporal decay rates of weak solutions of the Navier–Stokes equations, and for strong solutions. For the spatial decay rate of the weak solutions, the power of the weight given by He and Xin in 2001 does not exceed 3/2;. However, we show the power can be extended up to 5/2;.

We call a quasi-adequate semi-group whose set of idempotents forms a left [right] quasi-normal band a left [right] semi-perfect abundant semi-group. After obtaining some characterization theorems of such quasi-adequate semi-groups, we establish a structure for left [right] semi-perfect abundant semi-groups of type W. Our results generalize and strengthen the results of El-Qallali and Fountain on quasi-adequate semi-groups.

In this paper we consider Beurling-type distributions in the Hankel setting. The Hankel transform and Hankel convolution are studied on Beurling-type distributions. We also introduce a class of ultra-differential operators that allows us to show a Hankel version of the second structure theorem of Komatsu and Braun. Necessary and sufficient conditions are established in order that a Beurling distribution generates a surjective Hankel convolution operator.