Symbolic Computation Research Articles

Dynamical study and diverse soliton collisions for the Kraenkel–Manna–Merle system in ferrites

We investigate the rational solitons of the Kraenkel–Manna–Merle (KMM) system in ferrites via symbolic computation with the ansatz functions technique and logarithmic transformation. The KMM system describes the propagations of the ferromagnetic particles in ferrite materials. Multiwave solitons are reported by using the three-wave technique. Homocentric breathers (HBs) approach is also applied to build new forms of exact solutions. We construct M-shaped solitons and their dynamics are shown via the appropriate values of parameters. Two types of interactions for rational soliton with kink wave are investigated. Additionally, kink cross-rational solitons (KCRSs) and periodic cross-rational solitons (PCRSs) are examined via suitable transformation.

Higher rogue and rogue-soliton interaction solutions of a (2 + 1) dimensional nonlinear model in fluid mechanics

In this study, two new theorems are generalized. We obtained a new paradigm about the second order rogue wave and multiple exponential functions, and a new paradigm about the second order rogue wave and multiple hyperbolic cosine functions. Six sets of interaction solutions of the model are solved by means of symbolic calculation and two new theorems. Meaningful graphs of the propagation processes along time demonstrated the interaction phenomena for these solutions. The energy transfer process can be observed when the second order rogue waves interact with multiple exponential functions or multiple hyperbolic cosine functions. As a conclusion from our paper, the solitons’ energy transfers to the second order rogue wave at beginning, the rogue wave’s energy dissipates and transfers to the solitons along the time moving. It will contribute to the research on the generation of rogue waves.

On the Painlevé integrability and nonlinear structures to a (3 + 1)-dimensional Boussinesq-type equation in fluid mediums: Lumps and multiple soliton/shock solutions

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.

Dynamical exploration of optical soliton solutions for M-fractional Paraxial wave equation.

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.

Computing the Minimum Polynomial, the Function and the Drazin Inverse of a Matrix with Matlab

Aims/ Objectives: In this note we will discuss the known (but not well known) problem of finding the minimum polynomial and the function of a matrix providing the simplest proofs for undergraduate students. We will try to explain with fairly simple arguments how to compute the minimum polynomial of a matrix giving also the matlab code for its symbolic computation. Next we will describe the (symbolic) computation of the matrix of a function via the Hermite interpolation method which seems to be the simplest method for undergraduate students. Finally we shall see how we can compute the Drazin inverse given the nth power of a matrix A.

Mixed Subdivisions Suitable for the Greedy Canny–Emiris Formula

The Canny–Emiris formula (Canny and Emiris in International symposium on applied algebra, algebraic algorithms, and error-correcting codes, 1993) gives the sparse resultant as the ratio of the determinant of a Sylvester-type matrix over a minor of it, both obtained via a mixed subdivision algorithm. In Checa and Emiris (Proceedings of the 2022 international symposium on symbolic and algebraic computation, 2022), the same authors gave an explicit class of mixed subdivisions for the greedy approach so that the formula holds, and the dimension of the constructed matrices is smaller than that of the subdivision algorithm, following the approach of Canny and Pedersen (An algorithm for the Newton resultant, 1993). Our method improves upon the dimensions of the matrices when the Newton polytopes are zonotopes and the systems are multihomogeneous. In this text, we provide more such cases, and we conjecture which might be the liftings providing minimal size of the resultant matrices. We also describe two applications of this formula, namely in computer vision and in the implicitization of surfaces, while offering the corresponding JULIA code. We finally introduce a novel tropical approach that leads to an alternative proof of a result in Checa and Emiris (Proceedings of the 2022 international symposium on symbolic and algebraic computation, 2022).

Polynomial Solutions of Electric Field Equations in Anisotropic Media

The time-dependent system of partial differential equations of the
 second order describing the electric wave propagation in
 electrically and magnetically anisotropic homogeneous media is
 considered in the paper. A method for the computation of the polynomial solutions of the initial value problem for the considered system is proposed.
 Symbolic computations are used and these symbolic computations are implemented in Maple. It is proved also that these polynomial solutions are approximate solutions of the considered initial value problem with smooth initial data and the inhomogeneous term. The computational experiments confirm the robustness of the suggested method for the computation of electric fields in general electrically and magnetically anisotropic media.

