Nonlinear system of PT-symmetric excitations and Toda vibrations integrable by the Darboux–Bäcklund dressing method

The nonlinear dynamics of coupled P T -symmetric excitations and Toda-like vibrations on a one-dimensional lattice are studied analytically and elucidated graphically. The nonlinear exciton-phonon system as the whole is shown to be integrable in the Lax sense inasmuch as it admits the zero-curvature representation supported by the auxiliary linear problem of third order. Inspired by this fact, we have developed in detail the Darboux–Bäcklund integration technique appropriate to generate a higher-rank crop solution by dressing a lower-rank (supposedly known) seed solution. In the framework of this approach, we have found a rather non-trivial four-component analytical solution exhibiting the crossover between the monopole and dipole regimes in the spatial distribution of intra-site excitations. This effect is inseparable from the pronounced mutual influence between the interacting subsystems in the form of specific nonlinear superposition of two essentially distinct types of travelling waves. We have established the criterion of monopole-dipole transition based upon the interplay between the localization parameter of Toda mode and the inter-subsystem coupling parameter.

Removing mains interference from the mfERG by applying a post-processing digital notch filter: for the good or the bad?

Purpose Ideally, the multifocal electroretinogram (mfERG) is recorded without noticeable intrusion of mains interference. However, sometimes contamination is difficult to avoid. A post-processing digital notch filter can help to recover the retinal response even in severe cases of mains interference. While a digital filter can be designed to have little to no impact on peak times, filtering out mains interference also removes the retinal signal content of the same frequency, which may result in a change of amplitude. The present study addressed this issue in the standard first order kernel mfERG. Methods In 24 recordings from routine exams with no perceivable mains interference, the effects of 50-Hz and 60-Hz non-causal digital notch filters on amplitude and peak time were assessed. Furthermore, the effect of filtering on contaminated traces was demonstrated and simulated mains interference was used to provide an example of nonlinear superposition of retinal signal and mains interference. Results mfERG amplitudes were reduced by 0%–15% (median 6%) with the 50-Hz filter and remained virtually unaffected with the 60-Hz filter. Simulations illustrate that spurious high-frequency components can occur in the filtered signal if a strongly contaminated signal is clipped due to a limited input range of the analog-to-digital converter. Conclusion The application of a 50-Hz digital notch filter to mfERG traces causes a mild amplitude reduction which will not normally affect the clinical interpretation of the data. The situation is even more favorable with a 60-Hz digital notch filter. Caution is necessary if the assumption of linear additivity of retinal signal and mains interference is violated.

Generalized Darboux transformation and asymptotic analysis on the degenerate dark-bright solitons for a coupled nonlinear Schrödinger system

In this paper, our work is based on a coupled nonlinear Schr ̈odinger system in a two-mode nonlinear fiber. A (N,m)-generalized Darboux transformation is constructed to derive the Nth-order solutions, where the positive integers N and m denote the numbers of iterative times and of distinct spectral parameters, respectively. Based on the Nth-order solutions and the given steps to perform the asymptotic analysis, it is found that a degenerate dark-bright soliton is the nonlinear superposition of several asymptotic dark-bright solitons possessing the same profile. For those asymptotic dark-bright solitons, their velocities are z-dependent except that one of those velocities could become z-independent under the certain condition, where z denotes the evolution dimension. Those asymptotic dark-bright solitons are reflected during the interaction. When a degenerate dark-bright soliton interacts with a nondegenerate/degenerate dark-bright soliton, the interaction is elastic, and the asymptotic bound-state dark-bright soliton with z-dependent or z-independent velocity could take place under certain conditions. Our study extends the investigation on the degenerate solitons from the bright soliton case for the scalar equations to the dark-bright soliton case for a coupled system.

Solitary waves in magnetostrictive circular rods with temperature effect

A simple nonlinear model is constructed in this paper to study the solitary wave in an infinite circular magnetostrictive rod. Based on the constitutive relations for transversely isotropic magnetostrictive materials, considering the coupling of multiphysics, combined with Hamilton’s principle and Euler equation, the longitudinal wave equation (LWE) of the infinite circular rod is obtained. The nonlinearity considered is geometrically associated with the nonlinear normal strain in the longitudinal rod direction. The transverse Poisson’s effect is included by introducing the effective Poisson’s ratio. Solitary wave solution, non-topological bell-type soliton and singular periodic solutions of the LWE are obtained by the [Formula: see text]-expansion method. By using the reductive perturbation method, we derive the KdV equation, furthermore, the two-solitary solution is obtained. Numerical analysis results show that the increase of the magnetic field intensity or temperature will reduce the solitary wave’s propagation velocity. As the wave velocity ratio increases, the wave amplitude gradually increases; when the coupled physics parameter and the wave velocity ratio are constant, the increase of the dispersion parameter will make the wavelength longer. The dynamic behavior of the two-soliton solution in the magnetostrictive rod exhibits nonlinear superposition and has elastic collision characteristics.

