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In this work, we establish novel solitary wave solutions of three nonlinear evolution dynamical models, namely, Boussinesq model, (3 + 1) -dimensional generalized shallow water wave (SWW) and (4 + 1) -dimensional Fokas dynamical models by applying two-variable (G ′ ∕ G , 1 ∕ G) - expansion technique. The SWW equations are usually appropriate when the fluid is in a state where the vertical length scale is significantly smaller than the length of horizontal scale. In computer simulations for ocean engineering, Boussinesq-type models are frequently used to represent water waves in harbors and shallow seas. In order to understand the physical processes of waves inside and on the surface of water, the Fokas dynamical model plays a crucial part in wave theory. Analytical solutions in different forms such as solitons, solitary waves, trigonometric, hyperbolic, rational function solutions, breathers-type waves and more wave solutions are devised through using the proposed method. The exact solutions are also presented in graphical form having applications in engineering and other areas of applied sciences. The obtained results show that the given technique is universal and efficient. In addition, this technique can also be applied on lots of other nonlinear dynamical wave models occurring in many scientific real-world application domains, including engineering sciences, mathematics physics, and many more. [ABSTRACT FROM AUTHOR]

One of the most important physical models for describing the dynamics of optical soliton propagation in the theory of optical fibers is the nonlinear Schrödinger equation (NLSE). Since there are so many uses for ultrafast signal routing systems and brief light pulses in telecommunication, optical soliton propagation in nonlinear optical fibers is a subject of intense current interest. The conventional Khater approach and the extended direct algebraic method are used in this research to study a generalized unstable space–time fractional NLS model in mathematical physics. This leads to the discovery of solutions for the bright, dark, periodic, rational and elliptic functions. To illustrate the physical nature of the nonlinear model, the contour plots in 3D, 2D and 2D are produced by assigning the appropriate values to the arbitrary constants. Additionally, consideration is given to the stability analysis of the solutions to the governing equation. In the current era of communications network technology and nonlinear optics, the applied strategy appears to be a more potent and effective way for producing precise optical solutions to a number of various modern models of recent generations. [ABSTRACT FROM AUTHOR]

The study of spectral properties of natural geometric elliptic partial differential operators acting on smooth sections of vector bundles over Riemannian manifolds is a central theme in global analysis, differential geometry and mathematical physics. Instead of studying the spectrum of a differential operator L directly one usually studies its spectral functions, that is, spectral traces of some functions of the operator, such as the spectral zeta function ζ (s) = Tr L − s and the heat trace Θ (t) = Tr exp (− t L). The kernel U (t ; x , x ′) of the heat semigroup exp (− t L) , called the heat kernel, plays a major role in quantum field theory and quantum gravity, index theorems, non-commutative geometry, integrable systems and financial mathematics. We review some recent progress in the study of spectral asymptotics. We study more general spectral functions, such as Tr f (t L) , that we call quantum heat traces. Also, we define new invariants of differential operators that depend not only on the eigenvalues but also on the eigenfunctions, and, therefore, contain much more information about the geometry of the manifold. Furthermore, we study some new invariants, such as Tr exp (− t L +) exp (− s L −) , that contain relative spectral information of two differential operators. Finally, we show how the convolution of the semigroups of two different operators can be computed by using purely algebraic methods. [ABSTRACT FROM AUTHOR]

In this article, new traveling wave solutions are retrieved for two space-time fractional models: the resonant nonlinear Schrödinger equation and the Biswas–Milovic equation which illustrates the dynamics of optical soliton promulgation in optical fibers, both featuring Kerr law nonlinearity. These equations are explored via an efficient method namely, the extended simple equation method. The fractional derivative is used in the conformable sense to accomplish this analysis. The extracted solutions show dark, periodic, singular, and singular-periodic solitons behaviors, which are depicted graphically by using line, surface, and contour plots. The reported solutions are unique and novel. The proposed method distinguishes itself by its simplicity, reliability, and ability to generate novel soliton solutions for nonlinear Schrödinger equations within the realm of mathematical physics. [ABSTRACT FROM AUTHOR]

This work simulates the (1+1)-dimensional nonlinear perturbed Schrödinger model (NLPSM ). Hydrodynamics, elastic media, nonlinear optical fiber communication, and plasma physics are just a few of this model's mathematical physics and engineering applications. The study aims to accomplish two main objectives. First, it seeks to find unique soliton solutions such as solitary, dark, periodic, and plane wave solutions that haven't been found in the literature before using the modified Sardar sub-equation approach (MSSEA ). Second, a novel approach to analysis called bifurcation analysis is used to investigate the dynamic behavior of the model. Physical compatibility findings are supported by density, 3-D, and 2-D illustrations made with parametric variables. The analysis shows that the approach used to quickly acquire complete and typical answers was successful. This approach works well for solving challenging problems in physics, engineering, mathematics and fiber optic phenomena. [ABSTRACT FROM AUTHOR]

