Table of contents

We study reductions of the Hamiltonian flows restricted to their invariant submanifolds. As examples, we consider partial Lagrange–Routh reductions of the natural mechanical systems such as geodesic flows on compact Lie groups and n-dimensional variants of the classical Hess–Appel'rot case of a heavy rigid body motion about a fixed point.

We show that maps with homoclinic tangencies of arbitrarily high orders and, as a consequence, with arbitrarily degenerate periodic orbits are dense in the Newhouse regions in spaces of real-analytic area-preserving two-dimensional maps and general real-analytic two-dimensional maps (the result was earlier known only for the space of smooth non-conservative maps). Based on this, we show that a generic area-preserving map from the Newhouse region is 'universal' in the sense that its iterations approximate the dynamics of any other area-preserving map with arbitrarily good accuracy. In fact, we show that every dynamical phenomenon which occurs generically in any open set of symplectic diffeomorphisms of a two-dimensional disc, or in any open set of finite-parameter families of such diffeomorphisms, can be encountered at a perturbation of any area-preserving two-dimensional map with a homoclinic tangency.

We study equilibria of the Smoluchowski equation for rigid, dipolar rod ensembles where the intermolecular potential couples the dipole–dipole interaction and the Maier–Saupe interaction. We thereby extend previous analytical results for the decoupled case of the dipolar potential only (Fatkullin and Slastikov 2005 Nonlinearity18 2565–80; Ji et al Phys. Fluids at press; Wang et al 2005 Commun. Math. Sci.3 605–20) or the Maier–Saupe potential only (Constantin et al 2004 Arch. Ration. Mech. Anal.174 365–84; Constantin et al 2004 Discrete Contin. Dyn. Syst.11 101–12; Constantin and Vukadinovic 2005 Nonlinearity18 441–3; Constantin 2005 Commun. Math. Sci.3 531–44; Fatkullin and Slastikov 2005 Commun. Math. Sci.3 21–6; Liu et al 2005 Commun. Math. Sci.3 201–18; Luo et al2005 Nonlinearity18 379–89; Zhou et al2005 Nonlinearity18 2815–25; Zhou and Wang Commun. Math. Sci. at press), and prove certain numerical observations for equilibria of coupled potentials (Ji et al Phys. Fluids at press). We first derive stability conditions, on the magnitude of the polarity vector (the first moment of the orientational probability distribution function) and on the direction of the polarity. We then prove that all stable equilibria of rigid, dipolar rod dispersions are either isotropic or prolate uniaxial. In particular, all stable anisotropic equilibrium distributions admit the following remarkable symmetry: the peak axis of orientation is aligned with both the polarity vector (first moment) and the distinguished director of the uniaxial second moment tensor. The stability is essential in establishing the axisymmetry. To demonstrate that the stability is indeed required, we show that there exist unstable non-axisymmetric equilibria.

Finite time rupture for positive solutions of the thin film type equation in the presence of van der Waals forces is established for a class of initial data satisfying an energy criterion.

We study travelling waves on a two-dimensional lattice with linear and nonlinear coupling between nearest particles and a periodic nonlinear substrate potential. Such a discrete system can model molecules adsorbed on a substrate crystal surface. We show the existence of both uniform sliding states and periodic travelling waves as well in a two-dimensional sine-Gordon lattice equation using topological and variational methods.

In this paper, we survey some recent advances in the study of travelling wave solutions to the Burgers–Korteweg–de Vries equation. Some comments are given on the existing results. A class of travelling solitary wave solutions in terms of elliptic functions with arbitrary velocity is obtained by means of the first-integral method as well as the method of compatible vector fields.

In this paper, we study the existence and stability of pulse solutions in a system with interacting instability mechanisms, which is described by a Ginzburg–Landau equation for an A-mode, coupled to a diffusion equation for a B-mode. Our main question is whether this coupling may stabilize solutions of the Ginzburg–Landau equation that are unstable when the interactions with the neutrally stable B-mode are not included in the model. The spatially homogeneous B-mode is supposed to be neutrally stable. This implies that the pulse solutions cannot decay exponentially, but must decay with an algebraic rate as x → ±∞. As a consequence, the methods that exist in the literature by which the stability of pulses in singularly perturbed reaction–diffusion systems can be studied need to be extended. This results in an 'algebraic NLEP approach', which is expected to be relevant beyond the setting of this paper. As in the case of a (weakly) stable B-mode (Doelman et al 2004 J. Nonlin. Sci.14 237–78) we establish by the application of this approach that the B-mode indeed introduces a mechanism that may stabilize pulses that are unstable when the interactions with the B-mode are not taken into account.

Let f be an entire transcendental map, such that all the singularities of f⁻¹ are contained in a compact subset of the immediate basin B(z₀) of an attracting fixed point z₀. We study the structure of the Julia set of f, which is equal to the boundary of B(z₀), and the behaviour of the Riemann mapping φ onto B(z₀) using the technique of geometric coding trees of preimages of points from B(z₀). We show that for a given symbolic itinerary, if codes of the tracts of f are bounded and codes of the fundamental domains grow no faster than the iterates of an exponential function, then there exists a point in the Julia set with this itinerary. Moreover, we determine cluster sets for φ and show that φ has an unrestricted limit equal to ∞ at points of a dense uncountable set in the unit circle.

We consider the quasi-periodic dynamics of non-integrable perturbations of a family of integrable Hamiltonian systems with normally 1 : −1 resonant invariant tori. In particular, we focus on the supercritical quasi-periodic Hamiltonian Hopf bifurcation and address the persistence problem of the singular foliation into invariant quasi-periodic tori near the bifurcation point. With the help of KAM theory, we show that the singular torus foliation survives a small perturbation and that the persisting tori in this foliation form Cantor families. A leading example is the Lagrange top near gyroscopic stabilization weakly coupled with a quasi-periodic oscillator.

In this paper, we prove a criterion to determine if a black soliton solution (which is an odd solution that does not vanish at infinity) to a one-dimensional nonlinear Schrödinger equation is linearly stable or not. This criterion handles the sign of the limit at 0 of the Vakhitov–Kolokolov function. For some nonlinearities, we numerically compute the black soliton and the Vakhitov–Kolokolov function in order to investigate linear stability of black solitons. We then show that linearly unstable black solitons are also orbitally unstable. In the Gross–Pitaevskii case, we rigorously prove the linear stability of the black soliton. Finally, we numerically study the dynamical stability of these solutions solving both linearized and fully nonlinear equations with a finite differences algorithm.

We show that the only continuous, codimension one foliations which are invariant under the action of the geodesic flow of a compact surface without conjugate points are the central stable and the central unstable foliations. The uniqueness of central foliations as the only invariant, codimension one foliations is typical of Anosov systems and expansive geodesic flows in compact manifolds without conjugate points. We generalize this result to geodesic flows in compact surfaces without conjugate points with no further assumptions on the dynamics of the flow.

In this brief paper we consider two types of nonlocal perturbations of the 3D incompressible Euler equations with parameters ε > 0. When ε = 0 both of the equations reduce to the usual incompressible Euler system. For one perturbation we have global well-posedness for initial data in , m > 5/2, while for the other we have finite time singularity for initial data having sufficiently large vorticity. We also derive conditions of finite time blow-up/global regularity of the solutions of the Euler equations represented in terms of solutions of the perturbed systems.