On traveling wave solutions with stability and phase plane analysis for the modified Benjamin-Bona-Mahony equation.

The modified Benjamin-Bona-Mahony (mBBM) model is utilized in the optical illusion field to describe the propagation of long waves in a nonlinear dispersive medium during a visual illusion (Khater 2021). This article investigates the mBBM equation through the utilization of the rational [Formula: see text]-expansion technique to derive new analytical wave solutions. The analytical solutions we have obtained comprise hyperbolic, trigonometric, and rational functions. Some of these exact solutions closely align with previously published results in specific cases, affirming the validity of our other solutions. To provide insights into diverse wave propagation characteristics, we have conducted an in-depth analysis of these solutions using 2D, 3D, and density plots. We also investigated the effects of various parameters on the characteristics of the obtained wave solutions of the model. Moreover, we employed the techniques of linear stability to perform stability analysis of the considered model. Additionally, we have explored the stability of the associated dynamical system through the application of phase plane theory. This study also demonstrates the efficacy and capabilities of the rational [Formula: see text]-expansion approach in analyzing and extracting soliton solutions from nonlinear partial differential equations.

Investigating wave solutions and impact of nonlinearity: Comprehensive study of the KP-BBM model with bifurcation analysis.

In this paper, we investigate the (2+1)-dimensional Kadomtsev-Petviashvili-Benjamin-Bona Mahony equation using two effective methods: the unified scheme and the advanced auxiliary equation scheme, aiming to derive precise wave solutions. These solutions are expressed as combinations of trigonometric, rational, hyperbolic, and exponential functions. Visual representations, including three-dimensional (3D) and two-dimensional (2D) combined charts, are provided for some of these solutions. The influence of the nonlinear parameter p on the wave type is thoroughly examined through diverse figures, illustrating the profound impact of nonlinearity. Additionally, we briefly investigate the Hamiltonian function and the stability of the model using a planar dynamical system approach. This involves examining trajectories, isoclines, and nullclines to illustrate stable solution paths for the wave variables. Numerical results demonstrate that these methods are reliable, straightforward, and potent tools for analyzing various nonlinear evolution equations found in physics, applied mathematics, and engineering.

Vermont-wide assessment of anthropogenic background concentrations of perfluoroalkyl substances in surface soils.

Shallow surface soils from 66 suburban sampling locations across Vermont were analyzed for 17 different perfluoroalkyl acids (PFAA). PFAA were detected in all 66 surface soils, with a total concentration of PFAA ranging from 540 to 36,000 ng/kg dry soil weight (dw). Despite the complexity of site-specific factors, some general trends and correlations in PFAA concentrations were observed. For instance, perfluoro-1-octanesulfonate (PFOS) dominated in all soil samples while seven other PFAA, including perfluoro-n-nonanoic acid, perfluoro-n-octanoic acid, perfluoro-n-hexanoic acid, perfluoro-n-heptanoic acid, perfluoro-n-decanoic acid, perfluoro-n-undecanoic acid, perfluoro-1-butanesulfonate, and perfluoro-1-hexanesulfonate (PFNA, PFOA, PFHxA, PFHpA, PFDA, PFUnDA, and PFBS, respectively), were identified at more than 50 % of the locations. Perfluoroalkyl carboxylic acids (PFCA) showed a positive correlation with total organic carbon, whereas no clear correlation was observed for perfluoroalkyl sulfonate acids (PFSA). In addition, variations in geographical distributions of PFAA were observed, with relatively higher total PFAA in northern regions when compared to Southern Vermont. Moreover, PFHxA, PFNA, PFDA, PFUnDA, PFOS, and total PFAA were positively correlated to land-use types in Northern Vermont. These results are useful for understanding unique behaviors of PFCA vs. PFSA in geospatially distributed surface soils and for providing anthropogenic background data for setting PFAS cleanup standards for surface soils.

Invasive behaviour under competition via a free boundary model: a numerical approach.

