Similar Reduction Research Articles

Lie Group Classification of a Class of Variable Coefficient Boiti–Leon–Manna–Pempinelli Equations

The Boiti–Leon–Manna–Pempinelli (BLMP) equation with coefficients being functions of time is considered. Since the coefficient functions are arbitrary, we have a class of BLMP equations. Symmetry analysis is carried out for this class. We derive the equivalence group admitted by the class and we present the enhanced Lie group classification. Lie symmetries are used to construct similarity reductions. Reduction operators that are not equivalent to Lie ones are also constructed.

Signature of conservation laws and solitary wave solution with different dynamics in Thomas–Fermi plasma: Lie theory

We propose a Lie group method to discuss the modified KP equation appearing in Thomas–Fermi (TM) plasma, characterised by cold and hot electrons. The Lie method facilitates the identification of similarity reductions, infinitesimal symmetries, group-invariant solutions, and novel analytical solutions for nonlinear models. The similarity reduction method is carried out to transform the nonlinear partial differential equations (NLPDE) into the nonlinear ordinary differential equations (NLODE). This study focuses on solitary wave profiles due to their usefulness in various engineering applications, including monitoring public transportation systems, managing coastlines, and mitigating disaster risks. It also addresses the conservation laws associated with the modified KP equation. The generalised auxiliary equation (GAEM) scheme is used to compute the new solitary wave patterns of the modified KP equation, which explains the dynamics of nonlinear waves in Thomas–Fermi plasma. The idea of nonlinear self-adjointness is used to compute the conservation laws of the examined model. The graphical behaviour of some solutions is represented by adjusting the suitable value of the parameters involved.

In a river or an ocean: Similarity-reduction work on a (3+1)-dimensional extended shallow water wave equation

Researchers have recently become interested in a (3+1)-dimensional extended shallow water wave equation in a river or an ocean. This equation could be used to represent a variety of the physical phenomena that have some impacts on the environment, such as the floods and tsunamis. Two sets of the similarity reductions are discovered for that equation based on the variable coefficients, connected with the potential applications of the equation in a river or an ocean.

Lie symmetry analysis, exact solutions and conservation laws of (3+1)-dimensional modified Wazwaz-Benjamin-Bona-Mahony equation

In this paper, we study the (3+1)-dimensional modified Wazwaz-Benjamin-Bona-Mahony equation using Lie symmetry analysis. An optimal system is constructed with the help of a commutator table and an adjoint table. Similarity reductions are obtained on the basis of the optimal system, and the governing partial differential equation is transformed into another partial differential equations with a smaller number of independent variables. The solutions of these reduced partial differential equations give invariant solutions of the (3+1)-dimensional modified Wazwaz-Benjamin-Bona-Mahony equation. The 3D graphical plots are used to describe the dynamical characteristics of the solutions. Hence, by employing these graphs, physicists and mathematicians can more efficiently and successfully follow complex physical processes. Finally, self-adjoint is checked, and conservation laws are also deducted corresponding to each infinitesimal generator.

New insight to the shallow-water studies on a (2+1)-dimensional generalized Broer-Kaup system

Of current interest, there have recently appeared a series of the shallow-water papers in Mod. Phys. Lett. B. Illuminated by those papers and to make the story more complete, for the nonlinear long waves in the shallow water, this paper is scheduled to investigate a (2+1)-dimensional generalized Broer-Kaup system. As for the wave height and wave horizontal velocity, we employ symbolic computation, with the view of constructing out two groups of the similarity reductions.

In plasma physics and fluid dynamics: Symbolic computation on a (2+1)-dimensional variable-coefficient Sawada-Kotera system

Investigations on the real world have been facilitating the development of nonlinear science, and today, fluid dynamics and plasma physics attract people’s attention. This Letter studies a (2+1)-dimensional variable-coefficient Sawada-Kotera system in plasma physics and fluid dynamics. We build up a family of the similarity reductions, connecting that system with a known ordinary differential equation. Our similarity reductions depend on the plasma/fluid variable coefficients in that system, under certain variable-coefficient constraints.

Cortical Morphological Networks Differ Between Gyri and Sulci.

