Nonintegrable differential-difference equations are constructed which support two-kink and two-soliton solutions. These equations are related to the discrete Burgers hierarchy and a discrete form of the Korteweg-de Vries equation. In particular, discretisations of equations related to the Fitzhugh-Nagumo-Kolmogorov-Petrovskii-Piskunov, Satsuma-Burgers-Huxley equations are derived. Methods presented here can also be used to derive non-integrable differential-difference equations describing the elastic collision of more than two kinks or solitary waves.

Consider a Hele-Shaw cell with the fluid (liquid) confined to an angular region by a solid boundary in the form of two half-lines meeting at an angle απ; if 0 < α < 1 we have flow in a corner, while if 1 < α [les ] 2 we have flow around a wedge. We suppose contact between the fluid and each of the lines forming the solid boundary to be along a single segment that does not adjoin the vertex, so we have air at the vertex, and contemplate such a situation that has been produced by injection at a number of points into an initially empty cell. We show that, if we assume the pressure to be constant along the free boundaries, the region occupied by the fluid is the image of a rectangle under a conformal map that can be expressed in terms of elliptic functions if α = 1 or α = 2, and in terms of theta functions if 0 < α < 1 or 1 < α < 2. The form of the function giving the map can be written down, and the parameters appearing in it then determined as the solution to a set of transcendental equations. The procedure is illustrated by a number of examples.

This paper presents a new class of solutions for steady nonlinear capillary waves on a curved sheet of fluid in the plane. The solutions are exact in that the free surfaces of the sheet and the associated flow field can be found in closed form. The solutions are generalizations of the classic solutions for finite amplitude waves on fluid sheets [5] to the case where the fluid sheets are curved.

We show that a continuous, odd, regular (non-exceptional), ordinary, irreducible, two-dimensional, $l$-adic representation of the absolute Galois group of the rational numbers is modular over some totally real field. We deduce that it occurs in the $l$-adic cohomology of some variety over the rationals and that its $L$-function has meromorphic continuation to the whole complex plane and satisfies the expected functional equation.

AMS 2000 Mathematics subject classification: Primary 11F80. Secondary 11G40

In this paper, we construct explicitly endoscopic representations of $\widetilde{\mathrm{Sp}}_{2n}$, the metaplectic cover of a symplectic group of rank $n$. We do this in the automorphic case, and also in the local case, over a $p$-adic field.

AMS 2000 Mathematics subject classification: Primary 11F70

Let \bf{R}_+ be the set of all non-negative real numbers, \bf(I}\in \{ \bf(R}, \bf(R}_+\} and {\cal U}_\bf(I}=\{ U(t, s): t\ge s\in I\} be a strongly measurable and exponentially bounded evolution family of bounded linear operators acting on a Banach space X. Let \phi :\bf(R}_+\to \bf(R}_+ be a strictly increasing function and E be a normed function space over \bf(I} satisfying some properties; see Section 2. We prove that if

\phi \circ (\chi _{[s, \infty )}(\cdot )||U(\cdot , s)x||)

defines an element of the space E for every s\in \bf(I} and all x\in X and if there exists M>0 such that

\sup \limits _{s\in \bf(I}}|\phi \circ (\chi _{[s, \infty )} (\cdot )||U(\cdot , s)x||)|_{E}= M<\infty \quad \forall x\in X\ {\text with}\ ||x||\le 1,

then {\cal U}_\bf(I} is uniformly exponentially stable. In particular if \psi :\bf(R}_+\to \bf(R}_+ is a nondecreasing function such that \psi (t)>0, for all t>0, and if there exists K>0 such that

\sup \limits _{s\in \bf(I}}\int \limits _s^{\infty }\psi (||U(t, s)x||)dt=K <\infty ,\quad \forall x\in X\ \text(with)\ ||x||\le 1, then

{\cal U}_\bf(I} is uniformly exponentially stable. For \bf(I}=\bf(R}_+, \psi continuous and {\cal U}_{\bf(R}_+} strongly continuous this last result is due to S. Rolewicz. Some related results for periodic evolution families are also proved.

Compact hyperbolic surfaces of given genus g containing discs of the maximum radius have been studied from various points of view. In this paper we connect these different approaches and observe some properties of the Fuchsian groups uniformizing both compact and punctured extremal surfaces. We also show that extremal surfaces of genera g=2,3 may contain one or several extremal discs, while an extremal disc is necessarily unique for g \ge 4. Along the way we also construct explicit families of extremal surfaces, one of which turns out to be free of automorphisms.

We prove that if X is a real Banach space, with dim X\ge 3, which contains subspace of codimension 1 which is 1-complemented in X and whose group of isometries is almost transitive then X is isometric to a Hilbert space. This partially answers the Banach-Mazur rotation problem and generalizes some recent related results.

An exact control theorem is proven for nearly ordinary $p$-adic automorphic forms on symplectic and unitary groups over totally real fields if the algebraic group is split at $p$. In particular, a given nearly ordinary holomorphic Hecke eigenform can be lifted to a family of holomorphic Hecke eigenforms indexed by weights of the standard maximal split torus of the group. Their $q$-expansion coefficients are Iwasawa functions on the Iwasawa algebra of ${\mathbb{Z}}_p$-points of the split torus. The method is applicable to any reductive algebraic groups yielding Shimura varieties of PEL type under mild assumptions on the existence of integral toroidal compactification of the variety. Even in the Hilbert modular case, the result contains something new: freeness of the universal nearly ordinary Hecke algebra over the Iwasawa algebra, which eluded my scrutiny when I studied general theory of the $p$-adic Hecke algebra in the 1980s.

AMS 2000 Mathematics subject classification: Primary 11F03; 11F30; 11F33; 11F41; 11F60; 11G15; 11G18

We construct embedded periodic minimal surfaces with planar ends. We use the Weierstrass representation and solve the period problem using the implicit function theorem at a singular point.

AMS 2000 Mathematics subject classification: Primary 53A10