Integrable Abel equations and Vein's Abel equation

We first reformulate and expand with several novel findings some of the basic results in the integrability of Abel equations. Next, these results are applied to Vein's Abel equation whose solutions are expressed in terms of the third order hyperbolic functions and a phase space analysis of the corresponding nonlinear oscillator is also providedComment: 12 pages, 4 figures, 17 references, online at Math. Meth. Appl. Sci. since 7/28/2015, published 4/201

Ermakov-Lewis Invariants and Reid Systems

Reid's m'th-order generalized Ermakov systems of nonlinear coupling constant alpha are equivalent to an integrable Emden-Fowler equation. The standard Ermakov-Lewis invariant is discussed from this perspective, and a closed formula for the invariant is obtained for the higher-order Reid systems (m\geq 3). We also discuss the parametric solutions of these systems of equations through the integration of the Emden-Fowler equation and present an example of a dynamical system for which the invariant is equivalent to the total energyComment: 8 pages, published versio

Integrable equations with Ermakov-Pinney nonlinearities and Chiellini damping

We introduce a special type of dissipative Ermakov-Pinney equations of the form v_{\zeta \zeta}+g(v)v_{\zeta}+h(v)=0, where h(v)=h_0(v)+cv^{-3} and the nonlinear dissipation g(v) is based on the corresponding Chiellini integrable Abel equation. When h_0(v) is a linear function, h_0(v)=\lambda^2v, general solutions are obtained following the Abel equation route. Based on particular solutions, we also provide general solutions containing a factor with the phase of the Milne type. In addition, the same kinds of general solutions are constructed for the cases of higher-order Reid nonlinearities. The Chiellini dissipative function is actually a dissipation-gain function because it can be negative on some intervals. We also examine the nonlinear case h_0(v)=\Omega_0^2(v-v^2) and show that it leads to an integrable hyperelliptic caseComment: 15 pages, 5 figures, 1 appendix, 21 references, published versio

Second order coupling between excited atoms and surface polaritons

Casimir-Polder interactions between an atom and a macroscopic body are typically regarded as due to the exchange of virtual photons. This is strictly true only at zero temperature. At finite temperature, real-photon exchange can provide a significant contribution to the overall dispersion interaction. Here we describe a new resonant two-photon process between an atom and a planar interface. We derive a second order effective Hamiltonian to explain how atoms can couple resonantly to the surface polariton modes of the dielectric medium. This leads to second-order energy exchanges which we compare with the standard nonresonant Casimir-Polder energy.Comment: 7 pages, 2 figure

Pulses and Snakes in Ginzburg--Landau Equation

Using a variational formulation for partial differential equations (PDEs) combined with numerical simulations on ordinary differential equations (ODEs), we find two categories (pulses and snakes) of dissipative solitons, and analyze the dependence of both their shape and stability on the physical parameters of the cubic-quintic Ginzburg-Landau equation (CGLE). In contrast to the regular solitary waves investigated in numerous integrable and non-integrable systems over the last three decades, these dissipative solitons are not stationary in time. Rather, they are spatially confined pulse-type structures whose envelopes exhibit complicated temporal dynamics. Numerical simulations reveal very interesting bifurcations sequences as the parameters of the CGLE are varied. Our predictions on the variation of the soliton amplitude, width, position, speed and phase of the solutions using the variational formulation agree with simulation results.Comment: 30 pages, 14 figure