In this study, we use a systematic approach named the generalized unified method (GUM) to construct the general exact solutions of the derivative nonlinear Schrödinger (DNLS) family that also includes perturbed terms, which are the Kaup–Newell equation, the Chen–Lee–Liu equation, and the Gerdjikov–Ivanov equation. The GUM provides more general exact solutions with free parameters for nonlinear partial differential equations such that some solutions obtained by different exact solution methods, including the hyperbolic function solutions, the trigonometric function solutions, and the exponential solutions, are derived from these solutions by giving special values to these free parameters. Additionally, the used method reduces a large number of calculations compared to other exact solution methods, enabling computations to be made in a short, effortless, and elegant way. We investigate the DNLS family in this work because of its extensive applications in nonlinear optics. Particularly, the obtained optical soliton solutions of the DNLS family are useful for describing waves in optics and facilitating the interpretation of the propagation of solitons through optical fibers. Furthermore, this work not only contributes significantly to the advancement of soliton dynamics and their applications in photonic systems but also be productively used for more equations that occur in mathematical physics and engineering problems. Finally, 2D and 3D graphs of some derived solutions are plotted to illustrate behaviors of optical soliton. [ABSTRACT FROM AUTHOR]

In this research work, we utilized the auxiliary equation technique and extracted the optical soliton solutions to the nonlinear longitudinal wave equation (NLLWE) in magneto electro elastic (MEE) rod that was spread out in a circle. The NLLWE in MEE systems deals with the mathematical physics of transverse Poisson's effect dispersal and also very important in many engineering fields like sensors and actuators. As a result, we extracted the exact soliton solutions in bright solitons, dark solitons, kink wave solitons, anti-kink wave solitons, combined bright-dark solitons, solitary waves and periodic singular solitons. The physical structure of some extracted solutions visualizing in contour, two, and three dimensions through numerical simulation. The explored soliton solutions are interested, more general and having different physical structure, which may will be helpful to study of physical phenomena in the fields of optical fibers, plasma physics, soliton wave theory, nonlinear optics, ocean engineering, nonlinear dynamics and different branches of applied sciences. The successful extraction of exact solitons shows that this utilized approach is effective, straightforward, concise and powerful can also applicable to other nonlinear partial differential equations that involve in mathematical physics, engineering and applied sciences. [ABSTRACT FROM AUTHOR]

In this research, the solitary wave solutions, the periodic type, and single soliton solutions are attained. The coupled fractional Lakshmanan–Porsezian–Daniel (LPD) equation is depicted the wave pulses' physical properties in birefringent optical fibres containing two vector solitons. Here, the Paul–Painlevé operator is employed to investigate kink soliton solutions. Also, the new modified exponential Jacobi technique is used to find periodic wave and soliton, bright-dark soliton. By utilizing symbolic computation and the applied methods, the mentioned system is successfully investigated. The coupled fractional LPD model is exhibited the travelling waves, as shown by the research in the current paper. Through three-dimensional graph, contour graph, density graph, complex-plot, and two-dimensional design using Maple, the physical features of single soliton and periodic wave solutions are explained all right. The findings the investigated model's broad variety of explicit solutions are demonstrated. As a result, the exact solitary wave solutions to the studied issues, including solitary, single soliton, and periodic wave solution are found. It is shown that the approach is practical and flexible in mathematical physics. All outcomes in this work are necessary to understand the physical meaning and behavior of the explored results and shed light on the significance of the investigation of several nonlinear wave phenomena in sciences and engineering. [ABSTRACT FROM AUTHOR]

In this paper, we apply the fractional homotopy perturbation transform method (FHPTM) to deliver an effective semi-analytical technique for determining fractional-order Kuramoto–Sivashinsky equations. The project technique combines the Laplace transform with the Caputo–Fabrizio fractional derivative of order α where α ∈ (0 , 1 ] . Fractional-order Kuramoto–Sivashinsky equation is indeed important in the field of nonlinear physics and mathematics. It is a fractional partial differential equation that describes the behaviour of waves in certain dissipative media, such as flames and chemicals. The FHPTM is described to be fast and accurate. Illustrative examples are included to demonstrate the efficiency and reliability of the presented techniques. [ABSTRACT FROM AUTHOR]

This study retrieves optical soliton outcomes from the ion sound and Langmuir wave (ISLW) structure used in birefringent fibres. With the improved Kudryashov, novel Kudryashov and exp a -function approaches, we form soliton solutions to the proposed nonlinear structure. These structures are achieved by utilising the computational software Maple 18. The governing model is converted to an ordinary differential form using a variable transformation. The obtained optical soliton solutions demonstrate some waveforms, such as W -shaped periodic waves, dual-periodic waves and periodic waves with single and multibreather solitons. Optical dark–bright breather wave, optical dark soliton and optical bright multi-breathers are also obtained. The acquired results show that these three methods are more convenient and straightforward than any other technique for finding optical soliton solutions to the ISLW equation. These results are novel and have not been obtained in previous works. Therefore, the obtained results will play an important role in mathematical physics and engineering applications. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Using assembly theory, we investigate the assembly pathways of binary strings (bitstrings) of length N formed by joining bits present in the assembly pool and the bitstrings that entered the pool as a result of previous joining operations. We show that the bitstring assembly index is bounded from below by the shortest addition chain for N, and we conjecture about the form of the upper bound. We define the degree of causation for the minimum assembly index and show that, for certain N values, it has regularities that can be used to determine the length of the shortest addition chain for N. We show that a bitstring with the smallest assembly index for N can be assembled via a binary program of a length equal to this index if the length of this bitstring is expressible as a product of Fibonacci numbers. Knowing that the problem of determining the assembly index is at least NP-complete, we conjecture that this problem is NP-complete, while the problem of creating the bitstring so that it would have a predetermined largest assembly index is NP-hard. The proof of this conjecture would imply P ≠ NP since every computable problem and every computable solution can be encoded as a finite bitstring. The lower bound on the bitstring assembly index implies a creative path and an optimization path of the evolution of information, where only the latter is available to Turing machines (artificial intelligence). Furthermore, the upper bound hints at the role of dissipative structures and collective, in particular human, intelligence in this evolution. [ABSTRACT FROM AUTHOR]