This study explored how the human cortical folding pattern composed of convex gyri and concave sulci affected single-subject morphological brain networks, which are becoming an important method for studying the human brain connectome. We found that gyri-gyri networks exhibited higher morphological similarity, lower small-world parameters, and lower long-term test-retest reliability than sulci-sulci networks for cortical thickness- and gyrification index-based networks, while opposite patterns were observed for fractal dimension-based networks. Further behavioral association analysis revealed that gyri-gyri networks and connections between gyral and sulcal regions significantly explained inter-individual variance in Cognition and Motor domains for fractal dimension- and sulcal depth-based networks. Finally, the clinical application showed that only sulci-sulci networks exhibited morphological similarity reductions in major depressive disorder for cortical thickness-, fractal dimension-, and gyrification index-based networks. Taken together, these findings provide novel insights into the constraint of the cortical folding pattern to the network organization of the human brain.

Integrability, Similarity Reductions and New Classes of Exact Solutions for (3+1)-D Potential Yu–Toda–Sasa–Fukuyama Equation

Integrability, Similarity Reductions and New Classes of Exact Solutions for (3+1)-D Potential Yu–Toda–Sasa–Fukuyama Equation

Invariant analysis of the two-cell tumor growth model in the brain

In this study, we explore the invariance properties of a tumor growth model involving two distinct cell populations. These populations are characterized by different diffusion coefficients but share a common killing rate. This particular model serves as a representation of tumor growth within the brain. By employing the Lie group method, we unveil a two-dimensional symmetry algebra for cases where both diffusion coefficients are allowed to vary arbitrarily. Interestingly, this method reveals a nine-dimensional symmetry algebra when the diffusion coefficients are held constant. In both scenarios involving varying and constant diffusion coefficients, we conduct similarity reductions to deduce group invariant solutions, thus elucidating the model’s behavior. Notably, our findings demonstrate that the tumor’s growth remains exponential irrespective of the presence or absence of a killing rate. This remarkable phenomenon holds for various configurations of diffusion coefficients. To validate our observations, we employ Mathematica simulations, which corroborate the model’s exponential growth behavior and emphasize the role of killing rates, diffusion coefficients, and growth rate parameters in driving this exponential trend. Also, the conserved flows and conserved quantities of the model are demonstrated.

A study of self-adjointness, Lie analysis, wave structures, and conservation laws of the completely generalized shallow water equation

This article explores the analysis of the completely generalized Hirota–Satsuma–Ito equation through Lie symmetry analysis. The equation under consideration represents a more comprehensive form of the (2+1)-dimensional HSI equation, encompassing four additional second-order derivative terms: Δ3Hϖϖ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _{3}H_{\\varpi \\varpi }$$\\end{document}, Δ4Hϖι\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _{4}H_{\\varpi \\iota }$$\\end{document}, Δ3Hϖϖ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _{3}H_{\\varpi \\varpi }$$\\end{document}, Δ4Hϖι\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _{4}H_{\\varpi \\iota }$$\\end{document}, and Δ6Hιι\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _{6}H_{\\iota \\iota }$$\\end{document}, emerging from the inclusion of second-order dissipative-type elements. We calculate the infinitesimal generators and determine the symmetry group for each generator using the Lie group invariance condition. Employing the conjugacy classes of the Abelian algebra, we transform the considered equation into an ordinary differential equation through similarity reduction. Subsequently, we solve these ordinary differential equations to derive closed-form solutions for the completely generalized Hirota–Satsuma–Ito equation under certain conditions. For other scenarios, we utilize the extended direct algebraic method to obtain soliton solutions. Furthermore, we rigorously calculated the conserved quantities corresponding to each symmetry generator, the conservation laws of the model are established using the multiplier approach. Additionally, we present the graphical representation of selected solutions for specific values of the physical parameters of the equation under scrutiny.