The excitation of high-order localized waves in (3+1)-dimensional Kudryashov-Sinelshchikov equation

The aim of this work is to explore the excitation of high-order localized waves in the (3+1)-dimensional Kudryashov-Sinelshchikov equation, which is used to describe the dynamic of liquid with gas bubble. First of all, classical N-soliton solutions are constructed by means of Hirota bilinear form and symbolic calculation. What’s more, the high-order breather waves are derived through the degeneration process of the N-soliton solutions with conjugate parameter. Then, high-order lump waves are constructed by taking long wave limit technique on N-soliton solutions. Finally, the high-order mixed localized waves involving resonant Y-type solitons, high-order breather waves and high-order lump waves are obtained by utilizing some comprehensive methods. Abundant dynamical and evolutionary behaviors of these results are investigated specifically, some figures are presented to shed light on the nonlinear phenomena hidden in the high-order localized waves vividly.

Applications of two kinds of Kudryashov methods for time fractional (2 + 1) dimensional Chaffee–Infante equation and its stability analysis

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.

Numerical eigenanalysis of continuum geodesic acoustic mode by ideal magnetohydrodynamic model

In this study, we have developed a new eigenvalue code to solve the three-variables ideal magnetohydrodynamic (MHD) equation in flux coordinate system with the help of a symbolic vector analysis module and an automatic numerical discretization module, which were developed with the use of symbolic computation technique and were greatly facilitated the development of toroidal full MHD eigenvalue code SCELT. It has been used to solve for unstable modes and the present work is dedicated to the more difficult challenge of MHD continua, which ordinarily come with spatial singularities. With this new code, we were able to perform numerical calculations to determine the continuum spectrum of geodesic acoustic modes (GAMs), as well as obtain eigenfunctions for both the perturbed plasma displacement and perturbed magnetic field, incorporating a sufficient number of components. Through our global calculations, we have provided numerical evidence of singularities in the eigenfunctions, characterized by the type s−s0−1 and Ins−s0 . Moreover, we have examined the spatial variations of GAM fluctuations in regions both near and far from the singular surface. Here s represents a flux function that characterizes the magnetic flux surfaces and s=s0 defines the singular surface. Additionally, we have investigated the effects of finite aspect ratio, non-circular cross sections as well as the asymmetry on the frequency and eigenfunction of continuum GAMs through numerical analysis.

Non-singular complexiton, singular complexiton and the breather wave solutions to the (2 + 1)-dimensional Sawada-Kotera equation

This work aims to construct some novel exact solutions of the (2 + 1)-dimensional Sawada-Kotera equation (SKE). First, the resonant multiple soliton solutions (RMSSs) are discussed by employing the weight algorithm (WA) and linear superposition principle (LSP). Then, based on the RMSSs, the non-singular complexiton and singular complexiton solutions are developed by taking pairs of the conjugate parameters. Finally, the breather wave solutions (BWSs) of the sin-cosh type are explored through the symbolic computation and test function method. The dynamics of the corresponding extracted solutions are displayed graphically for showing the physical properties. The achievements of this article can provide some new insights to the (2 + 1)-dimensional SKE.

Multiquadric based RBF-HFD approximation formulas and convergence properties

RBF-FD formulas have received special attention due to their superior accuracy, well-conditioning, and ease of implementation in solving partial differential equations. To further improve the accuracy of RBF-FD formulas, without increasing the stencil size RBF-HFD formulas are proposed in the literature. In this article, we obtain leading/higher order analytical expressions for weights and local truncation errors associated with various such formulas using the multiquadric radial function. We also study the convergence properties of these formulas. Symbolic computation in Mathematica is employed for analytically solving linear systems for the unknown weights in the RBF-HFD formulas. Symmetry properties of central difference approximations help reduce the number of unknown weights and thereby the size of the linear system. We compute the analytical expressions of weights and local truncation errors for first and second-derivative approximation formulas till tenth order; and for two-dimensional Laplacian operator approximation formulas till sixth order. The obtained formulas are free from ill-conditioning. It is observed that the RBF-HFD formula weights converge to corresponding order classical compact FD scheme weights when the shape parameter (ϵ) tends to zero. The local truncation error has global minimum for an optimal value of the shape parameter. Further, the optimal shape parameter is independent of nodal distance but depends on choice of test function and its derivative values at a reference node.

