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showing 10 items of 4526 documents
Numerical study of blow-up in solutions to generalized Kadomtsev-Petviashvili equations
2013
We present a numerical study of solutions to the generalized Kadomtsev-Petviashvili equations with critical and supercritical nonlinearity for localized initial data with a single minimum and single maximum. In the cases with blow-up, we use a dynamic rescaling to identify the type of the singularity. We present a discussion of the observed blow-up scenarios.
Analytical Solution of the Richards Equation under Gravity-Driven Infiltration and Constant Rainfall Intensity
2020
In the field of soil hydrology, the Richards equation is commonly used to model water flow in unsaturated soils. The high nonlinearity of the Richards equation makes it very challenging to solve analytically for situations that are meaningful in practical applications. In this paper, an exact and simple analytical solution of the Richards equation under gravity-driven infiltration and constant rainfall intensity is derived. First, the solution is presented under Torricelli's law, which mimics the soil hydraulic conductivity function and describes the emptying or filling process of a nonlinear water reservoir. Then, following a similar approach, the solution is extended to the Brooks and Cor…
Temporal incoherent solitons supported by a defocusing nonlinearity with anomalous dispersion
2012
http://pra.aps.org/; International audience; We study temporal incoherent solitons in noninstantaneous response nonlinear media. Contrarily to the usual temporal soliton, which is known to require a focusing nonlinearity with anomalous dispersion, we show that a highly noninstantaneous nonlinear response leads to incoherent soliton structures which require the inverted situation: In the focusing regime (and anomalous dispersion) the incoherent wave packet experiences an unlimited spreading, whereas in the defocusing regime (still with anomalous dispersion) the incoherent wave packet exhibits a self-trapping. These counterintuitive results are explained in detail by a long-range Vlasov formu…
Universal charts for optical difference frequency generation in the terahertz domain
2010
We present a universal and rigorous approach to study difference frequency generation in the terahertz domain, keeping the number of degrees of freedom to a minimum, through the definition of a suitable figure of merit. The proposed method relies on suitably normalized charts, that enable to predict the optical-to-terahertz conversion efficiency of any system based on wave propagation in quadratic nonlinear materials. The predictions of our approach are found to be in good agreement with the best experimental results reported to date, enabling also to estimate the d22 nonlinear coefficient of high quality GaSe.
Spatio-temporal Characteristics of THz Emission at the Subwavelength Scale via Optical Rectification
2011
Highly localized THz emission via optical rectification in thin nonlinear crystals is a promising method for subwavelength microscopy. We present here the peculiar THz spatio-temporal characteristics induced by the non-paraxial generation regime.
Turing instability and traveling fronts for a nonlinear reaction–diffusion system with cross-diffusion
2012
In this work we investigate the phenomena of pattern formation and wave propagation for a reaction–diffusion system with nonlinear diffusion. We show how cross-diffusion destabilizes uniform equilibrium and is responsible for the initiation of spatial patterns. Near marginal stability, through a weakly nonlinear analysis, we are able to predict the shape and the amplitude of the pattern. For the amplitude, in the supercritical and in the subcritical case, we derive the cubic and the quintic Stuart–Landau equation respectively. When the size of the spatial domain is large, and the initial perturbation is localized, the pattern is formed sequentially and invades the whole domain as a travelin…
Propagation failure in discrete bistable reaction-diffusion systems: Theory and experiments
2001
International audience; Wave front propagation failure is investigated in discrete bistable reaction-diffusion systems. We present a theoretical approach including dissipative effects and leading to an analytical expression of the critical coupling beyond which front propagation can occur as a function of the nonlinearity threshold parameter. Our theoretical predictions are confirmed by numerical simulations and experimental results on an equivalent electrical diffusive lattice.
Elastic waves in random-fibre networks
1997
The propagation of the first displacement maximum of a semi-infinite wavetrain in a two-dimensional random-fibre network is analysed. Model calculations and numerical simulations are used for demonstrating that two qualitatively different wavefront velocities appear in the network. A transient wave, which travels fast and whose amplitude decreases exponentially, dominates the short-time behaviour when the bending stiffness of the fibres is small and the driving frequency is high. This mode can be described by a one-dimensional model. The transient-wave mode propagates even if the bending stiffness of the fibres vanishes, in which case the normal sound velocity is zero. The usual, and slower…
Pseudodifferential operators of Beurling type and the wave front set
2008
AbstractWe investigate the action of pseudodifferential operators of Beurling type on the wave front sets. More precisely, we show that these operators are microlocal, that is, preserve or reduce wave front sets. Some consequences on micro-hypoellipticity are derived.
Weakly nonlinear analysis of Turing patterns in a morphochemical model for metal growth
2015
We focus on the morphochemical reaction–diffusion model introduced in Bozzini et al. (2013) and carry out a nonlinear bifurcation analysis with the aim to characterize the shape and the amplitude of the patterns arising as the result of Turing instability of the physically relevant equilibrium. We perform a weakly nonlinear multiple scales analysis, and derive the normal form equations governing the amplitude of the patterns. These amplitude equations allow us to construct relevant solutions of the model equations and reveal the presence of multiple branches of stable solutions arising as the result of subcritical bifurcations. Hysteretic type phenomena are highlighted also through numerica…