We present a geometric formulation of existence of time quasi-periodic solutions. As an application, we prove the existence of quasi-periodic solutions of b frequencies, b ≤ d + 2, in arbitrary dimension d and for arbitrary non integrable algebraic nonlinearity p. This reflects the conservation of d momenta, energy and L norm. In 1d, we prove the existence of quasi-periodic solutions with arbit...

We introduce a discretization of the Lagrange-d’Alembert principle for Lagrangian systems with nonholonomic constraints, which allows us to construct numerical integrators that approximate the continuous flow. We study the geometric invariance properties of the discrete flow which provide an explanation for the good performance of the proposed method. This is tested on two examples: a nonholono...

We give a classical geometric optics description of incoherent solitons—those launched by a diffuse source. This method is intuitive, advances predictions such as the existence of solitons of arbitrary cross section, and importantly, it provides a simple (universal) analytical description for the incoherent solitons of any nonlinear medium. Previously, analytical results were known for the ln I...

We consider the semilinear parabolic equation ut = uxx + f(u), x ∈ R, t > 0, (A) where f is a bistable nonlinearity. It is well-known that for a large class of initial data, the corresponding solutions converge to traveling fronts. We give a new proof of this classical result as well as some generalizations. Our proof uses a geometric method, which makes use of spatial trajectories {(u(x, t), u...

In vitro studies have previously found a class of vestibular nuclei neurons to exhibit a bidirectional afterhyperpolarization (AHP) in their membrane potential, due to calcium and calcium-activated potassium conductances. More recently in vivo studies of such vestibular neurons were found to exhibit a boosting nonlinearity in their input-output tuning curves. In this paper, a Hodgkin-Huxley (HH...

We investigate traveling wave solutions in a family of reaction-diffusion equations which includes the Fisher–Kolmogorov–Petrowskii–Piscounov (FKPP) equation with quadratic nonlinearity and a bistable equation with degenerate cubic nonlinearity. It is known that, for each equation in this family, there is a critical wave speed which separates waves of exponential decay from those of algebraic d...