M -Breather, Lumps, and Soliton Molecules for the 2 + 1 -Dimensional Elliptic Toda Equation

The 2 + 1 -dimensional elliptic Toda equation is a higher dimensional generalization of the Toda lattice and also a discrete version of the Kadomtsev-Petviashvili-1 (KP1) equation. In this paper, we derive the M -breather solution in the determinant form for the 2 + 1 -dimensional elliptic Toda equation via Bäcklund transformation and nonlinear superposition formulae. The lump solutions of the 2 + 1 -dimensional elliptic Toda equation are derived from the breather solutions through the degeneration process. Hybrid solutions composed of two line solitons and one breather/lump are constructed. By introducing the velocity resonance to the N -soliton solution, it is found that the 2 + 1 -dimensional elliptic Toda equation possesses line soliton molecules, breather-soliton molecules, and breather molecules. Based on the N -soliton solution, we also demonstrate the interactions between a soliton/breather-soliton molecule and a lump and the interaction between a soliton molecule and a breather. It is interesting to find that the KP1 equation does not possess a line soliton molecule, but its discrete version—the 2 + 1 -dimensional elliptic Toda equation—exhibits line soliton molecules.

Relativistic Ermakov–Milne–Pinney Systems and First Integrals

The Ermakov–Milne–Pinney equation is ubiquitous in many areas of physics that have an explicit time-dependence, including quantum systems with time-dependent Hamiltonian, cosmology, time-dependent harmonic oscillators, accelerator dynamics, etc. The Eliezer and Gray physical interpretation of the Ermakov–Lewis invariant is applied as a guiding principle for the derivation of the special relativistic analog of the Ermakov–Milne–Pinney equation and associated first integral. The special relativistic extension of the Ray–Reid system and invariant is obtained. General properties of the relativistic Ermakov–Milne–Pinney are analyzed. The conservative case of the relativistic Ermakov–Milne–Pinney equation is described in terms of a pseudo-potential, reducing the problem to an effective Newtonian form. The non-relativistic limit is considered to be well. A relativistic nonlinear superposition law for relativistic Ermakov systems is identified. The generalized Ermakov–Milne–Pinney equation has additional nonlinearities, due to the relativistic effects.

Impacts of Heat and Drought on Gross Primary Productivity in China

Heat and drought stress, which often occur together, are the main environmental factors limiting the survival and growth of vegetation. Studies on the response of gross primary production (GPP) to extreme climate events such as heat and drought are highly significant for the identification of ecologically vulnerable regions, ecological risk assessments, and ecological environmental protection. We got 1982–2017 climatic data from the University of East Anglia Climatic Research Unit, Norwich, England, and GPP data from National Earth System Science Data Sharing Service Platform, Beijing, China. Using Theil–Sen median trend analysis and the Mann–Kendall test, we analyzed trends in temperature and the standardized precipitation/standardized precipitation evapotranspiration indices in the eight vegetation regions of China. Additionally, the response of GPP to the single and combined impacts of heat and drought were analyzed using multidimensional copula functions, and GPP reduction probabilities were estimated under different drought levels and heat intensities. The results showed that the probability of a drastic GPP reduction increases with increasing drought levels and heat intensities. The combined impacts of heat and drought on vegetation productivity is greater than the impacts of either drought or heat alone and presents a nonlinear superposition of the two extremes. The impact of heat on GPP is not evident when the drought level is high. The temperate grassland and warm temperate deciduous broad-leaved forest regions are the most sensitive regions to drought and heat in China. This study provides a scientific basis for the comprehensive evaluation of the risk of GPP reduction under the single and combined impacts of heat stress and drought stress.

Relativistic Ermakov-Milne-Pinney Systems and First Integrals

The Eliezer and Gray physical interpretation of the Ermakov-Lewis invariant is applied as a guiding principle for the derivation of the special relativistic analog of the Ermakov-Milne-Pinney equation and associated first integral. The special relativistic extension of the Ray-Reid system and invariant is obtained. General properties of the relativistic Ermakov-Milne-Pinney are analyzed. The conservative case of the relativistic Ermakov-Milne-Pinney equation is described in terms of a pseudo-potential, reducing the problem to an effective Newtonian form. The non-relativistic limit is considered as well. A relativistic nonlinear superposition law for relativistic Ermakov systems is identified. The generalized Ermakov-Milne-Pinney equation has additional nonlinearities, due to the relativistic effects.