Symmetry and complexity: a Lie symmetry method to bifurcation, chaos, multistability and soliton solutions of the nonlinear generalized advection-diffusion-reaction equation

This paper deals with the complexities of nonlinear dynamics within the nonlinear generalized advection-diffusion-reaction equation, which describes intricate transport phenomena involving advection, diffusion, and reaction processes occurring simultaneously. Through the utilization of the Lie symmetry approach, we thoroughly examine this proposed model, transforming the partial differential equation into an ordinary differential equation using similarity reduction techniques to facilitate a more comprehensive analysis. Exact solutions for this transformed equation are derived employing the extended simplest equation method and the new extended direct algebraic method. To enhance understanding, contour plots along with 2D and 3D visualizations of solutions are employed. Additionally, we explore bifurcation and chaotic behaviors through a qualitative analysis of the model. Phase portraits are meticulously scrutinized across various parameter values, offering insights into system behavior. The introduction of an external periodic strength allows us to utilize various tools including time series, 3D, and 2D phase patterns to discern chaotic and quasi-periodic behaviors. Furthermore, a multistability analysis is conducted to examine the impacts of diverse initial conditions. These findings underscore the efficacy and practicality of the proposed methodologies in evaluating soliton solutions and elucidating phase dynamics across a spectrum of nonlinear models, offering novel perspectives on intricate physical phenomena

Dynamics of invariant solutions of the DNA model using Lie symmetry approach

The utilization of the Lie group method serves to encapsulate a diverse array of wave structures. This method, established as a robust and reliable mathematical technique, is instrumental in deriving precise solutions for nonlinear partial differential equations (NPDEs) across a spectrum of domains. Its applications span various scientific disciplines, including mathematical physics, nonlinear dynamics, oceanography, engineering sciences, and several others. This research focuses specifically on the crucial molecule DNA and its interaction with an external microwave field. The Lie group method is employed to establish a five-dimensional symmetry algebra as the foundational element. Subsequently, similarity reductions are led by a system of one-dimensional subalgebras. Several invariant solutions as well as a spectrum of wave solutions is obtained by solving the resulting reduced ordinary differential equations (ODEs). These solutions govern the longitudinal displacement in DNA, shedding light on the characteristics of DNA as a significant real-world challenge. The interactions of DNA with an external microwave field manifest in various forms, including rational, exponential, trigonometric, hyperbolic, polynomial, and other functions. Mathematica simulations of these solutions confirm that longitudinal displacements in DNA can be expressed as periodic waves, optical dark solitons, singular solutions, exponential forms, and rational forms. This study is novel as it marks the first application of the Lie group method to explore the interaction of DNA molecules.

Sur une équation définie par une réduction de la hiérarchie de Drinfeld et Sokolov

A seventh order ordinary differential equation (ODE) arising by reduction of the Drinfeld-Sokolov hierarchyis shown to be identical to a similarity reduction of an equationin the hierarchy of Sawada-Kotera.We also exhibit its link with a particular F-VI,a fourth order ODE isolated by Cosgrove which is likely to define a higher order Painlev\'e function.

Similarity reduction, group analysis, conservation laws, and explicit solutions for the time-fractional deformed KdV equation of fifth order

Through this paper, we consider the time-fractional deformed fifth-order Korteweg–de Vries (KdV) equation. First of all, we detect its symmetries by Lie group analysis with the help of Riemann–Liouville (R-L) fractional derivatives. These symmetries are employed to convert the considered equation into a fractional ordinary differential (FOD) equation in the sense of Erdélyi-Kober (E-K) fractional operator. Also, a set of new analytical solutions for the equation under study are obtained via the power series method. We test the accuracy and effectiveness of this method by providing a numerical simulation of the obtained solution and studying the effect of [Formula: see text] which is represented graphically in 2D and 3D plots. Added to that, we prove the convergence of the power series solutions. Finally, the computation of the conservation laws is introduced in detail.

Auto-Bäcklund Transformation with the Solitons and Similarity Reductions for a Generalized Nonlinear Shallow Water Wave Equation

Auto-Bäcklund Transformation with the Solitons and Similarity Reductions for a Generalized Nonlinear Shallow Water Wave Equation

Novel nonlinear quantum dust acoustic waves for modified variable coefficients Zakharove–Kusnetsov equation in dusty plasma

In this paper, the direct similarity reduction method is used to reduce the modified variable coefficient Zakharove–Kusnetsov equation (vcmZK), which represents the propagation of quantum dust acoustic waves (QDAWs) in an inhomogeneous magnetized dusty plasma field, into a nonlinear ordinary differential equation (NODE). Moreover, the Jacobi elliptic method is used to solve the NODE, where many novel double periodic, kink, and solitary wave solutions are obtained. Additionally, we have chosen to plot periodic, kink, and soliton QDAWs solutions according to some choices for the variable coefficients. According to the a chieved results, we can conclude that the choices of the functional forms of the variable coefficients can control the QDAWs propagation. Finally, the new discovered results may have many important applications in astrophysics, atmospheric science, nanotechnology, and industrial plasma physics.

