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A computational procedure is developed for the efficient calculation of derivatives of integrals over non-separable Gaussian-type basis functions, used for the evaluation of gradients of the total energy in quantum-mechanical simulations. The approach, based on symbolic computation with computer algebra systems and automated generation of optimized subroutines, takes full advantage of sparsity and is here applied to first energy derivatives with respect to nuclear displacements and lattice parameters of molecules and materials. The implementation in the Crystal code is presented, and the considerably improved computational efficiency over the previous implementation is illustrated. For this purpose, three different tasks involving the use of analytical forces are considered: (i) geometry optimization; (ii) harmonic frequency calculation; and (iii) elastic tensor calculation. Three test case materials are selected as representatives of different classes: (i) a metallic 2D model of the Cu(111) surface; (ii) a wide-gap semiconductor ZnO crystal, with a wurtzite-type structure; and (iii) a porous metal-organic crystal, namely the ZIF-8 zinc-imidazolate framework. Finally, it is argued that the present symbolic approach is particularly amenable to generalizations, and its potential application to other derivatives is sketched. [ABSTRACT FROM AUTHOR]

In this paper, under the observation of extended modified rational expansion method based on symbolic computation, the multiple solitary wave solutions for nonlinear two-dimensional Jaulent–Miodek Hierarchy (JMH) equation are constructed. In this investigation, we use the computer software Mathematica for the construction of multiple solitary wave solutions. The interested and important things in this work are the multiple solitary wave solutions which have various kinds of physical structures such as kink soliton, periodic traveling wave, bright soliton, anti-kink soliton, dark soliton, combined bright and dark solitons, topological soliton and peakon soliton. We are sure that the various kinds of soliton solutions are found first time by using one method in the existing literature works. On the basis of this research, we can say that the applied technique is very efficient, reliable, fruitful and powerful. The constructed soliton solutions for nonlinear JMH equation will play an important role in the investigation of different physical phenomena in nonlinear sciences. [ABSTRACT FROM AUTHOR]

This study explores optical soliton solutions within the framework of the Caudrey–Dodd–Gibbon equation (CDGE) and the ( 1 + 1 )-dimensional Boussinesq equation, utilizing the modified Sardar sub-equation method (MSSEM). The derived solutions are meticulously verified using symbolic software Mathematica and encompass a rich array of mathematical functions, including trigonometric, hyperbolic, and exponential functions. Visualization techniques, such as 3D plots, 2D plots, density graphs, and contour graphs, effectively illustrate the diverse behaviors exhibited by these soliton solutions. These equations have been studied previously, their comprehensive traveling wave solutions are relatively unexplored. By employing the modified Sardar sub-equation method, which has demonstrated effectiveness in solving nonlinear evolution equations, this study aims to bridge this gap. The soliton solutions obtained encompass a wide spectrum of behaviors, including bright, dark, periodic, singular, dark-periodic, bright-periodic, dark-singular, compactons, and bright-singular soliton solutions, as well as other complex phenomena. The comprehensive exploration of soliton solutions not only enhances our understanding of fundamental wave phenomena but also provides valuable insights for addressing nonlinear equations across various scientific disciplines, including optical fiber communications, plasma physics, and nonlinear dynamics. [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]

The current research aims to look into the exact solutions (ESs) of the (2 + 1)-dimensional Boiti–Leon–Manna–Pempinelli equation (BLMPE), which describes the irrotational-incompressible fluid. The Hirota bilinear equation of the problem under consideration is developed by applying the Cole–Hopf transform. Then, the ansatz function, combined with the symbolic computation, is utilised to extract the ESs. Some new types of ESs like the multi-wave complexiton solution (MWCS), multi-wave solution (MWS) and the interaction wave solution (IWS) are successfully constructed. Furthermore, ESs such as the travelling wave solution (TWS) are also explored via the modified Kudryashov method. The dynamic characteristics of the ESs are described graphically using the Mathematica software and their physical explanations are presented. Compared with the publicly available literature, the obtained ESs are all new and have not yet been reported. The ideas in this work can be used to investigate the ESs of the other PDEs that appear in physics. [ABSTRACT FROM AUTHOR]

In this study, we study two generalized stochastic Korteweg-de Vries (KdV) equations. The Painlevé property of these nonlinear models is tested using Kruksal's method, which establishes the model's integrability. As a result, using Hirota's bilinear approach and symbolic computation, the N-soliton solutions are constructed. In addition, the extended hyperbolic function method (EHFM), the modified Kudryashov method (MKM), and the sub-equation method (SEM) are used to acquire the bright soliton, dark soliton, singular soliton, periodic, rational, and exponential solutions. To help understand the dynamic features of the derived soliton solutions, we present a number of 2D, 3D, and contour graphs using appropriate parametric values. [ABSTRACT FROM AUTHOR]

In this paper, the coupled system nonlinear (2 + 1)-dim Broer–Kaup–Kupershmit (BKK) equation under consideration is based on extension of modified rational expansion method. The various kinds of multiple soliton solutions named anti-kink soliton, traveling wave solutions, periodic solution, dark soliton, kink soliton, bright soliton, anti-kink bright solitons, kink bright solitons for nonlinear BKK system are successfully constructed under symbolic computation Mathematica. The calculated results are very interesting, different and novel which have not been investigated in past literature. The graphical demonstration of constructed solutions show by contour, 2D and 3D shape by using computer software Mathematica. The constructed multiple soliton solutions prove that EMRE approach is very straightforward, efficient, reliable and powerful for the investigation of other nonlinear partial differential equations. The investigated results will be described and play a keen role in the study of nonlinear physical phenomena in the area of physics and engineering. [ABSTRACT FROM AUTHOR]

