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The Mathieu partial differential equation (PDE) is analyzed as a prototypical model for pattern formation due to parametric resonance. After averaging and scaling, it is shown to be a perturbed nonlinear Schrödinger equation (NLS). Adiabatic perturbation theory for solitons is applied to determine which solitons of the NLS survive the perturbation...
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... A is a scalar variable and , , f , d , are constants for detuning, damping, parametric forcing, spatial coupling, and nonlinearity, respectively. The ordinary differential equation for the case of d ϭ 0 has been discussed in many papers ͑ e.g., Refs. ͓ 25–27 ͔͒ . Phase portraits for an undamped Mathieu equation are shown in Fig. 1. In the case of negative ␣ , the zero solution bifurcates subcritically at the bifurcation parameter f ϭ 0 when ␦ is negative, and bifurcates supercritically at f ϭ 2 ␦ when ␦ is positive. For 0 Ͻ f ϽϪ 2 ␦ and ␦ Ͻ 0, the system is bistable. Following the analysis in Ref. ͓ 5 ͔ , we can average Eq. ͑ 2 ͒ to ...
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Stationary spatially localized structures, sometimes called dissipative solitons, arise in many interesting and important applications, including buckling of slender structures under compression, nonlinear optics, fluid flow, surface catalysis, neurobiology and many more. The recent resurgence in interest in these structures has led to significant advances in our understanding of the origin and properties of these states, and these in turn suggest new questions, both general and system-specific. This paper surveys these results focusing on open problems, both mathematical and computational, as well as on new applications.
We investigate pattern selection at onset in a parametrically and inhomogeneously forced partial differential equation obtained by generalizing Mathieu's equation to include spatial interactions. No separation of scales is assumed. The proposed model is directly relevant to the case of parametrically forced surface waves, such as cross-waves, excited by the horizontal vibration of a fluid, where the forcing is localized to a finite region near the endwall or wavemaker. The availability of analytical solutions in the limit of piecewise constant forcing allows us investigate in detail the dependence of selected eigenfunctions on spatial detuning, forcing width, damping, boundary conditions, and container size. A wide range of onset patterns are located and described, many of which are rotated, modulated, or both, and deviate far from simple crosswise oriented standing waves. The linear selection mechanisms governing this multiplicity of potential onset patterns are discussed.
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Prospective experimental study.
University-based laboratory.
Imprinting control region female mice, 8-10 weeks old with regular estrous cycles.
Intraperitoneal injection of 10 IU of pregnant mare serum gonadotropin (PMSG) at noon followed by an additional injection of 10 IU hCG 48 hours later.
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The COH-induced supraphysiologic concentration of E2 aberrantly up-regulates CFTR, which leads to increased apoptosis in endometrial epithelial cells, resulting in impaired embryo implantation.
When a group of four states forms a subspace of the Hilbert space, i.e., appears to be strongly coupled with each other but very weakly interacts with all other states of the entire space, it is possible to express the nonadiabatic coupling (NAC) elements either in terms of s or in terms of electronic basis function angles, namely, mixing angles presumably representing the same sub-Hilbert space. We demonstrate that those explicit forms of the NAC terms satisfy the curl conditions--the necessary requirements to ensure the adiabatic-diabatic transformation in order to remove the NAC terms (could be often singular also at specific point(s) or along a seam in the configuration space) in the adiabatic representation of nuclear SE and to obtain the diabatic one with smooth functional form of coupling terms among the electronic states. In order to formulate extended Born-Oppenheimer (EBO) equations [J. Chem. Phys. 2006, 124, 074101] for a group of four states, we show that there should exist a coordinate independent ratio of the gradients for each pair of ADT/mixing angles leading to zero curls and, thereafter, provide a brief discussion on its analytical validity. As a numerical justification, we consider the first four eigenfunctions of the Mathieu equation to demonstrate the interesting features of nonadiabatic coupling (NAC) elements, namely, the validity of curl conditions and the nature of curl equations around CIs.
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The formation of oscillons in a synchronously oscillating background is studied in the context of both damped and self-exciting oscillatory media. Using the forced complex Ginzburg-Landau equation we show that such states bifurcate from finite amplitude homogenous states near the 2:1 resonance boundary. In each case we identify a region in parameter space containing a finite multiplicity of coexisting stable oscillons with different structure. Stable time-periodic monotonic and nonmonotonic frontlike states are present in an overlapping region. Both types of structure are related to the presence of a Maxwell point between the zero and finite amplitude homogeneous states.
A set of coupled complex Ginzburg-landau type amplitude equations which operates near a Hopf-Turing instability boundary is analytically investigated to show localized oscillatory patterns. The spatial structure of those patterns are the same as quantum mechanical harmonic oscillator stationary states and can have even or odd symmetry depending on the order of the state. It has been seen that the underlying Turing state plays a major role in the selection of the order of such solutions.