Invariant analysis, approximate solutions, and conservation laws for the time fractional higher order Boussinesq–Burgers system

In this paper, we employ the time fractional higher order nonlinear Boussinesq–Burgers system to investigate shallow water waves based on the Riemann–Liouville and Caputo fractional partial derivatives. The Lie algebra of infinitesimal generators associated with the symmetry groups that leave the studied system invariant is systematically constructed, and the similarity reductions are established. Furthermore, conserved quantities of the system under consideration are formulated. Next, the power series solution, including the convergence analysis, is obtained. Based on the fractional Sumudu–Adomian decomposition method, some approximate solutions are derived. Finally, in order to show the efficiency of the considered methods, the effect of the fractional order, as well as the dynamical behavior of solutions, numerical validation and graphical representations are designed.

A similarity reduction of the generalized Grad–Shafranov equation

We extend previous work [Y. E. Litvinenko, Phys. Plasmas 17, 074502 (2010)] on a direct method for finding similarity reductions of partial differential equations such as the Grad–Shafranov equation, to the case of the generalized Grad–Shafranov equation (GGSE) with arbitrary incompressible flow. Several families of analytic solutions are constructed, the generalized Solovév solution being a particular case, which contain both the classical and non-classical group-invariant solutions to the GGSE. Those solutions can describe a variety of equilibrium configurations pertinent to toroidal magnetically confined plasmas and planetary magnetospheres.

Bäcklund transformation, similarity reduction and new solutions for the (2+1)-dimensional graphene sheets thermophoretic motion equation with variable heat transmission

Graphene has appeared as one of the most promising nanomaterials with unique thermal, electronic and mechanical properties which make it suitable for many applications in materials science, energy storage, electronics, water filtration and medicine. In this paper, we have studied the (2+1)-dimensional graphene sheets (GS) equation with variable heat transmission thermophoretic motion by using both Bäcklund transformation and similarity reduction techniques. As a result, from Bäcklund transformation one soliton solution was obtained and from similarity reduction technique the GS equation is reduced to a nonlinear ordinary differential equation which can be solved using the trial equation method; where many novel wave solutions were obtained. As an additional remark, the Bäcklund transformation obtained can be found by the traditional Painléve test. Finally, some graphs are given to illustrate the effect of the variable coefficients on the behavior of the thermal wave transmission.

In silico agent-based modeling approach to characterize multiple in vitro tuberculosis infection models.

In vitro models of Mycobacterium tuberculosis (Mtb) infection are a valuable tool for examining host-pathogen interactions and screening drugs. With the development of more complex in vitro models, there is a need for tools to help analyze and integrate data from these models. To this end, we introduce an agent-based model (ABM) representation of the interactions between immune cells and bacteria in an in vitro setting. This in silico model was used to simulate both traditional and spheroid cell culture models by changing the movement rules and initial spatial layout of the cells in accordance with the respective in vitro models. The traditional and spheroid simulations were calibrated to published experimental data in a paired manner, by using the same parameters in both simulations. Within the calibrated simulations, heterogeneous outputs are seen for bacterial count and T cell infiltration into the macrophage core of the spheroid. The simulations also predict that equivalent numbers of activated macrophages do not necessarily result in similar bacterial reductions; that host immune responses can control bacterial growth in both spheroid structure dependent and independent manners; that STAT1 activation is the limiting step in macrophage activation in spheroids; and that drug screening and macrophage activation studies could have different outcomes depending on the in vitro culture used. Future model iterations will be guided by the limitations of the current model, specifically which parts of the output space were harder to reach. This ABM can be used to represent more in vitro Mtb infection models due to its flexible structure, thereby accelerating in vitro discoveries.