Cosmic plasmas are considered as the most abundant form of ordinary matter in the Universe while observations of the cosmic dust in different regions provide an insight into the Universe's recycling processes. For different types of the cosmic dusty plasmas, we hereby, with the symbolic computation and observational/experimental supports, study a (3+1)-dimensional generalized variable-coefficient Kadomtsev-Petviashvili-Burgers-type equation, which can describe the electron-acoustic, dust-acoustic, positron-acoustic, dust-magneto-acoustic, ion-acoustic, magneto-acoustic, ion, quantum-dust-ion-acoustic or dust-ion-acoustic waves in one of the cosmic/laboratory dusty plasmas. With respect to the fluctuation of the electron or ion density, or perturbation of the magnitude of the magnetic field, or electrostatic wave potential, or radial-direction component of the velocity of ions or dust particles, a set of the auto-Bäcklund transformations, several soliton families and a set of the similarity reductions are symbolically computed out, depending on the variable coefficients which represent the dispersion, nonlinearity, geometric effect, Burgers/dusty-fluid-viscosity dissipation and diffraction/transverse perturbation. Variable-coefficient constraints on the soliton solutions are presented. Our analytic results are in agreement with those dusty-plasma-experimentally reported. Future dusty-plasma experiments and observations might justify some other effects hereby offered. [ABSTRACT FROM AUTHOR]

A symbolic C++ library—Tinned—has been developed for symbolic differentiation and manipulation in response theory. By recognizing different key building blocks in the density matrix‐based (Thorvaldsen et al., J. Chem. Phys. 2008, 129, 214108) and coupled‐cluster response theories, we have implemented their corresponding C++ symbolic classes, including but not limited to one‐ and two‐electron operators, exchange‐correlation energy and potential, and coupled‐cluster operator. Formulas of response theory can be well expressed in terms of the symbolic classes in the library Tinned. Their high‐order perturbation‐strength derivatives can be straightforwardly computed and extracted afterwards for numerical evaluation. The library Tinned will greatly facilitate the development work of response theory and may lead to a unified framework for response theory at different levels of electronic structure theory. [ABSTRACT FROM AUTHOR]

Context. High-precision ephemerides are not only useful in supporting space missions, but also in investigating the physical nature of celestial bodies. This paper reports an update to the orbit and rotation model of the Martian moon Phobos. In contrast to earlier numerical models, this paper details a dynamical model that fully considers the rotation of Phobos. Here, Phobos' rotation is first described by Euler's rotational equations and integrated simultaneously with the orbital motion equations. We discuss this dynamical model, along with the differences with respect to the model now in use. Aims. This work is aimed at updating the physical model embedded in the ephemerides of Martian moons, considering improvements offered by exploiting high-precision observations expected from future missions (e.g., Japanese Martian Moons exploration, MMX), which fully supports future studies of the Martian moons. Methods. The rotational motion of Phobos can be expressed by Euler's rotational equations and integrated in parallel with the equations of the orbital motion of Phobos around Mars. In order to investigate the differences between the two models, we first reproduced and simulated the dynamical model that is now used in the ephemerides, but based on our own parameters. We then fit the model to the newest Phobos ephemeris published by Institut de Mécanique Céleste et de Calcul des Éphémérides (IMCCE). Based on our derived variational equations, the influence of the gravity field, the Love number, k2, and the rotation behavior were studied by fitting the full model to the simulated simple model. Our revised dynamic model for Phobos was constructed as a general method that can be extended with appropriate corrections (mainly rotation) to systems other than Phobos, such as the Saturn and Jupiter systems. Results. We present the variational equation for Phobos' rotation employing the symbolic Maple computation software. The adjustment test simulations confirm the latitude libration of Phobos, suggesting gravity field coefficients obtained using a shape model and homogeneous density hypothesis should be re-examined in the future in the context of dynamics. Furthermore, the simulations with different k2 values indicate that it is difficult to determine k2 efficiently using the current data. [ABSTRACT FROM AUTHOR]

The main purpose of this work is to study the optical soliton solutions and single traveling wave solutions of the generalized stochastic Schrödinger–Hirota equation in magneto-optic waveguides. With the help of the complete discriminant system technique and symbolic computation, a range of new single traveling wave solutions and optcial solitons are derived, which include Jacobian elliptic function solutions, dark solitons, trigonometric function solutions, singular solitons, rational function solutions, hyperbolic function solutions, periodic wave solutions and solitary wave solutions. Lastly, in order to understand mechanisms of complex physical phenomena and dynamical processes for the generalized stochastic Schrödinger–Hirota equation in magneto-optic waveguides, two-dimensional and three-dimensional diagrams are also drawn. [ABSTRACT FROM AUTHOR]

