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We prove several generic existence results for infinitely many periodic orbits of Hamiltonian diffeomorphisms or Reeb flows. For instance, we show that a Hamiltonian diffeomorphism of a complex projective space or Grassmannian generically…
We find explicit solutions of the Laplacian coflow of $G_2-$structures on seven-dimensional almost-abelian Lie groups. Moreover, we construct new examples of solitons for the Laplacian coflow which are not eigenforms of the Laplacian and we…
In this paper, we compare components of the horizontal flow below the solar surface in and around regions consisting of rotating and non-rotating sunspots. Our analysis suggests that there is a significant variation in both components of…
A section of a Hamiltonian system is a hypersurface in the phase space of the system, usually representing a set of one-sided constraints (e.g. a boundary, an obstacle or a set of admissible states). In this paper we give local…
Formation of oblique solitons by a flow of polariton condensate past an obstacle is considered. The flow is non-uniform due to a finite life-time of polaritons what changes drastically the conditions of formation of oblique solitons…
We study the spinor flow on homogeneous spin manifolds. After providing the general setup we discuss the homogeneous spinor flow in dimension 3 and on almost abelian Lie groups in detail. As a further example the flag manifold in dimension…
A Floquet quantum system is governed by a Hamiltonian that is periodic in time. Consider the space of piecewise time-independent Floquet systems with (geometrically) local interactions. We prove that for all but a measure zero set of…
We prove that the orbit of a non-periodic point at prime values of the horocycle flow in the modular surface is dense in a set of positive measure. For some special orbits we also prove that they are dense in the whole space (assuming the…
We analyse the intersection of positively and negatively sectional-hyperbolic sets for flows on compact manifolds. First we prove that such an intersection is hyperbolic if the intersecting sets are both transitive (this is false without…
The Ricci flow on the 2-sphere with marked points is shown to converge in all three stable, semi-stable, and unstable cases. In the stable case, the flow was known to converge without any reparametrization, and a new proof of this fact is…
We analyze the dichotomy between {\em sectional-Axiom A flows} (c.f. \cite{memo}) and flows with points accumulated by periodic orbits of different indices. Indeed, this is proved for $C^1$ generic flows whose singularities accumulated by…
There are many topological faces of the superfluid phases of $^3$He. These superfluids contain various topological defects and textures. The momentum space topology of these superfluids is also nontrivial, as well as the topology in the…
We introduce a quantitative condition on orbits of dynamical systems which measures their aperiodicity. We show the existence of sequences in the Bernoulli-shift and geodesics on closed hyperbolic manifolds which are as aperiodic as…
In this paper we investigate the structure of certain solutions of the fully nonlinear Yamabe flow, which we call quotient almost Yamabe solitons because they extend quite naturally those called quotient Yamabe solitons. We then present…
We prove global well-posedness and scattering in $H^1$ for the defocusing nonlinear Schr\"{o}dinger equations \begin{equation*} \begin{cases} &(i\partial_t+\Delta_\g)u=u|u|^{2\sigma}; &u(0)=\phi, \end{cases} \end{equation*} on the…
Given a smooth compact surface without focal points and of higher genus, it is shown that its geodesic flow is semi-conjugate to a continuous expansive flow with a local product structure such that the semi-conjugation preserves…
The considered continuous-and-discrete hybrid system is a cyclic relay of smooth flows on an $n$-dimensional manifold $M$, where the discrete process of switching from each flow to the next takes place on the boundaries of the corresponding…
We consider real analytic Hamiltonians whose flow depends linearly on time. Trivial examples are Hamiltonians $H(q,p)$ that do not depend on the coordinate $q$. By a theorem of Moser, every polynomial Hamiltonian of degree 3 reduces to such…
We report an observation of a stable soliton-like structure on the surface of a ferrofluid, generated by a local perturbation in the hysteretic regime of the Rosensweig instability. Unlike other pattern-forming systems with localized 2D…
Quadratic flows have the unique property of uniform strain and are commonly used in turbulence modeling and hydrodynamic analysis. While previous application focused on two-dimensional homogeneous fluid, this study examines the geometric…