Perturbative expansions have played a peculiarly central role in quantum field theory, not only in extracting empirical predictions but also in investigations of the theory's mathematical and conceptual foundations. This paper brings the special status of QFT perturbative expansions into focus by tracing the history of mathematical physics work on perturbative QFT and situating a contemporary approach, perturbative algebraic QFT, within this historical context. Highlighting the role that perturbative expansions have played in foundational investigations helps to clarify the relationships between the formulations of QFT developed in mathematical physics and high-energy phenomenology. [ABSTRACT FROM AUTHOR]

With the breakthroughs made by physicists in high-dimensional mathematics, it has become possible to represent and solve a number of classical mathematical physics problems using the split quaternion algebra. In this paper, we study the least squares approximation of a class of Schrödinger equations in split quaternionic mechanics and propose two algebraic algorithms to the generalized right eigen-problem for an i-Hermitian split quaternion matrix pencil by using two isomorphic mappings. Numerical examples show the effectiveness of the proposed theories and algorithms. [ABSTRACT FROM AUTHOR]

In this work, the new nonlinear (1 + 1) -dimensional Schrödinger (NLS) equation, which describes wave propagation in optical fibers, has been successfully studied using the (G ′ / G) -expansion method and the Unified method (UM). These techniques have been applied for the first time in extracting new analytic solutions for the mentioned NLS equation. The obtained solutions exhibit various types of optical solitons, including singular solitons, solitary waves, periodic solitons, and wave collisions. These solutions have wide-ranging applications in various fields such as fiber optics, telecommunication systems, plasma physics, hydrodynamics, and nonlinear optics. The behavior of the derived solutions has been visually represented using three-dimensional, two-dimensional, and contour plots, considering appropriate choices of the involved parameters within specific time intervals. These plots provide insights into the dynamics and characteristics of the solutions. The effectiveness, reliability, and ease of application of the (G ′ / G) -expansion method and the UM are demonstrated through the obtained solutions. These mathematical tools have proven to be powerful in solving nonlinear partial differential equations, offering valuable contributions to the field of mathematical physics, soliton theory and nonlinear sciences. [ABSTRACT FROM AUTHOR]

In this paper, within the scope of the local fractional derivative theory, the (4+1)-dimensional local fractional Fokas equation is researched. The study of exact solutions of high-dimensional nonlinear partial differential equations plays an important role in understanding complex physical phenomena in reality. In this paper, the exact traveling wave solution of generalized functions is analyzed defined on Cantor sets in high-dimensional integrable systems. The results of non-differentiable solutions in different cases are numerically simulated when the fractal dimension is equal to ρ = ln 2/ln 3. The results show that the exact solution of the local fractional Fokas equation represents the fractal waves on the shallow water surface. Through numerical simulation, we find that the exact solution of the local fractional Fokas equation can describe the fractal waves and waves characteristics of shallow water surface. It also shows that the study of traveling wave solutions of nonlinear local fractional equations has important significance in mathematical physics. [ABSTRACT FROM AUTHOR]

Feynman diagrams represent one of the most powerful and fascinating tools developed in theoretical physics in the last century. Introduced within the framework of quantum electrodynamics as a suitable method for computing the amplitude of a physical process, they rapidly became a fundamental mathematical object in quantum field theory. However, their abstract nature seems to suggest a wider usage, which actually exceeds the physical context. Indeed, as mathematical objects, they could simply be considered graphs that depict not only physical quantities but also biological or economic entities. We survey the analytical and algebraic properties of such diagrams to understand their utility in several areas of science, eventually providing some examples of recent applications. [ABSTRACT FROM AUTHOR]

The (3+1)-dimensional double sine-Gordon equation plays a crucial role in various physical phenomena, including nonlinear wave propagation, field theory, and condensed matter physics. However, obtaining exact solutions to this equation faces significant challenges. In this article, we successfully employ a modified G ′ G 2 -expansion and improved tan ϕ ξ 2 -expansion methods to construct new analytical solutions to the double sine-Gordon equation. These solutions can be divided into four categories like trigonometric function solutions, hyperbolic function solutions, exponential solutions, and rational solutions. Our key findings include a rich spectrum of soliton solutions, encompassing bright, dark, singular, periodic, and mixed types, showcasing the (3+1)-dimensional double sine-Gordon equation ability to model diverse wave behaviors. We uncover previously unreported complex wave structures, revealing the potential for complex nonlinear interactions within the (3+1)-dimensional double sine-Gordon equation framework. We demonstrate the modified G ′ G 2 -expansion and improved tan ϕ ξ 2 -expansion methods effectiveness in handling higher-dimensional nonlinear partial differential equations, expanding their applicability in mathematical physics. These method offers enhanced flexibility and broader solution categories compared to conventional approaches. [ABSTRACT FROM AUTHOR]