This study provides an innovative and attractive analytical strategy to examine the numerical solution for the time-fractional Schrödinger equation (SE) in the sense of Caputo fractional operator. In this research, we present the Elzaki transform residual power series method (ET-RPSM), which combines the Elzaki transform (ET) with the residual power series method (RPSM). This strategy has the advantage of requiring only the premise of limiting at zero for determining the coefficients of the series, and it uses symbolic computation software to perform the least number of calculations. The results obtained through the considered method are in the form of a series solution and converge rapidly. These outcomes closely match the precise results and are discussed through graphical structures to express the physical representation of the considered equation. The results showed that the suggested strategy is a straightforward, suitable, and practical tool for solving and comprehending a wide range of nonlinear physical models. [ABSTRACT FROM AUTHOR]

The representation of elements of free non-associative algebras as a set of multidimensional tables of coefficients is defined. An operation for finding partial derivatives for elements of free non-associative algebras in the same form is considered. Using this representation, a criterion of primitivity for elements of lengths 2 and 3 in terms of matrix ranks, as well as a primitivity test for elements of arbitrary length, is derived. This test makes it possible to estimate the number of primitive elements in free non-associative algebras with two generators over a finite field. The proposed representation allows us to optimize algorithms for symbolic computations with primitive elements. Using these algorithms, we find the number of primitive elements of length 4 in a free non-associative algebra of rank 2 over a finite field. [ABSTRACT FROM AUTHOR]

This paper considers the problem of constructing compartmental models of dynamic systems by using a software package for symbolic calculation written in Julia. The software package is aimed at unifying the formalized construction of compartmental models, taking into account the meaningful description of possible interactions among compartments and the influence of various factors on the evolution of systems. An approach to the development of the instrumental and methodological basis for modeling the dynamic systems the behavior of which can be described by one-step processes is developed. The proposed software package enables the symbolic representation of the differential equations of the model in both stochastic and deterministic cases. It is implemented in Julia and uses the Julia Symbolics computer algebra library. A comparison between the Julia Symbolics tools and some other computer algebra systems is carried out. The application of the developed software package to a compartmental model is considered. The results can be used to solve problems of constructing and studying dynamic models in natural sciences that are represented by one-step processes. [ABSTRACT FROM AUTHOR]

The computation of the normal form as well as its unfolding is a key step to understand the topological structure of a bifurcation. Though a lot of results have been obtained, it still remains unsolved for higher co-dimensional bifurcations. The main purpose of this paper is devoted to the computation of a codimension-3 zero-Hopf–Hopf bifurcation, at which a zero as well as two pairs of pure imaginary eigenvalues can be found from the matrix evaluated at the equilibrium point. Different distributions of eigenvalues are considered, which may behave in a non-semisimple form for 1:1 internal resonance. Based on the combination of center manifold and normal form theory, all the coefficients of normal forms and nonlinear transformations are derived explicitly in terms of parameters of the original vector field, which are obtained via a recursive procedure. Accordingly, a user friendly computer program using a symbolic computation language Maple is developed to compute the coefficients up to an arbitrary order for a specific vector field with zero-Hopf–Hopf bifurcation. Furthermore, universal unfolding parameters are derived in terms of the perturbation of physical parameters, which can be employed to investigate the local behaviors in the neighborhood of the bifurcation point. Here, we emphasize that though different norm forms based on different choices may exist, their topological structures are the same, corresponding to qualitatively equivalent dynamics. [ABSTRACT FROM AUTHOR]

This present paper is inspired by one of the questions posed by Okeke (Results Math 78(96):1-30, 2023, see Remark 2.10b). In particular, we aim to develop a robust computer code based on the theoretical results obtained in Okeke (2023), which determines the polynomial solutions of the following functional equation, 0.1 ∑ i = 1 n γ i F (a i x + b i y) = ∑ j = 1 m (α j x + β j y) f (c j x + d j y) , for all x , y ∈ R , γ i , α j , β j ∈ R , and a i , b i , c j , d j ∈ Q , and their special forms. The primary motivation for writing such a computer code is that solving even simple equations belonging to class (0.1) needs long and tiresome calculations. Therefore, one of the advantages of such a computer code is that it allows us to solve complicated problems quickly, easily, and efficiently. Additionally, the computer code will significantly improve the level of accuracy in calculations. Along with that, there is also the factor of speed. We point out that the computer code will operate with symbolic calculations provided by the programming language Python, which means that it does not contain any numerical or approximate methods, and it yields the exact solutions of the equations considered. [ABSTRACT FROM AUTHOR]

For the electromagnetic waves in an isotropic charge-free infinite thin film with the potential application in magneto-optic recording, we investigate a (2 + 1)-dimensional variable-coefficient modified Kadomtsev-Petviashvili system. Under the help of symbolic computation, different from those in the existing literatures, we find two branches of the similarity reductions, able to reduce that system to a solvable ordinary differential equation. [ABSTRACT FROM AUTHOR]

In this paper, we explore how to generate solitary, peakon, periodic, cuspon and kink wave solution of the well-known partial differential equation Korteweg–de Vries (KdV) by using exp-function and modified exp-function methods. The presented methods construct more efficiently almost all types of soliton solution of KdV equation that can be rarely seen in the history. These methods appear to be straightforward and symbolic calculations are used to solve the problem. All resulting answers are verified for accuracy using the symbolic computation program with M a p l e. To show the physical appearance of the model, 3D plots of all the generated solutions are then displayed. The obtained solutions revealed the compatibility of the proposed techniques which provide the general solution with some free parameters. This is the key benefit of these methods over the other methods. [ABSTRACT FROM AUTHOR]

