By Christos H Skiadas, Ioannis Dimotikalis, Charilaos Skiadas
This quantity comprises the simplest papers provided on the CHAOS 2008 overseas convention on Chaotic Modeling, Simulation and purposes. It presents a worthy choice of new principles, tools, and strategies within the box of nonlinear dynamics, chaos, fractals and their purposes more often than not technological know-how and in engineering sciences. It touches on many fields similar to chaos, dynamical structures, nonlinear platforms, fractals and chaotic attractors. It additionally covers mechanics, hydrofluid dynamics, chaos in meteorology and cosmology, Hamiltonian and quantum chaos, chaos in biology and genetics, chaotic regulate, and chaos in economic system and markets, and chaotic simulations; therefore, containing state of the art interdisciplinary study with high-interest purposes. those contributions current new options by way of examining the suitable info and during using contemporary advances in several fields, particularly in chaotic simulation equipment and methods.
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Extra info for Topics on Chaotic Systems: Selected Papers from Chaos 2008 International Conference
A. Cornell, Phys. Rev. Lett. 77, 4984 (1996) 11. A. S. Rokhsar, Phys. Rev. A 55, 4346 (1997) 12. F. London, Nature 141, 643 (1938) Chapter 3 Gross-Pitaevskii Model of the Condensate Abstract The Gross-Pitaevskii equation (GPE) is a successful and well-established model for describing an atomic Bose-Einstein condensate. Here we introduce this model, along with its assumptions. Throughout the rest of this chapter we explore its properties and key time-independent solutions. 1 The Gross-Pitaevskii Equation We assume that the gas is at zero temperature, such that the thermal gas and thermallydriven excitations of the condensate are non-existent.
13) The continuity equation expresses conservation of the number of atoms (or, when written in terms of ρ(r, t), conservation of mass). By integrating the equation over a given volume, we see that, if the number of atoms changes in that volume, it is because fluid has moved in or out of it. The second equation is, ∂v = −∇ m ∂t √ 2 1 2 ∇2 n mv + V + gn − √ 2 2m n . 14) √ √ The ∇ 2 n/ n term is termed the quantum pressure term (see below). 15) where P and P are respectively the pressure and the quantum pressure, P= gn 2 , 2 P =− 2 4m n∇ 2 (ln n).
23) V (r) = ωr2 r 2 , 2 where r 2 = x 2 + y 2 + z 2 . The characteristic length scale of this potential is the harmonic oscillator length, /mωr . 24) r = There is no general analytic solution for the ground state (lowest energy) solution of the BEC in a harmonic trap; usually the ground state is found by numerically solving Eq. 9). However, there exist useful analytic results for certain regimes which we describe below. It is useful to work in terms of the interaction parameter,2 N as / r . Below we distinguish the following cases: no interactions, strong repulsive 1) and weak interactions (|N as / r | 1).