This paper examines the complex characteristics of the modified Benjamin–Bona–Mahony equation ( m BBM ) and the Klein–Gordon ( K G ) equation in the field of mathematical physics. The m BBM equation is a basic model used to describe surface water waves, especially in shallow water situations. It provides valuable information on wave propagation, stability, and the formation of solitons. The applications of this instrument are wide-ranging, including fields such as oceanography, where it plays a crucial role in comprehending wave behavior. On the other hand, the K G equation is of utmost importance in quantum field theory since it sheds light on the dynamics and interactions of scalar fields such as mesons. Within the field of particle physics, it offers substantial insights into basic concepts, acting as a fundamental basis for comprehending particle behavior. The primary goal of our work is to develop strong analytical techniques for solving these problems. In order to tackle these issues, we use two novel methodologies: the extended simple equation technique and the generalized Kudryashov method. Furthermore, we validate our results by using the extended cubic–B–spline approach for numerical computations. The work effectively solves these intricate equations, resulting in encouraging results. The presented approaches demonstrate their effectiveness, providing significant advances to mathematical physics. This work has inherent worth by presenting innovative analytical methods and perspectives, particularly designed to solve complex nonlinear equations such as the m BBM and K G equations. The finding has significant ramifications that extend across several scientific fields, offering novel approaches to tackle complex issues in mathematical physics. To summarize, our paper introduces innovative analytical techniques designed to solve nonlinear equations in the field of mathematical physics, with a particular emphasis on the m BBM and K G equations. [ABSTRACT FROM AUTHOR]

Extremely short pulse propagation in optical fiber is modelled via using the higher order nonlinear Schrödinger equation (NLSE) with cubic quintic nonlinearity. To construct singular bright, kink, and anti-kink solitons, periodic waves as well as multi-peak and breather type waves of generalized NLSE in a cubic quintic non-Kerr medium, we used a generalised exponential approach. In physics and applied mathematics, the obtained solutions have significant applications. We have also discussed the solitary wave parameters under which dark and bright solitons may develop in this medium. In order to visualise the physical phenomena of this model, we have provided a graphic representation of the movements of the created solitary wave and soliton solutions. This reliable, effective, and successful strategy may also be used to solve many other similar sorts of models that arise in applied sciences. [ABSTRACT FROM AUTHOR]

The Hamiltonian amplitude equation that describes the transmissions of wire optical technology along wave propagation in the physics linked disciplines can be solved by utilizing the unique analytical and semi-analytical approaches namely the polynomial expansion method and the Adomian decomposition technique. Various types of soliton solutions are attained that express different behaviours including singular and v-shaped soliton solutions behaviour. The developed solutions are highly important in dealings the wave behaviour in mathematical physics related fields and in the transmission of optical fibers. The approaches applied in this paper are not utilized in given model so we apply these strategies to attain the soliton solutions. Hence we conclude the the results attained are unique, robust and newly made that proves novelty of the work. Furthermore, we study the stability plus the modulation instability of the given model. The 3D, contours, and 2D graphics are visualised to demonstrate the dynamical structures of the established results more efficiently. [ABSTRACT FROM AUTHOR]

The primary objective of this article is to construct novel solitary wave solutions for the nonlinear (3+1)-dimensional modified Korteweg-de Vries-Zakharov-Kuznetsov equation with a conformable derivative by employing the subsidiary ordinary differential equation method and 1 φ (ς) , φ ′ (ς) φ (ς) method. The solutions obtained encompass diverse bright, dark, singular, and periodic solitary waves. Additionally, the article thoroughly discusses the sufficient conditions for the existence of these solutions. To visually represent the obtained solutions, we include 2D and 3D graphical images using the computational software Maple 18. We also included a comparison of fractional derivatives at different values of fractional order and conducted an analysis to understand the influence of nonlinear parameters on wave behaviors. The significance of these solutions lies in their wide-ranging applications in applied sciences and mathematical physics. Furthermore, the proposed methods are invaluable in efficiently solving nonlinear partial differential equations prevalent in engineering and physical science. [ABSTRACT FROM AUTHOR]

The extended (2 + 1) -dimensional Calogero–Bogoyavlenskii–Schiff equation is investigated in this study. The extended (2 + 1) -dimensional Calogero–Bogoyavlenskii–Schiff equation is an extension of Calogero–Bogoyavlenskii–Schiff equation that describes the movement of Riemann waves along y-axis while long waves moves along the x-axis. The dynamics of Riemann waves is one of the most significant applications including tsunami in rivers, internal waves in oceans and magento-sound waves in plasmas. Finding new precise solutions with the assistance of a relatively new extended G ′ G 2 -expansion approach and exp (- φ (ζ)) -expansion technique is the primary objective of this effort. The suggested techniques are important tools in the fields of mathematical physics. Successful extraction of hyperbolic, rational, and trigonometric function solutions are achieved by using the proposed analytical methods. The extended (2 + 1) -dimensional Calogero–Bogoyavlenskii–Schiff equation is studied for the first time using extended G ′ G 2 -expansion approach and exp (- φ (ζ)) -expansion technique in this work and novel solutions are observed. 3D plots, contour plots and 2D plots are used to depict the dynamics of the extracted solutions. [ABSTRACT FROM AUTHOR]

