local equation
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2021 ◽  
Author(s):  
Jiang-Yan Song ◽  
Yu Xiao ◽  
Chi-Ping Zhang

Abstract In this paper, we firstly deduce a reverse space-time Fokas-Lenells equation which can be derived from a rather simple but extremely important symmetry reduction of corresponding local equation. Next, the determinant representations of one-fold Darboux transformation and N-fold Darboux transformation are expressed in detail by special eigenfunctions of spectral problem. Depending on zero seed solution and nonzero seed solution, exact solutions, including bright soliton solutions, kink solutions, periodic solutions, breather solutions, rogue wave solutions and several types of mixed soliton solutions, can be presented. Furthermore, the dynamical behaviors are discussed through some figures. It should be mentioned that the solutions of nonlocal Fokas-Lenells equation possess new characteristics different from the ones of local case. Besides, we also demonstrate the integrability by providing infinitely many conservation laws. The above results provide an alternative possibility to understand physical phenomena in the field of nonlinear optics, and related fields.


2021 ◽  
Vol 2021 (11) ◽  
Author(s):  
Wen-Yuan Ai ◽  
Marco Drewes ◽  
Dražen Glavan ◽  
Jan Hajer

Abstract We study how oscillations of a scalar field condensate are damped due to dissipative effects in a thermal medium. Our starting point is a non-linear and non-local condensate equation of motion descending from a 2PI-resummed effective action derived in the Schwinger-Keldysh formalism appropriate for non-equilibrium quantum field theory. We solve this non-local equation by means of multiple-scale perturbation theory appropriate for time-dependent systems, obtaining approximate analytic solutions valid for very long times. The non-linear effects lead to power-law damping of oscillations, that at late times transition to exponentially damped ones characteristic for linear systems. These solutions describe the evolution very well, as we demonstrate numerically in a number of examples. We then approximate the non-local equation of motion by a Markovianised one, resolving the ambiguities appearing in the process, and solve it utilizing the same methods to find the very same leading approximate solution. This comparison justifies the use of Markovian equations at leading order. The standard time-dependent perturbation theory in comparison is not capable of describing the non-linear condensate evolution beyond the early time regime of negligible damping. The macroscopic evolution of the condensate is interpreted in terms of microphysical particle processes. Our results have implications for the quantitative description of the decay of cosmological scalar fields in the early Universe, and may also be applied to other physical systems.


2021 ◽  
pp. 2150314
Author(s):  
Cui-Lian Yuan ◽  
Xiao-Yong Wen

In this paper, we construct a discrete nonlocal integrable lattice hierarchy related to a reverse space-time nonlocal nonlinear self-dual network equation which may have the potential applications in designing nonlocal electrical circuits and understanding the propagation of electrical signals. By means of nonlocal version of [Formula: see text]-fold Darboux transformation (DT) technique, discrete multi-soliton solutions in determinant form are constructed for the reverse space-time nonlocal nonlinear self-dual network equation. Through the asymptotic and graphic analysis, unstable soliton structures of one-, two- and three-soliton solutions are discussed graphically. We observe that the single components in this nonlocal equation display instability while the combined potential terms with nonlocal [Formula: see text]-symmetry show stable soliton structures. It is shown that these nonlocal solutions are clearly different from those of its corresponding local equation. The results given in this paper may explain the soliton propagation in electrical signals.


2021 ◽  
Vol 6 (10) ◽  
pp. 11046-11075
Author(s):  
Wen-Xin Zhang ◽  
◽  
Yaqing Liu

<abstract><p>In this paper, the reverse space cmKdV equation, the reverse time cmKdV equation and the reverse space-time cmKdV equation are constructed and each of three types diverse soliton solutions is derived based on the Hirota bilinear method. The Lax integrability of three types of nonlocal equations is studied from local equation by using variable transformations. Based on exact solution formulae of one- and two-soliton solutions of three types of nonlocal cmKdV equation, some figures are used to describe the soliton solutions. According to the dynamical behaviors, it can be found that these solutions possess novel properties which are different from the ones of classical cmKdV equation.</p></abstract>