This work focuses on the utilization of the generalized exponential rational function method (GERFM) to analyze wave propagation of the extended (3 + 1)-dimensional Sakovich equation. The demonstrated effectiveness and robustness of the employed method underscore its relevance to a wider spectrum of nonlinear partial differential equations (NPDEs) in physical phenomena. An examination of the physical characteristics of the generated solutions has been conducted through two- and three-dimensional graphical representations. [ABSTRACT FROM AUTHOR]

The A4BX6 molecular halide perovskites have received attention owing to their interesting optoelectronic properties at the molecular scale; however, a comprehensive dataset of their atomic structures and electronic properties and associated data-driven investigation are still unavailable now, which makes it difficult for inverse materials design for semiconductor applications (e.g. wide band gap semiconductor). In this manuscript, we employ data-driven methods to predict band gaps of A4BX6 molecular halide perovskites via machine learning. A large virtual design database including 246 904 A4BX6 perovskite samples is predicted via machine learning, based on the model trained using 2740 first-principles results of A4BX6 molecular halide perovskites. In addition, symbolic regression-based machine learning is employed to identify more physically intuitive descriptors based on the starting first-principles dataset of A4BX6 molecular halide perovskites. In addition, different ranking methods are employed to offer a comprehensive feature importance analysis for the halide perovskite materials. This study highlights the efficacy of machine learning-assisted compositional design of A4BX6 perovskites, and the multi-dimensional database established here is valuable for future experimental validation toward perovskite-based wide band gap semiconductor materials. [ABSTRACT FROM AUTHOR]

The modified unstable nonlinear Schrödinger equation governs specific instabilities observed in modulated wave-trains and also describes the time evolution of disturbances in marginally stable or unstable media. In this article, we explore the modified unstable dynamical model analytically using three symbolic computational techniques: the positive quadratic function approach, the three-wave approach, and the double exponential approach. As a result of our analysis, we have derived novel exact solutions, including rational solitons, multi-wave solutions, and other types of wave solutions for this dynamical model. These solutions are obtained through symbolic computation and the ansatz function method, incorporating both traveling wave and logarithmic transformations. The derived wave solutions hold practical significance, contributing significantly to understanding the physical phenomena of this complex model. We also computed conservative quantities such as power, momentum, and energy associated with solitons. To evaluate the stability of this modified unstable equation, we conducted a comprehensive modulational instability analysis, confirming the stability and exactness of all soliton solutions. By selecting appropriate parameter values, we generated 3D visual representations in various forms, including breather-type waves, lump waves, multi-peak solitons, etc. Furthermore, our observations revealed intriguing phenomena arising from the interactions among these multi-waves, with applications spanning a wide range of scientific and engineering disciplines. [ABSTRACT FROM AUTHOR]

This work examines the Painlevé integrability of a (3 + 1)-dimensional Boussinesq-type equation. Using the Mathematica program, we rigorously establish Painlevé's integrability for the suggested problem. By utilizing Hirota's bilinear technique, we obtain the dispersion relations and phase shifts, which enable us to derive multiple soliton solutions. In addition, we systematically derive a wide range of lump solutions using the Maple symbolic computation. The investigation extends to encompass a variety of exact solutions with distinct structural features, including kink, periodic, singular, and rational solutions. This comprehensive analysis illustrates the profound richness of the model's dynamics and its potential to elucidate diverse nonlinear wave phenomena across various physical contexts. Therefore, the results that we will obtain play a vital role in understanding the mechanism of generation and propagation of many mysterious phenomena that arise in various scientific fields, including plasma physics, fluid mechanics, and the propagation of waves on the surfaces of seas and oceans to optical fibers. [ABSTRACT FROM AUTHOR]

Three-dimensional voxel models are widely applied in various fields such as 3D imaging, industrial design, and medical imaging. The advancement of 3D modeling techniques and measurement devices has made the generation of three-dimensional models more convenient. The exponential increase in the number of 3D models presents a significant challenge for model retrieval. Currently, these models are numerous and typically represented as point clouds or meshes, resulting in sparse data and high feature dimensions within the retrieval database. Traditional methods for 3D model retrieval suffer from high computational complexity and slow retrieval speeds. To address this issue, this paper combines spatial-filling curves with octree structures and proposes a novel approach for representing three-dimensional voxel model sequence data features, along with a similarity measurement method based on symbolic operators. This approach enables efficient similarity calculations and rapid dimensionality reduction for the three-dimensional model database, facilitating efficient similarity calculations and expedited retrieval. [ABSTRACT FROM AUTHOR]