Bell polynomials and superposition wave solutions of Hirota–Satsuma coupled KdV equations

In this article, the bilinear form, the bilinear Bäcklund transformation, Lax pair, the integrability, infinite conservation laws, nonlinear superposition formula and superposition wave solutions of Hirota–Satsuma coupled KdV (HSCKdV) equations are studied, which can help us get more properties, increase the diversity of solutions and get more new phenomena. The bilinear form, the bilinear Bäcklund transformation and Lax pair of HSCKdV equations are constructed by using the Bell polynomials approach. Based on this, HSCKdV system is integrable in the sense of Lax pair. According to the obtained bilinear Bäcklund transformation, infinite conservation laws and nonlinear superposition formula are derived. And some superposition wave solutions of HSCKdV equations can be attained by the nonlinear superposition formula. With the help of the symbolic calculation system Mathematica, the properties of the superposition wave solutions are analyzed. What is more, in addition to many properties obtained, the relationship between the solutions is found via the derived nonlinear superposition formula, and mixed solutions containing different functions and arbitrary functions are constructed. By using the Bell polynomial method and studying the integrability and other properties of HSCKdV equations, as well as their exact solutions, it can be found that solitons maintain their shape unchanged during motion and are not destroyed. Such waves have very important applications as a communication method. Furthermore, it has been observed that the results attained in this work have not been presented before.

Novel closed-form analytical solutions and modulation instability spectrum induced by the Salerno equation describing nonlinear discrete electrical lattice via symbolic computation

A nonlinear electrical lattice is an effective experimental tool for dealing with nonlinear dispersive media and creating possibilities for the realistic modeling of electrical soliton. In this work, we discuss the dynamical behavior of the governing (1+1)-dimensional time–space Salerno equation, which describes the discrete electrical lattice with nonlinear dispersion. The different kinds of solutions in the form of periodic, Jacobi elliptic, and exponential functions are obtained by the two analytical approaches, namely the modified generalized exponential rational function method (MGERFM) and the extended sinh-Gordon equation expansion method (shGEEM). The MGERF approach is applied to construct exact solitary wave solutions containing trigonometry, hyperbolic and exponential functions, while the extended shGEE technique is employed to obtain periodic wave solutions containing hyperbolic and Jacobi elliptic functions via performing the symbolic computation in the software package MATHEMATICA. Moreover, by means of standard linear-stability analysis, the modulation instability (MI) and the MI gain spectrum is analytically computed for the considered equation. By selecting suitable parametric values, solution profiles are portrayed in order to make the solutions physically relevant. These results yield periodic waves, kink waves, multi-soliton, and their interactions. By comparing the obtained solutions to the previous findings, we can reveal the novelty of the solutions. In addition, all the obtained solutions were verified by substituting them back into the governing equation using the Mathematica software. The acquired results are significant for the study of wave propagation, signal transmission, and applications of super-transmission phenomena.

Dynamical behavior of solitons of the (2+1)-dimensional Konopelchenko Dubrovsky system.

Utilizing nonlinear evolution equations (NEEs) is common practice to establish the fundamental assumptions underlying natural phenomena. This paper examines the weakly dispersed non-linear waves in mathematical physics represented by the Konopelchenko-Dubrovsky (KD) equations. The [Formula: see text]-expansion method is used to analyze the model under consideration. Using symbolic computations, the [Formula: see text]-expansion method is used to produce solitary waves and soliton solutions to the [Formula: see text]-dimensional KD model in terms of trigonometric, hyperbolic, and rational functions. Mathematica simulations are displayed using two, three, and density plots to demonstrate the obtained solitary wave solutions' behavior. These proposed solutions have not been documented in the existing literature.

