Heteroclinic connections between triple collisions and relative periodic orbits in the isosceles three-body problem

We study the isosceles three-body problem and show that there exist infinitely many families of relative periodic orbits converging to heteroclinic cycles between equilibria on the collision manifold in Devaney's blown-up coordinates. Towards this end, we prove that two types of heteroclinic orbits exist in much wider parameter ranges than previously detected, using self-validating interval arithmetic calculations, and we appeal to the previous results on heteroclinic orbits. Moreover, we give numerical computations for heteroclinic and relative periodic orbits to demonstrate our theoretical results. The numerical results also indicate that the two types of heteroclinic orbits and families of relative periodic orbits exist in wider parameter regions than detected in the theory and that some of them are related to Euler orbits

Monotonicity of the first eigenvalue and the global bifurcation diagram for the branch of interior peak solutions

AbstractLet B⊂Rn (n⩾1) be a unit ball and D⊂Rm be a bounded domain. We study the global branch consisting of interior single-peak solutions of the elliptic Neumann problemΔu+λ(−u+up)=0in B,∂νu=0on ∂B and the monotonicity of the first eigenvalue along the branch for large λ, where1<p<pS:={n+2n−2,n⩾3,∞,n=1,2. When the domain is replaced with B×D, using this monotonicity, we show that the branch of solutions concentrating on {0}×D has secondary bifurcation points. For n=1 and p⩾2 an integer, we determine the global bifurcation diagram by showing the monotonicity of the time-map (the period function). The monotonicity of the first eigenvalue along the whole branch is also proved for n=1 and p=3

Discrete embedded solitons

We address the existence and properties of discrete embedded solitons (ESs), i.e., localized waves existing inside the phonon band in a nonlinear dynamical-lattice model. The model describes a one-dimensional array of optical waveguides with both the quadratic (second-harmonic generation) and cubic nonlinearities. A rich family of ESs was previously known in the continuum limit of the model. First, a simple motivating problem is considered, in which the cubic nonlinearity acts in a single waveguide. An explicit solution is constructed asymptotically in the large-wavenumber limit. The general problem is then shown to be equivalent to the existence of a homoclinic orbit in a four-dimensional reversible map. From properties of such maps, it is shown that (unlike ordinary gap solitons), discrete ESs have the same codimension as their continuum counterparts. A specific numerical method is developed to compute homoclinic solutions of the map, that are symmetric under a specific reversing transformation. Existence is then studied in the full parameter space of the problem. Numerical results agree with the asymptotic results in the appropriate limit and suggest that the discrete ESs may be semi-stable as in the continuous case.Comment: A revtex4 text file and 51 eps figure files. To appear in Nonlinearit