In this article, two most prominent models namely the Camassa–Holm nonlinear Schrödinger equation and the (3 + 1) -dimensional Boussinesq equation have been investigated for extracting new and novel solitary wave solutions. The Camassa–Holm nonlinear Schrödinger (CHNLS) equation is a mathematical model that combines the features of two important equations in physics: the Camassa–Holm equation and the nonlinear Schrödinger equation. The Camassa–Holm equation describes the propagation of shallow water waves over a flat bottom, and has soliton solutions with sharp peaks called peakons. The nonlinear Schrödinger equation describes the evolution of wave packets in nonlinear and dispersive media, and has soliton solutions with smooth profiles. The CHNLS equation has been addressed analytically to determine the exact solutions by implementing extended G ′ / G 2 -expansion approach. The Boussinesq equation is a mathematical model that is capable of simulating weakly nonlinear and long-wave approximations is also solved analytically by applying extended G ′ / G 2 -expansion approach. This model finds its applications in various fields such as coastal engineering, and numerical models for water wave simulation in harbors and shallow seas. The two considered equations have significant applications in mathematical physics and their exact wave solutions are essential to understand their dynamical behavior. Dark solitons, bright solitons, and periodic waves are observed from the obtained results. It is reasonable to say that our approach provides an impressive mathematical mechanism for producing traveling wave solutions for numerous mathematical and physical models. Also our proposed method improves the accuracy of the solution. The technique employed here is basic and concise. Graphs are presented to depict the behavior of some of the retrieved dynamical wave structures. All computations are done using the mathematical software Maple. [ABSTRACT FROM AUTHOR]

This paper explore the new solitary wave solutions of the Biswas–Milovic equation and resonant nonlinear Schr o ¨ dinger equations with Kerr–Law nonlinearity. In mathematical physics, the Biswas–Milovic equation plays an important role. The dynamics of these solitons are well known to propagate over large distances in a few femto-seconds through these fibres. These dynamics are governed by the nonlinear Schrödinger's equation. The resonant nonlinear Schrödinger equation can be used to model the propagation of waves in fiber optics. The powerful analytical approach known as the extended simple equation method is employed to explore the solitary wave solutions of the Biswas–Milovic and resonant nonlinear Schr o ¨ dinger equations. These equations are primarily explored in the realm of solitons in nonlinear fiber optics. New dark, kink, anti-kink and singular periodic solitons are secured. Furthermore, 3D surface graphs, contour plots and parametric graphs are drawn to show dynamical behavior of some obtained solutions with the aid of symbolic software such as Mathematica. The constructed solutions will helpful to understand the dynamical framework of nonlinear Schr o ¨ dinger equations in the related physical phenomena. [ABSTRACT FROM AUTHOR]

In this article, we discuss Micro Assignment Guided Inquiry and Collaboration (MAGIC), an active learning method that draws on the merits of inquiry-based learning in STEM courses. We describe the use of Micro Assignments (MAs) consisting of a series of short, instructive guiding questions that scaffold the course material. Students work through these questions and present solutions to the class. The instructor facilitates learning as well as collaboration among the students during face-to-face interactions, providing the Guided Inquiry and Collaboration (GIC). We present a qualitative discussion about this active learning approach that achieves some of the documented benefits, such as deep student engagement with the material and familiarity with communicating arguments. This method also helps students develop the skills they need as they move from lower- to upper-level courses, such as collaborating, active reading, and breaking down involved assignments and problems into manageable steps. Through GIC and use of the MAs, the instructor guides students to become more abstract thinkers who desire evidence, evaluate arguments, and no longer follow step-by-step formulas but rather produce logical thought processes. [ABSTRACT FROM AUTHOR]

Debonding of an elastic layer with a circular cylindrical cavity 0 ≤ ζ ≤ h , ρ ≥ 1 , from a rigid substrate under action of a rigid cylinder is the object of this study. The annular debonding zone 1 ≤ ρ < 1 + η is caused by rotation of a cylinder bonded to the cavity surface. The problem is reformulated as dual integral equations with Weber integral transforms kernels. A Volterra operator transforming a Weber transforms kernel into a Bessel function of the first kind and Hankel integral transforms allow us to reduce dual equations to a Fredholm integral equation of the second kind and then, by some transformation, to another Fredholm integral equation which is more suitable for approximate methods. As h ≥ 0. 5 and 0 < η ≤ 1 , a highly accurate analytic approximate solution of the problem is suggested. The asymptotic solution of the problem is obtained as the width of debonding zone is very small while the thickness is not small. When the thickness is small, the Fredholm integral equations are computationally inefficient. A new method based on an operator transforming a Bessel function of the first kind into the kernel of Mehler–Fock integral transforms enabled us to convert one of the above-mentioned Fredholm equations into an equivalent Fredholm integral equation of the second kind that is effective for a small thickness. The asymptotic solution of the problem is obtained when both the layer thickness and the debonding zone width are small. Accurate mathematical methods, in particular investigations, and transformations of equations, developed in this study can be interesting to researchers employing dual integral equations technique in problems of mechanics and mathematical physics with mixed boundary conditions. [ABSTRACT FROM AUTHOR]

This article is a tribute to Oleksandr Mykolayovych Sharkovsky, a renowned mathematician and one of the founders of the mathematical concept of deterministic chaos. Sharkovsky's groundbreaking work, including his famous Sharkovsky theorem, has had a significant impact on the theory of dynamical systems. The article highlights Sharkovsky's early life, education, and career, as well as his contributions to the field of mathematics. It also mentions his achievements, such as publishing 10 monographs and over 250 research works, and receiving several prestigious awards. Sharkovsky's work continues to be influential in the study of nonlinear dynamics and chaos theory. [Extracted from the article]

