Search results for " Nonlinear"
showing 10 items of 1224 documents
Multiple positive normalized solutions for nonlinear Schrödinger systems
2018
We consider the existence of multiple positive solutions to the nonlinear Schr\"odinger systems sets on $H^1(\mathbb{R}^N) \times H^1(\mathbb{R}^N)$, \[ \left\{ \begin{aligned} -\Delta u_1 &= \lambda_1 u_1 + \mu_1 |u_1|^{p_1 -2}u_1 + \beta r_1 |u_1|^{r_1-2} u_1|u_2|^{r_2}, -\Delta u_2 &= \lambda_2 u_2 + \mu_2 |u_2|^{p_2 -2}u_2 + \beta r_2 |u_1|^{r_1} |u_2|^{r_2 -2} u_2, \end{aligned} \right. \] under the constraint \[ \int_{\mathbb{R}^N}|u_1|^2 \, dx = a_1,\quad \int_{\mathbb{R}^N}|u_2|^2 \, dx = a_2. \] Here $a_1, a_2 >0$ are prescribed, $\mu_1, \mu_2, \beta>0$, and the frequencies $\lambda_1, \lambda_2$ are unknown and will appear as Lagrange multipliers. Two cases are studied, the first …
Pattern formation in hyperbolic reaction-transport systems and applications to dryland ecology
2023
Pattern formation and modulation is an active branch of mathematics, not only from the perspective of fundamental theory but also for its huge applications in many fields of physics, ecology, chemistry, biology, and other sciences. In this thesis, the occurrence of Turing and wave instabilities, giving rise to stationary and oscillatory patterns, respectively, is theoretically investigated by means of two-compartment reaction-transport hyperbolic systems. The goal is to elucidate the role of inertial times, which are introduced in hyperbolic models to account for the finite-time propagation of disturbances, in stationary and transient dynamics, in supercritical and subcritical regimes. In p…
System identification via optimised wavelet-based neural networks
2003
Nonlinear system identification by means of wavelet-based neural networks (WBNNs) is presented. An iterative method is proposed, based on a way of combining genetic algorithms (GAs) and least-square techniques with the aim of avoiding redundancy in the representation of the function. GAs are used for optimal selection of the structure of the WBNN and the parameters of the transfer function of its neurones. Least-square techniques are used to update the weights of the net. The basic criterion of the method is the addition of a new neurone, at a generic step, to the already constructed WBNN so that no modification to the parameters of its neurones is required. Simulation experiments and compa…
An Inverse Problem for the Relativistic Boltzmann Equation
2020
We consider an inverse problem for the Boltzmann equation on a globally hyperbolic Lorentzian spacetime $(M,g)$ with an unknown metric $g$. We consider measurements done in a neighbourhood $V\subset M$ of a timelike path $\mu$ that connects a point $x^-$ to a point $x^+$. The measurements are modelled by a source-to-solution map, which maps a source supported in $V$ to the restriction of the solution to the Boltzmann equation to the set $V$. We show that the source-to-solution map uniquely determines the Lorentzian spacetime, up to an isometry, in the set $I^+(x^-)\cap I^-(x^+)\subset M$. The set $I^+(x^-)\cap I^-(x^+)$ is the intersection of the future of the point $x^-$ and the past of th…
An operatorial approach to stock markets
2009
We propose and discuss some toy models of stock markets using the same operatorial approach adopted in quantum mechanics. Our models are suggested by the discrete nature of the number of shares and of the cash which are exchanged in a real market, and by the existence of conserved quantities, like the total number of shares or some linear combination of cash and shares. The same framework as the one used in the description of a gas of interacting bosons is adopted.
A hybrid stimulation strategy for suppression of spiral waves in cardiac tissue
2011
International audience; Atrial fibrillation (AF) is the most common cardiac arrhythmia whose mechanisms are thought to be mainly due to the self perpetuation of spiral waves (SW). To date, available treatment strategies (antiarrhythmic drugs, radiofrequency ablation of the substrate, electrical cardioversion) to restore and to maintain a normal sinus rhythm have limitations and are associated with AF recurrences. The aim of this study was to assess a way of suppressing SW by applying multifocal electrical stimulations in a simulated cardiac tissue using a 2D FitzHugh-Nagumo model specially convenient for AF investigations. We identified stimulation parameters for successful termination of S…
Recurrence quantification analysis as a tool for complex fractionated atrial electrogram discrimination
2012
International audience; Atrial fibrillation is the most encountered pathology of the heart rate. The reasons of its occurrence and its particular characteristics remain unknown, resulting from complex phenomena interaction. From these interactions emerges Complex Fractionated Atrial Electrograms (CFAE) which are useful for the ablation procedure. This study presents a method based on nonlinear data analysis, the Recurrence Quantification Analysis (RQA) applied on intracardiac atrial electrograms to detect CFAE particularities. The results obtained on areas previously tagged by a cardilogist show a good sensitivity to CFAE. Combination of RQA features offers a larger discrimination potential…
Noise effect in a FitzHugh-Nagumo circuit driven by a bichromatic signal
2013
We analyze the response of a nonlinear circuit exactly ruled by the FitzHugh-Nagumo equations. This circuit is submitted to a bichromatic signal including a high frequency and a low frequency. In absence of noise, we show that for an appropriate amplitude of the high frequency driving, the response of the circuit estimated at the low frequency can be optimized via the phenomenon of vibrational resonance. Next, we show that under certain conditions, noise can contribute to the effect of vibrational resonance. Colored noise is also considered. Our experimental results are confirmed by a numerical analysis.
Increasing the Astrophysical Reach of the Advanced Virgo Detector via the Application of Squeezed Vacuum States of Light
2019
Current interferometric gravitational-wave detectors are limited by quantum noise over a wide range of their measurement bandwidth. One method to overcome the quantum limit is the injection of squeezed vacuum states of light into the interferometer's dark port. Here, we report on the successful application of this quantum technology to improve the shot noise limited sensitivity of the Advanced Virgo gravitational-wave detector. A sensitivity enhancement of up to 3.2±0.1 dB beyond the shot noise limit is achieved. This nonclassical improvement corresponds to a 5%-8% increase of the binary neutron star horizon. The squeezing injection was fully automated and over the first 5 months of the thi…
Noise effect in a sine-Gordon Lattice
2013
International audience; This paper is devoted to the influence of internal noise in a sine-Gordon chain exhibiting the well known nonlinear supratransmission phenomenon. It is shown that spatiotemporal noise can trigger breather modes with a given probability in a range of parameters where they do not occur without noise. A frequency analysis is carried out to quantify the degree of coherence of the emitted breather. It is shown that there exists an appropriate amount of noise which ensures the existence of breather modes with the best coherence.