Orbital analysis in the planar circular Copenhagen problem using polar coordinates

We examine the orbital dynamics of the planar Copenhagen problem, by classifying sets of starting conditions of orbits, expressed in polar coordinates. Specifically, we examine how the Jacobi constant influences several aspects of the overall dynamics of the test particle, such as its final state, as well as the time of escape/collision. By using modern diagrams with color codes, we manage to present the different types of basins. It is shown that both the character of the orbits and the fractal degree of the system are highly dependent on the Jacobi constant.     

Tranversality of the invariant manifolds associated to the Lyapunov family of periodic orbits near L2 in the restricted three-body problem

The restricted three-body problem is considered for values of the Jacobi constant C near the value C₂ associated to the Euler critical point L₂. A Lyapunov family of periodic orbits near L₂, the so-called family (c), is born for C = C₂ and exists for values of C less than C₂. These periodic orbits are hyperbolic. The corresponding invariant manifolds meet transversally along homoclinic orbits. In this paper the variation of the transversality is analyzed as a function of the Jacobi constant C and of the mass parameter μ. Asymptotical expressions of the invariant manifolds for C ? C₂ and μ ? 0 are found. Several numerical experiments provide accurate information for the manifolds and a good agreement is found with the asymptotical expressions. Symbolic dynamic techniques are used to show the existence of a large class of motions. In particular the existence of orbits passing in a random way (in a given sense) from the region near one primary to the region near the other is proved.     

Representation hypergéométrique des singularités de la solution du problème de Cauchy caractéristique à données holomorphes

We consider the Cauchy problem for the wave equation in a case where density is variable and can tend to zero at infinity. Examples are provided showing that, unlike the case of constant density, there is, generally speaking, neither uniqueness nor smoothness of solutions, if we donot impose any restrictions on them at infinity. We have found a class of unique solvability of the problem, containing physically interesting finite energy solutions.     

Global structure for a class of dynamical systems

In this paper, we shall consider the global structure of positive bounded systems on the plane which have m singular points, but not any closed orbits and singular closed orbits. We shall prove that these systems have at least m−1 connecting orbits; and all the connecting orbits, homoclinic orbits and singular points constitute a compact simply connected set. Each of other orbits tends to a singular point as t→+∞, and approaches to the infinity as t→−∞.     

Long-term behaviour of a cyclic catalytic branching system

We investigate the long-term behaviour of a system of SDEs for d≥2 types, involving catalytic branching and mutation between types. In particular, we show that the overall sum of masses converges to zero but does not hit zero in finite time a.s. We shall then focus on the relative behaviour of types in the limit. We prove weak convergence to a unique stationary distribution that does not put mass on the set where at least one of the coordinates is zero. Finally, we provide a complete analysis of the case d=2.     

Equations of Camassa‐Holm type and Jacobi ellipsoidal coordinates

We consider the integrable Camassa‐Holm (CH) equation on the line with positive initial data rapidly decaying at infinity. On such a phase space we construct a one‐parameter family of integrable hierarchies that preserves the mixed spectrum of the associated string spectral problem. This family includes the CH hierarchy. We demonstrate that the constructed flows can be interpreted as Hamiltonian flows on the space of Weyl functions of the associated string spectral problem. The corresponding Poisson bracket is the Atiyah‐Hitchin bracket. Using an infinite dimensional version of the Jacobi ellipsoidal coordinates, we obtain a one‐parameter family of canonical coordinates linearizing the flows. © 2005 Wiley Periodicals, Inc.     

Computing complex and real tropical curves using monodromy

Tropical varieties capture combinatorial information about how coordinates of points in a classical variety approach zero or infinity. We present algorithms for computing the rays of a complex and real tropical curve defined by polynomials with constant coefficients. These algorithms rely on homotopy continuation, monodromy loops, and Cauchy integrals. Several examples are presented which are computed using an implementation that builds on the numerical algebraic geometry software Bertini.     