This investigation underscores the paramount imperative of discerning network intrusions as a pivotal measure to fortify digital systems and shield sensitive data from unauthorized access, manipulation, and potential compromise. The principal aim of this study is to leverage a publicly available dataset, employing a Genetic Programming Symbolic Classifier (GPSC) to derive symbolic expressions (SEs) endowed with the capacity for exceedingly precise network intrusion detection. In order to augment the classification precision of the SEs, a pioneering Random Hyperparameter Value Search (RHVS) methodology was conceptualized and implemented to discern the optimal combination of GPSC hyperparameter values. The GPSC underwent training via a robust five-fold cross-validation regimen, mitigating class imbalances within the initial dataset through the application of diverse oversampling techniques, thereby engendering balanced dataset iterations. Subsequent to the acquisition of SEs, the identification of the optimal set ensued, predicated upon metrics inclusive of accuracy, area under the receiver operating characteristics curve, precision, recall, and F1-score. The selected SEs were subsequently subjected to rigorous testing on the original imbalanced dataset. The empirical findings of this research underscore the efficacy of the proposed methodology, with the derived symbolic expressions attaining an impressive classification accuracy of 0.9945. If the accuracy achieved in this research is compared to the average state-of-the-art accuracy, the accuracy obtained in this research represents the improvement of approximately 3.78%. In summation, this investigation contributes salient insights into the efficacious deployment of GPSC and RHVS for the meticulous detection of network intrusions, thereby accentuating the potential for the establishment of resilient cybersecurity defenses. [ABSTRACT FROM AUTHOR]

Formal, mathematically rigorous programming language semantics are the essential prerequisite for the design of logics and calculi that permit automated reasoning about concurrent programs. We propose a novel modular semantics designed to align smoothly with program logics used in deductive verification and formal specification of concurrent programs. Our semantics separates local evaluation of expressions and statements performed in an abstract, symbolic environment from their composition into global computations, at which point they are concretised. This makes incremental addition of new language concepts possible, without the need to revise the framework. The basis is a generalisation of the notion of a program trace as a sequence of evolving states that we enrich with event descriptors and trailing continuation markers. This allows to postpone scheduling constraints from the level of local evaluation to the global composition stage, where well-formedness predicates over the event structure declaratively characterise a wide range of concurrency models. We also illustrate how a sound program logic and calculus can be defined for this semantics. [ABSTRACT FROM AUTHOR]

Kadomtsev-Petviashvili (KP)-typed models are used to elucidate certain shallow water waves that arise in plasma physics, marine engineering, ocean physics, fluid dynamics, etc. In this paper, we investigate a (3+1)-dimensional KP equation for the shallow water waves: (1) Via some exchange formulae, a bilinear Bäcklund transformation and the corresponding soliton-like solutions are constructed. (2) Via symbolic computation, fission/fusion solutions are derived with some parameter conditions in the N-soliton solutions, which are different from those in the existing literature, where N is a positive integer. We graphically display the spatial structures of the fission/fusion waves to supplement the existing literature. (3) Periodic-wave solutions are worked out via the Hirota-Riemann method. Via the asymptotic properties, relation between the periodic-wave and one-soliton solutions is discussed. [ABSTRACT FROM AUTHOR]

This work explores diverse novel soliton solutions due to fractional derivative, dispersive, and nonlinearity effects for the nonlinear time M-fractional paraxial wave equation. The advanced exp [-φ(ξ)] expansion method integrates the nonlinear M-fractional Paraxial wave equation for achieving creative solitonic and traveling wave envelopes to reconnoiter such dynamics. As a result, trigonometric and hyperbolic solutions have been found via the proposed method. Under the conditions of the constraint, fruitful solutions are gained and verified with the use of the symbolic software Maple 18. For any chosen set of the allowed parameters 3D, 2D and density plots illustrate, this inquisition achieved kink shape, the collision of kink type and rogue wave, periodic rogue wave, some distinct singular periodic soliton waves for time M-fractional Paraxial wave equation. As certain nonlinear effects cancel out dispersion effects, optical solitons typically can travel great distances without dissipating. We have constructed reasonable soliton solutions and managed the actual meaning of the acquired solutions of action by characterizing the particular advantages of the summarized parameters by the portrayal of figures and by interpreting the physical occurrences. New precise voyaging wave configurations are obtained using symbolic computation and the previously described methodologies. However, the movement role of the waves is explored, and the modulation instability analysis is used to describe the stability of waves in a dispersive fashion of the obtained solutions, confirming that all created solutions are precise and stable. [ABSTRACT FROM AUTHOR]

This paper presents an investigation into original analytical solutions of the (2+1)-dimensional combined potential Kadomtsev–Petviashvili and B-type Kadomtsev–Petviashvili equations. For this purpose, the generalized Kudryashov technique (GKT) and exponential rational function technique (ERFT) have been applied to deal with the equation. These two methods have been applied to the model for the first time, and the the generalized Kudryashov method has an important place in the literature. The characteristics of solitons are unveiled through the use of three-dimensional, two-dimensional, contour, and density plots. Furthermore, we conducted a stability analysis on the acquired results. The results obtained in the article were seen to be different compared to other results in the literature and have not been published anywhere before. [ABSTRACT FROM AUTHOR]

Waves advantageous properties have recently been exploited to enable a wide range of applications, including impact reduction, switching, focussing, and asymmetric transmission. We find that the motion of lumps, rogue waves, lump periodic waves, Ma-breather waves, lump one-strip, and generalised breather waves may also be used to build flexible constructions crawl. Symbolic computation and the Hirota bilinear method are used to create these nonlinear waves. Combining the aforementioned approaches, we demonstrate that such pulse-driven locomotion achieves optimal effectiveness when the pulses are solitons. Our research shows how nonlinear waves can be used in a variety of new ways, including as a platform for developing adaptable robots that can move. [ABSTRACT FROM AUTHOR]