The nonlinear Schrödinger equation (NLSE) is a fundamental equation in quantum mechanics with applications in optical fibers, plasma physics, and biomolecule dynamics. The focus of this paper is on four types of nonlinear Schrödinger equations, including the cubic nonlinear Schrödinger equation (CNLSE) and the Chen–Lee–Liu equation (CLLE). We present the existence of a Lagrangian and the invariant variational principle for two coupled equations of these types. Our investigation involves the approximation solutions of these equations, achieved by selecting a trial function with single or two nontrivial variational parameters in the rectangular box region in three cases. By using these trial functions, we found the functional integral and the Lagrangian of the system without loss. The general case for the two-box potential can be obtained based on a different ansatz. We approximate the Jost function by quadratic polynomials instead of a piece-wise linear function and then approximate it by the tanh function. Using the amplitude ansatz method, we derive the exact bright soliton, dark soliton, bright–dark solitary wave solutions, rational dark–bright solutions, and the periodic solitary wave solutions for the proposed equations. The solutions obtained from these techniques have a wide range of applications in various areas of physics and other applied sciences. The results will be presented in graphical representations including 2D, 3D, and contour plots, which highlight their effectiveness. These techniques can be utilized to solve numerous other nonlinear models that arise in mathematical physics and different other applied sciences fields. [ABSTRACT FROM AUTHOR]

We reveal the continuous change of information dynamics patterns in anyonic-PT symmetric systems that originates from the continuity of anyonic-PT symmetry. We find there are three information dynamics patterns for anyonic-PT symmetric systems: damped oscillations with an overall decrease (increase) and asymptotically stable damped oscillations, which are three-fold degenerate and are distorted using the Hermitian quantum Rényi entropy or distinguishability. It is the normalization of the non-unitary evolved density matrix that causes the degeneracy and distortion. We give a justification for non-Hermitian quantum Rényi entropy being negative. By exploring the mathematics and physical meaning of the negative entropy in open quantum systems, we connect negative non-Hermitian quantum Rényi entropy and negative quantum conditional entropy, paving the way to rigorously investigate negative entropy in open quantum systems. [ABSTRACT FROM AUTHOR]

The utilization of machine learning advantages in solving engineering problems has proven to be beneficial and accurate. Dealing with partial differential equations (PDEs), on the other hand, PINN could be enhanced using additional information from the physical mathematics of the problems. In this regard, a physics-informed neural network (PINN) is exploited in the current study to extract phase and group velocities of the multi-scale hybrid composite plate structure. In doing so, the exact differential equations of homogenized plate structure are extracted using Hamilton's principle based on modified couple stress theory and shear deformation displacement field. The boundary conditions along with initial conditions are also considered in the equations. The PINN tries to minimize the loss function which is defined using extracted PDEs, boundary conditions, and initial conditions. Moreover, the results of PINN are further compared with the analytical solution using harmonic expansion of the displacement variables. The results show highly accurate and reliable results obtained using PINN in comparison to analytical results. [ABSTRACT FROM AUTHOR]

Operated algebras, that is, algebras equipped with linear operators, have important applications in mathematics and physics. Two primary instances of operated algebras are the Rota–Baxter algebra and TD-algebra. In this paper, we introduce a λ -TD algebra that includes both the Rota–Baxter algebra and the TD-algebra. The explicit construction of free commutative λ -TD algebra on a commutative algebra is obtained by a generalized shuffle product, called the λ -TD shuffle product. We then show that the free commutative λ -TD algebra possesses a left counital bialgebra structure by means of a suitable 1-cocycle condition. Furthermore, the classical result that every connected filtered bialgebra is a Hopf algebra, is extended to the context of left counital bialgebras. Given this result, we finally prove that the left counital bialgebra on the free commutative λ -TD algebra is connected and filtered, and thus is a left counital Hopf algebra. [ABSTRACT FROM AUTHOR]

In this study, we examine three essential types of nonlinear Schrödinger equation (NLSE), which are NLSE with high nonlinearity, resonant NLSE with parabolic law nonlinearity, and fourth-order dispersive NLSE. We use two powerful and easily understandable techniques, namely the variational principle method and amplitude ansatz method, to investigate these equations. We derive the functional integral and the Lagrangian of these equations to describe the system's dynamics. Additionally, we utilize different forms of trial ansatz functions based on the Jost function with one and two nontrivial variational parameters to substitute them in the Lagrangian and determine the amount of eigenvalue. We also obtain new solitary wave solutions of the proposed equations, which include bright soliton, dark soliton, bright–dark solitary wave solutions, rational dark–bright solutions, and periodic solitary wave solutions. The solutions accepted from these techniques have a broad scope of applications in various areas of physics and other applied sciences. The results will be illustrated in graphical representations such as 2D, 3D, and contour plots, which underline their effectiveness. These techniques can be employed to solve numerous other nonlinear models that appear in mathematical physics and various other applied sciences fields. [ABSTRACT FROM AUTHOR]