Burgers turbulence initialized by a regenerative impulse

In this article we look at a one-dimensional infinitesimal particle system governed by the completely inelastic collision rule. Considering uniformly spread mass, we feed the system with initial velocities, so that when time evolves the corresponding velocity field fulfils the inviscid Burgers equation. More precisely, we suppose here that the initial velocities are zero, except for particles located on a stationary regenerative set for which the velocity is some given constant number. We give results of a large deviation type. First, we estimate the probability that a typical particle is located at time 1 at distance at least D from its initial position, when D tends to infinity. Its behaviour is related to the left tail of the gap measure of the regenerative set. We also show the same asymptotics for the tail of the shock interval length distribution. Second, we analyse the event that a given particle stands still at time T as T tends to infinity. The data to which we relate its behaviour are the right tail of the gap measure of the regenerative set. We conclude with some results on the shock structure.     

Existence and nonexistence of ground state solutions for elliptic equations with a convection term

We deal with the existence of positive solutions u decaying to zero at infinity, for a class of equations of Lane-Emden-Fowler type involving a gradient term. One of the main points is that the differential equation contains a semilinear term σ(u) where σ:(0,∞)→(0,∞) is a smooth function which can be both unbounded at infinity and singular at zero. Our technique explores symmetry arguments as well as lower and upper solutions.     

Coupled-mode envelope solitary waves in a pair of cubic Schrödinger equations with cross modulation: Analytical solution and collisions with application to Rossby waves

Coupled-Nonlinear Schrödinger equations, linked by cross modulation terms, arise in both nonlinear optics and in Rossby waves in the atmosphere and ocean. In this paper, we derive exact, analytic solutions for the “bright” coupled-mode soliton, for which the envelope in each mode asymptotes to zero at spatial infinity, and for its spatially periodic generalization. We then numerically study the collisions of the coupled-mode solitary wave both with a conventional envelope soliton, confined to a single mode, and also with a second coupled-mode solitary wave. The collisions are sensitive to both the relative speed and phase of the solitons. In some parameter ranges, the collisions are nearly elastic, but in others, one or both solitons fission.     

Optimal dividend and capital injection strategy with a penalty payment at ruin: Restricted dividend payments

In this paper, we study the optimal dividend and capital injection problem with the penalty payment at ruin. The dividend strategy is assumed to be restricted to a small class of absolutely continuous strategies with bounded dividend density. By considering the surplus process killed at the time of ruin, we transform the problem to a combined stochastic and impulse control one up to ruin with a free boundary at zero. We illustrate the theoretical verifications for different types of capital injection strategies comparing to the conventional results given in the literature, where the capital injections are made before the time of ruin. Under the assumption of restricted dividend density, the value function is proved as the unique increasing, bounded, Lipschitz continuous and upper semi-continuous at zero viscosity solution to the corresponding quasi-variational Hamilton–Jacobi–Bellman (HJB) equation. The uniqueness of such class of viscosity solutions is shown by considering its boundary condition at infinity. The optimality of a specific band-type strategy is proved for the case when the premium rate is (i) greater than or (ii) less than the ceiling dividend rate respectively. Some numerical examples are presented under the exponential and gamma claim size assumptions.     

Sampling Theorems Associated with Singular q-Sturm Liouville Problems

In this paper we introduce three sampling theorems for transformations defined in terms of Jackson q-integration when the kernel of the transformation is a solution or the Green’s function of singular q-Sturm–Liouville problems. We consider the problem when the q-Sturm–Liouville problem is singular either at infinity or at zero with detailed investigations when the singular point is infinity. This approach allows the derivation of sampling representations for transforms whose kernels are linear combinations of q-Bessel functions, not just a single one as previously established.     

Nontrivial solutions of superlinear p-Laplacian equations

We consider p-Laplacian equations on a bounded domain, where the nonlinearity is superlinear but dose not satisfy the usual Ambrosetti-Rabinowitz condition near infinity, or its dual version near zero. Nontrivial solutions are obtained by computing the critical groups and Morse theory.