This manuscript employs the (2+1)-dimensional Zoomeron model extensively used in laser physics, fluids, optical fibre communication, and other mathematical physics and engineering fields. We utilize the extended Jacobian elliptic function and the modified extended tanh techniques to derive the solution for this model explicitly. Furthermore, the linear stability process is applied to investigate the stability analysis of the governing model. By employing symbolic computation, we can obtain periodic waves, anti-kink waves, kink waves, bright-dark breather waves, and kinky breather waves within this model. Additionally, we identify singular breather waves and singular kink waves. The propagation of these outcomes is visually represented through 3D, 2D and density images. Furthermore, we present the novelty of the obtained solutions and compare them to other solutions in the literature. Finally, we discuss the physical implications of our results and the potential applications of this model. These results enhance our understanding of nonlinear dynamics and pave the way for further research in this area. [ABSTRACT FROM AUTHOR]

The main goal of the current work is to study dynamical behavior and dispersive optical solitons for the fractional Ginzburg–Landau equation in optical fibers. Starting with the traveling wave transformations, the fractional Ginzburg–Landau model is converted into an equivalent ordinary differential traveling wave system. Then, the Hamiltonian function and orbits phase portraits of this system are found. Here, we derived explicit fractional periodic wave solutions, bell-shaped solitary wave solutions and kink-shaped solitary wave solutions through the bifurcation theory of differential dynamical system. In addition to, some other traveling wave solutions are obtained by using the polynomial complete discriminant method and symbolic computation. Most notably, we give the classification of all single traveling wave solutions of fractional Ginzburg–Landau equation at the same time. The obtained optical soliton solutions in this work may substantially improve or complement the corresponding results in the known references. Finally, we give the comparison between our solutions and other's results. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to obtain some variable separation solitary solutions of (2 + 1)-dimensional nonlinear long water wave (LWW) equation through symbolic computation. By applying Riccati equation mapping method, the localized dromion excitations are constructed and different number of M × N multi-dromions are described by selecting appropriate functions for the LWW equations, which provide a model of the interaction between two waves propagation consequently, the solition solutions are obtained in different forms of dynamical structures. Moreover, the collision phenomena between two multidromions are explored he fractal structure was also investigated. The results have rich physical structures that are helpful to explain the nonlinear solitary phenomena in nonlinear physics. [ABSTRACT FROM AUTHOR]

In present paper, multiple rogue wave solutions of (2+1)-dimensional Yu–Toda–Sasa–Fukuyama equation were studied by applying the traveling wave transformation and the Hirota bilinear method. We obtained its 1-rogue wave, 3-rogue wave, 6-rogue wave and 8-rogue wave solutions through symbolic computation. Three-dimensional, contour and density plots of corresponding solutions vividly shown the physical structure and properties. Furthermore, these solutions have greatly enriched the exact solutions of (2+1)-dimensional Yu–Toda–Sasa–Fukuyama equation on the existing literature. [ABSTRACT FROM AUTHOR]

This paper is concerned with the exact solution of mixed-integer programs (MIPs) over the rational numbers, i.e., without any roundoff errors and error tolerances. Here, one computational bottleneck that should be avoided whenever possible is to employ large-scale symbolic computations. Instead it is often possible to use safe directed rounding methods, e.g., to generate provably correct dual bounds. In this work, we continue to leverage this paradigm and extend an exact branch-and-bound framework by separation routines for safe cutting planes, based on the approach first introduced by Cook, Dash, Fukasawa, and Goycoolea in 2009 [INFORMS J. Comput., 21 (2009), pp. 641-649]. Constraints are aggregated safely using approximate dual multipliers from an LP solve, followed by mixed-integer rounding to generate provably valid, although slightly weaker inequalities. We generalize this approach to problem data that is not representable in floating-point arithmetic, add routines for controlling the encoding length of the resulting cutting planes, and show how these cutting planes can be verified according to the VIPR certificate standard. Furthermore, we analyze the performance impact of these cutting planes in the context of an exact MIP framework, showing that we can solve 21.5% more instances to exact optimality and reduce solving times by 26.8% on the MIPLIB 2017 benchmark test set. [ABSTRACT FROM AUTHOR]

A (2 + 1)-dimensional fourth order Burgers–KdV equation with variable coefficients (vcBKdV) is studied here and interesting wave-type solutions with variable amplitudes and velocities are reported. The model has been not previously studied in the chosen form and it presents a twofold interest: as a model describing a rich variety of phenomena and as a higher-order equation solving difficulties generated by the presence of the variable coefficients. The novelty of our approach is related to the use of the functional expansion, a solving method based on an auxiliary equation that generalizes other approaches, such as, for example, the G ′ G proved here. We use a similarity reduction with a nonlinear wave variable that leads to a determining system that it is not usually algebraic, but an over-determined system of partial differential equations. It depends on 14 constant or functional parameters and can generate much richer classes of solutions. Three such classes of solutions, corresponding to the case when a specific form of the generalized reaction–diffusion equation is used as auxiliary equation, are considered. The influence on the dynamical behavior of two important factors, the choices of the auxiliary equation and the form of solution, are studied by providing graphical representations of specific solutions for various values of the parameters. [ABSTRACT FROM AUTHOR]

