Physically meaningful periodic solutions to certain integrable partial differential equations are given in terms of multi-dimensional theta functions associated with real Riemann surfaces. Typical analytical problems in the numerical evaluation of these solutions are studied. In the case of hyperelliptic surfaces efficient algorithms exist even for almost degenerate surfaces. This allows the numerical study of solitonic limits. For general real Riemann surfaces, the choice of a homology basis adapted to the anti-holomorphic involution is important for a convenient formulation of the solutions and smoothness conditions. Since existing algorithms for algebraic curves produce a homology basis not related to automorphisms of the curve, we study symplectic transformations to an adapted basis and give explicit formulae for M-curves. As examples we discuss solutions of the Davey–Stewartson and the multi-component nonlinear Schrödinger equations.

This paper, which deals with internal waves in the two-layer formulation, consists of three parts. The first part is devoted to the derivation of asymptotic models in the intermediate long wave (ILW) regime for internal waves with a free upper surface and with a flat bottom. Using a method similar to that introduced by Bona et al (2008 J. Math. Pures Appl. 89 538–66), we obtain a one-parameter ILW system which is consistent with the full Euler system. In the second part we investigate the well-posedness of the ILW system under the rigid lid assumption. The local well-posedness on the time scale in lower order Sobolev spaces and the large time well-posedness on the long time scale in higher order Sobolev spac...

This paper concerns the vanishing viscosity limit of the two-dimensional degenerate viscous lake equations when the Navier slip conditions are prescribed on the impermeable boundary of a simply connected bounded regular domain. When the initial vorticity is in the Lebesgue space L ^q with 2 < q ≤ ∞, we show that the degenerate viscous lake equations possess a unique global solution and the solution converges to a corresponding weak solution of the inviscid lake equations. In a special case when the vorticity is in L ^∞, an explicit convergence rate is obtained.

A coupled cell network is a model for many situations such as food webs in ecosystems, cellular metabolism and economic networks. It consists in a directed graph G, each node (or cell) representing an agent of the network and each directed arrow representing which agent acts on which. It yields a system of differential equations , where the component i of f depends only on the cells x _j ( t) for which the arrow j → i exists in G. In this paper, we investigate the observation problems in coupled cell networks: can one deduce the behaviour of the whole network (oscillations, stabilization, etc) by observing only one of the cells? We show that the natural observation properties hold for almost all the interactions f.

Using the H ⁻¹ norm as a measure of mixing, we prove that 2D Euler flows on the torus mix passive scalars at most exponentially. The mixing rate is bounded linearly by the BMO norm of the vorticity (and thus by its L ^∞ norm). We also give an analogous bound on the growth rate of scalar gradients.

We study a class of smooth maps of the form



where h : [0, 1] → [0, 1] is a unimodal map exhibiting an attracting periodic point of prime period 3, and ω is irrational ( ). We show that the following phenomenon can occur for certain h and : There exists a single measurable function

Weierstrass' example of an everywhere continuous but nowhere differentiable function is given by where λ (0, 1), b ≥ 2, λ b > 1. There is a well-known and widely accepted, but as yet unproven, formula for the Hausdorff dimension of the graph of w. Hunt [H] proved that this formula holds almost surely on the addition of a random phase shift. The graphs of Weierstrass-type functions appear as repellers for a certain class of dynamical system; in this paper we prove formulae analogous to those for random phase shifts of w( x) but in a dynamic context. Let T : S ¹ → S ¹ be a uniformly expanding map of the circle. Let λ : S

A theory for noise-driven subcritical instabilities in spatially extended systems is put forward. The theory allows one to calculate the critical bifurcation parameter for a first-order phase transition in such non-equilibrium systems in the thermodynamic limit and analyse the mechanism of phase transition. Two examples with distinctive features are studied in detail to demonstrate the usefulness of the theory and the different scenarios that can occur in the thermodynamic limit of non-equilibrium systems.

One-dimensional scalar conservation laws with convex flux and Markov initial data are now known to yield a completely integrable Hamiltonian system. In this paper, we rederive the analogue of Loitsiansky's invariant in hydrodynamic turbulence from the perspective of integrable systems. Other relevant physical notions such as energy dissipation and spectrum are also discussed.

Recently, the phase space structures governing reaction dynamics in Hamiltonian systems have been identified and algorithms for their explicit construction have been developed. These phase space structures are induced by saddle type equilibrium points which are characteristic for reaction type dynamics. Their construction is based on a Poincaré–Birkhoff normal form. Using tools from the geometric theory of Hamiltonian systems and their reduction, we show in this paper how the construction of these phase space structures can be generalized to the case of the relative equilibria of a rotational symmetry reduced N-body system. As rotations almost always play an important role in the reaction dynamics of molecules, the approach presented in this paper is of great relevance for applications.