The nonlinear evolution equations (NLEEs) play a significant role in applied mathematics, including ordinary and partial differential equations, which are frequently used in many disciplines of applied sciences. The Drinfeld–Sokolov–Wilson (DSW) equation and the Burgers equation are the fundamental equations occurring in various areas of physics and applied mathematics, such as nonlinear acoustics and fluid mechanics. The new generalized (G ′ / G) -expansion method is an effective and more powerful mathematical tool for solving NLEEs arising in applied mathematics and mathematical physics. In this article, we investigate further exact solutions as well as soliton solutions to these couple of nonlinear evolution equations by executing the new generalized (G ′ / G) -expansion method. A large number of soliton solutions, including single soliton, bell-shaped soliton, kink-shaped soliton, singular kink soliton, singular soliton, periodic soliton, irregular periodic soliton solutions, and others, have been retrieved. Each of the derived solutions includes an explicit function of the variables in the equations under consideration. We provide some 3D plots to visualize and realize the characteristics of these solutions. It has been established that the suggested techniques are more potential and successful at obtaining soliton solutions for nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

In this study, the beta time fractional (2 + 1) dimensional Chaffee–Infante equation used to describe the behavior of gas diffusion in a homogeneous medium is discussed. Generalized Kudryashov and modified Kudryashov procedures were used to discovered solitons of the equation. These methods can be easily applied and offer different solutions checked to other methods in the literature. At the same time, these two methods use symbolic calculations to better understand various nonlinear wave models and offer a powerful and effective mathematical approach. The solutions created in this article are different from those in the literature and will guide those working in the field of physics and engineering to better understand this model. Figures of the results were made values different from each other. The stability of the equations in applications has been demonstrated by testing the stability feature on some solutions obtained using the features of the Hamilton system. This work demonstrates the power and effectiveness of the methods discussed in applying many different forms of fractional-order nonlinear equations. The results obtained in this paper are original to our research and have the potential to be helpful in the fields of mathematical engineering and physics. [ABSTRACT FROM AUTHOR]

The goal of this paper is to find exact solutions to the improved modified Korteweg-de Vries (mKdV) equation with a conformable derivative using the Jacobi elliptic function expansion method. The improved mKdV equation is a prominent mathematical model in the realm of nonlinear partial differential equations, with widespread applicability in diverse scientific and engineering domains. This study leverages the well-known effectiveness of the Jacobi elliptic function expansion method in solving nonlinear differential equations, specifically focusing on the intricacies of the improved mKdV problem. The investigation reveals innovative and explicit solutions, providing insight into the dynamics of the related physical processes. This paper provides a comprehensive examination of these solutions, emphasizing their distinct features and depictions using Jacobi elliptic functions. These findings are especially advantageous for specialists in the fields of nonlinear science and mathematical physics, providing significant insights into the behavior and development of nonlinear waves in various physical situations. This work also contributes to our knowledge of the improved mKdV equation and shows that the Jacobi elliptic function expansion method is a useful tool for solving complex nonlinear situations. The study is enhanced with graphical illustrations of various solutions, which further enhance its analytical complexity. [ABSTRACT FROM AUTHOR]

This research provides the first known solution to the nonlinear Hamiltonian amplitude equation using the combined Bernoulli sub-equation function and trigonometric-quintic-B-spline methods. While previous works have analyzed this model through limited linear approximations, an exact solution accounting for nonlinearity remains lacking. We address this gap by employing innovative techniques to reveal and scrutinize solitary wave solutions, elucidating complex behaviors inaccessible through linear methods. Our findings show the unequivocal effectiveness of our approach, comprising the first successful solution to this equation. This breakthrough fundamentally advances the comprehension of nonlinear physical systems across domains encompassing wave propagation, quantum mechanics, and optical phenomena. [ABSTRACT FROM AUTHOR]

A minimalist approach to the linear stability problem in fluid dynamics is developed that ensures efficiency by utilizing only the essential elements required to find the eigenvalues for given boundary conditions. It is shown that the problem is equivalent to a single first-order ordinary differential equation, and that studying the argument of the unknown complex function in the eigenvalue space is sufficient to find the dispersion relation. The method is applied to a model for relativistic magnetized astrophysical jets. [ABSTRACT FROM AUTHOR]

A Slinky is a loose helical coil spring and is a well-known educational toy. In this paper a model for a Slinky is presented. The Slinky is represented as a sequence of rigid half coils connected by torsional springs. A range of Slinky configurations in static equilibrium are calculated. Where possible the torsion spring model is compared with the much simpler point mass model of a Slinky, and a more sophisticated model that includes axial and shear deformation. The torsional spring model is shown for the most part to produce similar qualitative behaviour as the more complex model. The simplicity of the torsional spring model makes it a candidate for a discrete stair walking dynamic model of a Slinky, The point mass model is straightforward to derive and solve and makes it possible for an undergraduate engineering student in the early part of their degree course to understand. The torsional spring model is a useful tool for showing what can be done in a more advanced mechanics course to model a system that most students would have previous experience of. [ABSTRACT FROM AUTHOR]

Physics with its mathematical equations using Hooke's Law, Einstein's General Theory of Relativity, and Cauchy's Law of Mechanics show evidence of God's existence for the first time as a boundary value problem. Philosophical arguments for God's existence have been based upon cosmological and ontological reasons such as those from Socrates to Thomas Aquinas. However, the evidence herein does not invoke any aforementioned philosophical or logical arguments. Instead, a purely physics argument and the associated set of mathematical equations using the General Theory of Relativity and its use for the universe are shown as evidence that the Cauchy traction vector, t, operates on the outside of the universe, when the continuum body is assumed to be the universe. Bible verses are presented that correlate the traction force, t, with the hand(s) of God. Finally, since the mathematical equations provide a boundary value problem and not an initial value problem, the activity of the Cauchy traction force, t, is acting now and not just in the past, thus refuting atheism and deism. [ABSTRACT FROM AUTHOR]