Shallow water waves are seen in oceanography, atmospheric science, and other fields. In this paper, we investigate an extended (3+1)-dimensional shallow water wave equation. We get the travelling-wave solutions via the polynomial-expansion method. Applying the Hirota method and symbolic computation, we derive some mixed-lump-kink and mixed-rogue-wave-kink solutions. Based on the mixed-lump-kink solutions, we graphically show the interaction between a lump and a kink soliton, and find two different cases: (1) the lump merges into the kink soliton; (2) the lump separates from the kink soliton. Based on the mixed-rogue-wave-kink solutions, we graphically analyze the interaction between the rogue wave and two-kink solitons, and find that the rogue wave emerges from the one kink soliton and merges into the other kink soliton. [ABSTRACT FROM AUTHOR]

The study of formal stability of equilibrium positions of a multiparametric Hamiltonian system in a generic case is traditionally carried out using its normal form under the condition of the absence of resonances of small orders. In this paper we propose a method of symbolic computation of the condition of existence of a resonance of arbitrary order for a system with three degrees of freedom. It is shown that this condition for each resonant vector can be represented as a rational algebraic curve. By methods of computer algebra the rational parametrization of this curve for the case of an arbitrary resonance is obtained. A model example of some two-parameter system of pendulum type is considered. [ABSTRACT FROM AUTHOR]

In this paper, a (2+1)-dimensional generalized Calogero-Bogoyavlenskii-Konopelchenko-Schiff system in a fluid or plasma is investigated. Via the Hirota method and symbolic computation, we work out some two-resonance Y-type soliton solutions as well as some hybrid solutions composed of the two-resonance Y-type solitons and solitons/breathers. Graphically, we display some two-resonance Y-type solitons. We present the interactions between the two-resonance Y-type soliton and one soliton, among the two-resonance Y-type soliton and two solitons, between the two-resonance Y-type soliton and first-order breather as well as among the two-resonance Y-type soliton and second-order breathers. [ABSTRACT FROM AUTHOR]

The behavior of optical soliton solutions in optical fiber theory is described by the Schrödinger–Hirota equation, a nonlinear partial differential equation with multiple applications in science and engineering. In this work, a highly effective approach for solving the Schrödinger–Hirota equation is utilized based on the modified generalized Riccati equation mapping approach. The remarkable achievement of obtaining over one hundred plus newly-formed distinct analytical soliton solutions for the governing Schrödinger–Hirota equation is accomplished through the employment of this technique. The technique employed is characterized by its power, simplicity, and adaptiveness, enabling the extraction of various forms of soliton solutions into a unified framework. As a consequence of the applications, traveling waves, bell-shaped solitons, kink waves, and combo-multisoliton wave profiles are generated. To provide valuable insights into the behavior and characteristics of the solutions, two-dimensional, three-dimensional, and corresponding contour graphs are plotted for selected solutions. These findings contribute to the improvement of theoretical knowledge regarding the Schrödinger–Hirota equation and lay the groundwork for practical implementations. The abundant soliton solutions discovered through this method have the potential to find utility in a variety of fields, including electronics, quantum physics, and optical fibers. [ABSTRACT FROM AUTHOR]

As artificial intelligence (AI) becomes progressively more engaged with society, its shift from technical tool to participating in society raises questions about AI personhood. Drawing upon developmental psychology and systems theory, a mediating structure for AI proto‐personhood is defined analogous to an early stage of human development. The proposed AI bridges technical, psychological, and theological perspectives on near‐future AI and is structured by its hardware, software, computational, and sociotechnical systems through which it experiences its world as embodied (even for putatively disembodied AI). Further social and moral construction can occur building upon a simple "self" for AI synthesized from symbolic and statistical approaches to AI. [ABSTRACT FROM AUTHOR]

To date, with respect to water waves, researchers have studied certain systems, including a generalized Whitham–Broer–Kaup–Boussinesq–Kupershmidt system that describes, e.g., the dispersive long waves in the oceanic shallow water, which we study here. With respect to, e.g., the horizontal velocity of the water wave as well as the height of the deviation from the equilibrium position of the water, with symbolic computation, on the one hand, the system is found to pass the Painlevé test under some coefficient constraints, while on the other hand, two families of the bilinear forms and two families of the N-soliton solutions are constructed, with N as a positive integer. Related constraints are shown. Our bilinear forms and N-soliton solutions depend on the coefficients in the system. [ABSTRACT FROM AUTHOR]

In this paper, we focus our attention on a (3 + 1)-dimensional variable-coefficient Hirota bilinear system in a fluid with symbolic computation. The Painlevé integrable property is derived. Via the Ablowitz–Kaup–Newell–Segur procedure, we obtain a Lax pair under the coefficient constraints. Based on the Hirota method, we obtain a bilinear form and a bilinear Bäcklund transformation under the coefficient constraints. We derive the auto-Bäcklund transformations based on the truncated Painlevé expansions. According to the bilinear form, we give the two-soliton solutions under the coefficient constraints. We also discuss the relation between the variable coefficients and soliton solutions, i.e., how the two solitons present different types with the different forms of the variable coefficients. [ABSTRACT FROM AUTHOR]