What will happen when two invasive species are competing and invading the environment at the same time? In this paper, we try to find all the possible scenarios in such a situation based on the diffusive Lotka-Volterra competition system with free boundaries. In a recent work, Du and Wu (Calc Var Partial Differ Equ, 57(2):52, 2018) considered a weak-strong competition case of this model (with spherical symmetry) and theoretically proved the existence of a "chase-and-run coexistence" phenomenon, for certain parameter ranges when the initial functions are chosen properly. Here we use a numerical approach to extend the theoretical research of Du and Wu (Calc Var Partial Differ Equ, 57(2):52, 2018) in several directions. Firstly, we examine how the longtime dynamics of the model changes as the initial functions are varied, and the simulation results suggest that there are four possible longtime profiles of the dynamics, with the chase-and-run coexistence the only possible profile when both species invade successfully. Secondly, we show through numerical experiments that the basic features of the model appear to be retained when the environment is perturbed by periodic variation in time. Thirdly, our numerical analysis suggests that in two space dimensions the population range and the spatial population distribution of the successful invader tend to become more and more circular as time increases no matter what geometrical shape the initial population range possesses. Our numerical simulations cover the one space dimension case, and two space dimension case with or without spherical symmetry. The numerical methods here are based on that of Liu et al. (Mathematics, 6(5):72, 2018, Int J Comput Math, 97(5): 959-979, 2020). In the two space dimension case without radial symmetry, the level set method is used, while the front tracking method is used for the remaining cases. We hope the numerical observations in this paper can provide further insights to the biological invasion problem, and also to future theoretical investigations. More importantly, we hope the numerical analysis may reach more biologically oriented experts and inspire applications of some refined versions of the model tailored to specific real world biological invasion problems.

Factors influencing recovery of SARS-CoV-2 RNA in raw sewage and wastewater sludge using polyethylene glycol-based concentration method.

Wastewater surveillance for monitoring severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is an important epidemiologic tool for the assessment of population-wide coronavirus disease 2019 (COVID-19). This tool can be successfully implemented only if SARS-CoV-2 RNA in wastewater can be accurately recovered and quantified. The lack of standardized procedure for wastewater virus analysis has resulted in varying SARS-CoV-2 concentrations for the same sample. This study reports the effect of 4 key factors-sample volume, percentage polyethylene glycol (PEG)-NaCl, incubation period, and storage duration at 4°C-on the recovery of spiked noninfectious SARS-CoV-2 RNA in raw sewage and sludge samples. N1 and N2 genes of SARS-CoV-2 were quantified using the reverse transcription-quantitative polymerase chain reaction (RT-qPCR) and digital droplet PCR (RT-ddPCR) techniques. Results indicate that 1) for raw sewage, 50-ml sample volume, 30% PEG-NaCl addition, 6-h incubation, and sample analysis within 24 h of collection can result in much better RNA recovery (RT-qPCR: 72% for N1 and 82% for N2; RT-ddPCR: 55% for N1 and 85% for N2) when compared with commonly used PEG-based method; 2) for sludge, the sample analysis using raw sewage protocol and all other variations of each factor mostly resulted in false negatives for both N1 and N2. The absence of N1 and N2 suggests that sludge samples probably need a pretreatment step that releases RNA entrapped in sludge solids back into bulk solution. In conclusion, our modified PEG-based concentration method can cut down the analysis time at least by half, which in turn helps to implement early detection system for SARS-CoV-2 in wastewater.

Reaction Time Improvements by Neural Bistability.

The often reported reduction of Reaction Time (RT) by Vision Training) is successfully replicated by 81 athletes across sports. This enabled us to achieve a mean reduction of RTs for athletes eye-hand coordination of more than 10%, with high statistical significance. We explain how such an observed effect of Sensorimotor systems' plasticity causing reduced RT can last in practice for multiple days and even weeks in subjects, via a proof of principle. Its mathematical neural model can be forced outside a previous stable (but long) RT into a state leading to reduced eye-hand coordination RT, which is, again, in a stable neural state.

Exact traveling wave solutions for system of nonlinear evolution equations.

In this work, recently deduced generalized Kudryashov method is applied to the variant Boussinesq equations, and the (2 + 1)-dimensional breaking soliton equations. As a result a range of qualitative explicit exact traveling wave solutions are deduced for these equations, which motivates us to develop, in the near future, a new approach to obtain unsteady solutions of autonomous nonlinear evolution equations those arise in mathematical physics and engineering fields. It is uncomplicated to extend this method to higher-order nonlinear evolution equations in mathematical physics. And it should be possible to apply the same method to nonlinear evolution equations having more general forms of nonlinearities by utilizing the traveling wave hypothesis.