Analytical solutions to time-space fractional Kuramoto-Sivashinsky Model using the integrated Bäcklund transformation and Riccati-Bernoulli sub-ODE method

<abstract><p>This paper solves an example of a time-space fractional Kuramoto-Sivashinsky (KS) equation using the integrated Bäcklund transformation and the Riccati-Bernoulli sub-ODE method. A specific version of the KS equation with power nonlinearity of a given degree is examined. Using symbolic computation, we find new analytical solutions to the current problem for modeling many nonlinear phenomena that are described by this equation, like how the flame front moves back and forth, how fluids move down a vertical wall, or how chemical reactions happen in a uniform medium while they oscillate uniformly across space. In the field of mathematical physics, the Riccati-Bernoulli sub-ODE approach is shown to be a valuable tool for producing a variety of single solutions.</p></abstract>

Soliton cluster solutions of nonlinear Schrödinger equations with variable coefficients in Bessel lattice

The goal of this article is to obtain soliton cluster solutions of (2 + 1)-dimensional nonlinear variable coefficient Schrödinger equations through computerized symbolic computation. By applying the self-similarity method, one soliton solution is constructed for the NLS equations with variable coefficient, which provide a model of the propagation of the soliton waves. Consequently, the solitonary cluster solution is achieved in different structures. Additionally, the propagation of the obtaining solitonary cluster solutions is analyzed and discussed. The results are useful to explain the soliton phenomena in nonlinear optics.

Bifurcation solitons, Y-type, distinct lumps and generalized breather in the thermophoretic motion equation via graphene sheets

Learned from wrinkle wave motions, we concentrated on bifurcation phenomena in substrate-supported graphene sheets by obtaining the bifurcation solitons of thermophoretic motion equation. We find new soliton solutions by applying the bilinear technique and correctly choosing the auxiliary function involved in the bilinear form. Use of the Hirota bilinear technique combined with symbolic computation yields lump solutions as well as lump-type solutions. The governing model is subject to computation of the lump one strip, lump two strip, Y-type, lump periodic solution, generalised breather, and lump periodic solution. When simulating waves in different physical systems, such as water waves, fibre optics, and plasma physics, lump and lump type solutions are crucial. Rogue waves are studied in relation to large and unforeseen ocean wave phenomena that might affect coastal buildings and ship safety. In communication systems, including optical fibres, generalised breathers are used to comprehend and regulate pulse propagation.

A Novel Approximation Approach for the Analytical Solution of the Flow of Micropolar Fluid Through a Permeable Channel

An attempt is made in this study to investigate the problem of micropolar fluid flow in a porous medium theoretically. Employing the Berman’s similarity solution, the model equations governing the flow is transformed into a set of nonlinear ordinary differential equation and solved using Temimi-Ansari method. Expressions for the velocity and micro-rotation profiles are obtained under the impressions of diverse parameters affecting the flow problem. Using symbolic computation software Mathematica, the nondimensional equations are solved numerically using the Keller Box scheme. Comparison between the analytical solution obtained by TAM and the numerical result are compared with results in literature to observe rapid convergence. Findings from the study showed in the presence of

Analysis of peakon-like soliton solutions: (3+1)-dimensional Fractional Klein-Gordon equation

<abstract><p>In this study, we investigate the fundamental properties of ($ 3+1 $)-$ D $ Fractional Klein-Gordon equation using the sophisticated techniques of Riccatti-Bornoulli sub-ODE approach with Backlund transformation. Using a more stringent criterion, our study reveals new soliton solutions that have peakon-like properties and unique cusp features. This research provides significant understanding of the dynamic behaviours and odd events related to these solutions. This work is important because it helps to elucidate the complex dynamics that exist within physical systems, which will benefit many different scientific fields. Our method is used to examine the existence and stability of compactons and kinks in the context of actual physical systems. Under a double-well on-site potential, these structures are made up of a network of connected nonlinear pendulums. Both $ 2D $ and contour plots produced by parameter changes provide as clear examples of the efficiency, simplicity, and conciseness of the computational method used. The results highlight how flexible this approach is, and demonstrate how symbolic calculations broaden its application to more complex events. This work offers a useful framework and studying intricate physical systems, as well as a flexible computational tool that may be used in a variety of scientific fields.</p></abstract>