2020 ◽  
Vol 31 (1) ◽  
Author(s):  
Nikos I. Kavallaris ◽  
Raquel Barreira ◽  
Anotida Madzvamuse

AbstractThe main purpose of the current paper is to contribute towards the comprehension of the dynamics of the shadow system of a singular Gierer–Meinhardt model on an isotropically evolving domain. In the case where the inhibitor’s response to the activator’s growth is rather weak, then the shadow system of the Gierer–Meinhardt model is reduced to a single though non-local equation whose dynamics is thoroughly investigated throughout the manuscript. The main focus is on the derivation of blow-up results for this non-local equation, which can be interpreted as instability patterns of the shadow system. In particular, a diffusion-driven instability (DDI), or Turing instability, in the neighbourhood of a constant stationary solution, which then is destabilised via diffusion-driven blow-up, is observed. The latter indicates the formation of some unstable patterns, whilst some stability results of global-in-time solutions towards non-constant steady states guarantee the occurrence of some stable patterns. Most of the theoretical results are verified numerically, whilst the numerical approach is also used to exhibit the dynamics of the shadow system when analytical methods fail.


2020 ◽  
Vol 68 (4) ◽  
Author(s):  
NATHAN CASTRO FONSÊCA ◽  
Isabelle Maria Jacqueline Meunier ◽  
Ana Carolina Borges Lins-e-Silva

Introduction: Estimating aboveground biomass (AGB) in protected forests is a challenge, due to high costs and legal restrictions for direct assessments, and also to frequently weak estimation provided by general AGB equations. Objective: We propose a new approach that uses dead fallen trees (DFTs) to improve AGB estimation. We aim to analyse if the adjustment of allometric models based on DFTs provides a suitable local equation for AGB estimation or helps to validate existing pantropical or regional allometric equations. Methods: The study was carried out at the Dois Irmãos State Park (PEDI), Pernambuco, Northeast Brazil. Along 4 000 m, using the line intersect technique, we sampled 37 recent dead fallen trees ranging from 7.6 to 92.3 cm in diameter and from 8.6 to 29.4 m in height. Nine models were adjusted, and the best equation (local) was compared to regional and pantropical equations. Results: One equation (AGBkg= 1.5292* DBH2.0601*TH-0.2187) produced biomass estimates which did not differ from the observed values (P > 0.05). Differently from the others, this equation overestimated AGB in only 2.8 % and, along with other parameters of analysis (R2adj and Syx %) had the best overall performance. Analysing the performance of the local equation versus available equations for estimating AGB, we found that only the local and the pantropical equations by Pearson et al. (2005) estimated values that were not significantly different from observed values. However, the local equation is more appropriate for local estimation of AGB, since it has the lowest RSME and CV. Conclusions: Based on the proposed approach, we were able to offer a local equation for tree AGB estimation based on fallen trees. The DFT method is efficient regarding time and costs, avoiding tree logging in protected forests and helps to guide the choice of a proper equation for forest AGB demands.


2019 ◽  
Vol 11 (6) ◽  
pp. 129
Author(s):  
Adriano Castelo dos Santos ◽  
Eleneide Doff Sotta ◽  
Marcelino Carneiro Guedes ◽  
Lilian Blanc

Tropical forests play an important role in maintaining the regional rainfall regime and global climate, besides representing a significant stock of carbon. This study aimed at evaluate above-ground biomass (AGB) recovery, after reduced-impact logging (RIL) in a managed forest on the Jari River valley. The data were collected in 15 plots (100 m &times; 100 m) in the management area of the Jari Florestal Company. To estimate AGB we used a local equation adjusted for forests in the eastern Amazon. AGB before logging ranged from 157.9 Mg ha-1 to 619.9 Mg ha-1, with an average of 362.5 Mg ha-1. AGB after logging ranged from 151.2 Mg ha-1 to 632.8 Mg ha-1, with an average of 322.4 Mg ha-1. The time of monitoring of the plots and logging intensity were the main factors that influenced the recovery of the AGB. In 12 years after the RIL, the forest was able to recover its initial stocks of AGB, in places of low exploitation intensity.


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