Study of coupled nonlinear partial differential equations for finding exact analytical solutions.

Exact solutions of nonlinear partial differential equations (NPDEs) are obtained via the enhanced (G'/G)-expansion method. The method is subsequently applied to find exact solutions of the Drinfel'd-Sokolov-Wilson (DSW) equation and the (2+1)-dimensional Painlevé integrable Burgers (PIB) equation. The efficiency of this method for finding these exact solutions is demonstrated. The method is effective and applicable for many other NPDEs in mathematical physics.

Exact solutions of unsteady Korteweg-de Vries and time regularized long wave equations.

In this paper, we implement the exp(-Φ(ξ))-expansion method to construct the exact traveling wave solutions for nonlinear evolution equations (NLEEs). Here we consider two model equations, namely the Korteweg-de Vries (KdV) equation and the time regularized long wave (TRLW) equation. These equations play significant role in nonlinear sciences. We obtained four types of explicit function solutions, namely hyperbolic, trigonometric, exponential and rational function solutions of the variables in the considered equations. It has shown that the applied method is quite efficient and is practically well suited for the aforementioned problems and so for the other NLEEs those arise in mathematical physics and engineering fields. PACS numbers: 02.30.Jr, 02.70.Wz, 05.45.Yv, 94.05.Fq.

Study of analytical method to seek for exact solutions of variant Boussinesq equations.

ABSTRACT: In this paper, we have been acquired the soliton solutions of the Variant Boussinesq equations. Primarily, we have used the enhanced (G'/G)-expansion method to find exact solutions of Variant Boussinesq equations. Then, we attain some exact solutions including soliton solutions, hyperbolic and trigonometric function solutions of this equation. MATHEMATICS SUBJECT CLASSIFICATION: 35 K99; 35P05; 35P99.

Exact traveling wave solutions of modified KdV-Zakharov-Kuznetsov equation and viscous Burgers equation.

ABSTRACT: Mathematical modeling of many physical systems leads to nonlinear evolution equations because most physical systems are inherently nonlinear in nature. The investigation of traveling wave solutions of nonlinear partial differential equations (NPDEs) plays a significant role in the study of nonlinear physical phenomena. In this article, we construct the traveling wave solutions of modified KDV-ZK equation and viscous Burgers equation by using an enhanced (G '/G) -expansion method. A number of traveling wave solutions in terms of unknown parameters are obtained. Derived traveling wave solutions exhibit solitary waves when special values are given to its unknown parameters. MATHEMATICS SUBJECT CLASSIFICATION: 35C07; 35C08; 35P99.

Exact solutions for (1 + 1)-dimensional nonlinear dispersive modified Benjamin-Bona-Mahony equation and coupled Klein-Gordon equations.

ABSTRACT: In this work, recently developed modified simple equation (MSE) method is applied to find exact traveling wave solutions of nonlinear evolution equations (NLEEs). To do so, we consider the (1 + 1)-dimensional nonlinear dispersive modified Benjamin-Bona-Mahony (DMBBM) equation and coupled Klein-Gordon (cKG) equations. Two classes of explicit exact solutions-hyperbolic and trigonometric solutions of the associated equations are characterized with some free parameters. Then these exact solutions correspond to solitary waves for particular values of the parameters. PACS NUMBERS: 02.30.Jr; 02.70.Wz; 05.45.Yv; 94.05.Fg.

A note on improved F-expansion method combined with Riccati equation applied to nonlinear evolution equations.

The purpose of this article is to present an analytical method, namely the improved F-expansion method combined with the Riccati equation, for finding exact solutions of nonlinear evolution equations. The present method is capable of calculating all branches of solutions simultaneously, even if multiple solutions are very close and thus difficult to distinguish with numerical techniques. To verify the computational efficiency, we consider the modified Benjamin-Bona-Mahony equation and the modified Korteweg-de Vries equation. Our results reveal that the method is a very effective and straightforward way of formulating the exact travelling wave solutions of nonlinear wave equations arising in mathematical physics and engineering.