Strong untestable assumptions are almost universal in causal point estimation. In particular settings, bounds can be derived to narrow the possible range of a causal effect. Symbolic bounds apply to all settings that can be depicted using the same directed acyclic graph and for the same effect of interest. Although the core of the methodology for deriving symbolic bounds has been previously developed, the means of implementation and computation have been lacking. Our Rpackage causaloptim aims to solve this usability problem by providing the user with a graphical interface through Shiny. This interface takes input in a form that most researchers with an interest in causal inference will be familiar: a graph drawn in the user's web browser and a causal query written in text using common counterfactual notation. [ABSTRACT FROM AUTHOR]

In this paper, the M-lump solutions, the periodic type, and cross-kink wave solutions are acquired. Here, the Hirota bilinear operator is employed. By utilizing the symbolic computation and employing the utilized method, the (3+1)-dimensional Jimbo–Miwa (JM) equation is investigated. Based on the Hirota bilinear form, the soliton solution and periodic wave solution to the mentioned equation, respectively, are obtained. We gained plenty of multiple collisions of lumps. Next, the periodic wave and cross-kink wave have greatly enriched the existing literature on the JM equation. Through the three-dimensional designs, contour design, density design, and two-dimensional design by using Maple, the physical features of these soliton solutions are explained all right. The forms of the attained solutions are one-lump, two-lumps, and three-lumps wave solutions. Then, a class of rogue waves-type solutions to the (3+1)-dimensional JM equation within the frame of the bilinear equation is found. These results can help us better understand interesting physical phenomena and mechanisms. [ABSTRACT FROM AUTHOR]

This paper aims to search for lump waves in a spatial symmetric (2+1)-dimensional dispersive wave model. Through an ansatz on positive quadratic functions, we conduct symbolic computations with Maple to generate lump waves for the proposed nonlinear model. A line of critical points of the lump waves is computed, whose two spatial coordinates travel at constant speeds. The corresponding maximum and minimum values are evaluated in terms of the wave numbers, and interestingly, all those extreme values do not change with time, either. The last section is the conclusion. [ABSTRACT FROM AUTHOR]

The main features of scientific efforts in physics and engineering are the development of models for various physical issues and the development of solutions. In order to solve the time-fractional coupled Korteweg–De Vries (KdV) equation, we combine the novel Yang transform, the homotopy perturbation approach, and the Adomian decomposition method in the present investigation. KdV models are crucial because they can accurately represent a variety of physical problems, including thin-film flows and waves on shallow water surfaces. The fractional derivative is regarded in the Caputo meaning. These approaches apply straightforward steps through symbolic computation to provide a convergent series solution. Different nonlinear time-fractional KdV systems are used to test the effectiveness of the suggested techniques. The symmetry pattern is a fundamental feature of the KdV equations and the symmetrical aspect of the solution can be seen from the graphical representations. The numerical outcomes demonstrate that only a small number of terms are required to arrive at an approximation that is exact, efficient, and trustworthy. Additionally, the system's approximative solution is illustrated graphically. The results show that these techniques are extremely effective, practically applicable for usage in such issues, and adaptable to other nonlinear issues. [ABSTRACT FROM AUTHOR]

An innovative semianalytical solution was introduced to accurately assess the uplift bearing capacity of spread foundations in sand, which is important for the design of transmission towers. The research addresses the shortcomings of previous analyses, which relied on a 1g laboratory test and consequently overestimated uplift bearing capacities due to scale effects. To overcome these limitations, a series of centrifuge tests were conducted under stress conditions representative of field conditions. The test results led to the identification of a bilinear failure surface, effectively capturing distinct uplift failure mechanisms along the embedment depth, including a passive mode near the ground surface and an active mode downward. A novel application of genetic programming analysis, based on symbolic regression techniques, was employed to optimize the determination of parameters forming the bilinear failure surface. The net uplift bearing capacity corresponding to the bilinear failure surface was evaluated using a limit equilibrium approach combined with a slice procedure. The proposed semianalytical solution was validated with both the current test results and previously published data, illustrating its reasonable applicability and representing a significant innovation in determining the net capacity of spread foundations in sand. [ABSTRACT FROM AUTHOR]

In this manuscript, we implement analytical multiple soliton wave and singular soliton wave solutions for coupled mKdV with a time-dependent variable coefficient. Based on the similarity transformation and Hirota bilinear technique, we construct both multiple wave kink and wave singular kink solutions for coupled mKdV with a time-dependent variable coefficient. We implement the Hirota bilinear technique to compute analytical solutions for the coupled mKdV system. Such calculations are made by using a software with symbolic computation software, for instance, Maple. Recently some researchers used Maple in order to show that the bilinear method of Hirota is a straightforward technique which can be used in the approach of differential, nonlinear models. We analyzed whether the experiments proved that the procedure is effective and can be successfully used for many other mathematical models used in physics and engineering. The results of this study display that the profiles of multiple-kink and singular-kink soliton types can be efficiently controlled by selecting the particular form of a similar time variable. The changes in the solitons based on the changes in the arbitrary function of time allows for more applications of them in applied sciences. [ABSTRACT FROM AUTHOR]