Hermann Weyl's philosophical reflections remain a topic of considerable interest in the history and philosophy of science. In particular, Weyl's commitment to a form of idealism, as it pertains to his reading of Husserl and Fichte, has garnered much discussion. However, much less attention has been given to Weyl's later, and at that only partial, turn towards a form of empiricism (i.e. from the late 1920s onward). This lack of focus on Weyl's later philosophy has tended to obscure some of the most significant lessons that Weyl sought to draw from his decades of research in the foundations of mathematics and physics. In this paper, I develop some aspects of what I will term as Weyl's 'modest' empiricism. I will argue that Weyl's turn toward empiricism can be read in the context of a development of Helmholtz's epistemological program and his unique form of 'Kantianism'. The hope is that this reading will not only provide a better understanding of Weyl's later thought, especially his (1954) criticism of Cassirer, but that it may also provide the basis for a novel 'Weylian' account of the mathematization of nature underwriting the group-theoretic methodology of parts of modern physics. [ABSTRACT FROM AUTHOR]

A novel method is presented for explicitly solving inhomogeneous initial-boundary-value problems (IBVPs) on the half-line for a well-known coupled system of evolution partial differential equations. The so-called double-diffusion model, which is based on a simple, yet general, inhomogeneous diffusion configuration, describes accurately several important physical and mechanical processes and thus emerges in miscellaneous applications, ranging from materials science, heat-mass transport and solid–fluid dynamics, to petroleum and chemical engineering. For instance, it appears in nanotechnology and its inhomogeneous version has recently appeared in the area of lithium-ion rechargeable batteries. Our approach is based on the extension of the unified transform (also called the Fokas method), so that it can be applied to systems of coupled equations. First, we derive formally effective solution representations and then justify a posteriori their validity rigorously. This includes the reconstruction of the prescribed initial and boundary conditions, which requires careful analysis of the various integral terms appearing in the formulae, proving that they converge in a strictly defined sense. The novel solution formulae are also utilized to rigorously deduce the solution's regularity properties near the boundaries of the spatiotemporal domain. In particular, we prove uniform convergence of the solution to the data, its rapid decay at infinity as well as its smoothness up to (and beyond) the boundary axes, provided certain data compatibility conditions at the quarter-plane corner are satisfied. As a sample of important applications of our analysis and investigation of the boundary behavior of the solution and its derivatives, we both prove a novel uniqueness theorem and construct a 'non-uniqueness counterexample'. These supplement the preceding 'constructive existence' result, within the framework of well-posedness. Moreover, one of the advantages of the unified transform is that it yields representations which are defined on contours in the complex Fourier λ -plane, which exhibit exponential decay for large values of λ . This important characteristic of the solutions is expected to allow for an efficient numerical evaluation; this is envisaged in future numerical-analytic investigations. The new formulae and the findings reported herein are also expected to find utility in the study of questions pertaining to well-posedness for nonlinear counterparts too. In addition, our rigorous approach can be extended to IBVPs for other significant models of mathematical physics and potentially also to higher-dimensional and variable-coefficient cases. [ABSTRACT FROM AUTHOR]

This article explores the phenomenon of adiabatic shear bands (ASB) in materials under dynamic loading. It discusses the initiation and growth of ASB, as well as the influence of initial defects and thermoplastic instability. The study uses statistical theory and self-similar solutions to understand the behavior of microshears and the scaling properties of shear bands. The findings contribute to a better understanding of ASB phenomena and their impact on material properties. The article includes experimental analysis and numerical simulations to support its conclusions. [Extracted from the article]

In this short note, we investigate a linear stochastic differential equation from mathematical physics, driven by parametric uncertainty. Given the Legendre differential equation with random inputs, the goal is to give a proof of a conjecture posed in a recent paper, concerning the power‐series solution in a Lebesgue sense. By using the finiteness of the moment‐generating function of the equation's coefficient, mean‐square convergence is derived. [ABSTRACT FROM AUTHOR]

The most important physical structure that is assumed to illustrate the geometry of optical soliton replication in optical fiber theory is the nonlinear Schrödinger equation (NLSE). Optical soliton generation in nonlinear optical fibers is a topic of great contemporary interest because of the numerous applications of ultrafast signal routing systems and short light pulses in communications. This analysis's main goal is to create a large number of soliton solutions for the dynamical model using a variety of contemporary analytical methods. This paper studies different soliton solutions to the perturbed NLSE with Kerr law nonlinearity using two sets of two distinct integration strategies: the mapping approach and the unified auxiliary equation method. The majority of solutions have been found as Jacobi elliptic functions with limiting ellipticity modulus values. Solitons like dark, bright, optical, lonely, and others are also retrieved. We were able to create various single‐type solutions with the help of these strategies. As a result, there is a variety of optical, bell‐shaped, single periodic, and multi‐periodic solutions. In order to validate the computations, the stability of the acquired findings must also be proven. The study provides a highly stunning and suitable strategy for combining numerous exciting wave demonstrations for more advanced models of the present era. Furthermore, we can assert that the outcomes reported here are unique and novel. [ABSTRACT FROM AUTHOR]