Weierstrass Elliptic Function Research Articles

New exact solitary waves for the Sasa-Satsuma model with variable coefficients

In this paper, we investigate the variable coefficients Sasa-Satsuma model, which can describe the propagation of a light pulse in a cylindrical fiber. We study this model and obtain rich solutions using two separate methods. We obtain analytical Weierstrass elliptic function solutions using the Weierstrass elliptic function expansion method. Some Jacobi elliptic function solutions are obtained using the modified Jacobi elliptic function expansion method. When the Jacobi elliptic function degenerates, we obtain the corresponding trigonometric, hyperbolic function solutions and periodic solutions. We also try to take the coefficients of the equation as some functions and obtain some more complicated exact solutions, which have not appeared in previous studies. Finally, we simulate some waveform diagrams of the solutions using the computer software Mathematica and obtain periodic waves, bright and dark soliton, double solitons and some complex periodic waves. With these waveform diagrams, we can observe the dynamical behavior of the solutions more clearly.

Deep and hand-crafted features based on Weierstrass elliptic function for MRI brain tumor classification

Abstract Advances in medical imaging and artificial intelligence have led to improvements in diagnosis and non-invasive patient examination accuracy. The use of the fundamental method for Magnetic resonance imaging (MRI) brain scans as a screening tool has increased in recent years. Numerous studies have proposed a variety of feature extraction methods to classify the abnormal growths in MRI scans. Recently, the MRI texture analysis and the use of deep features have resulted in remarkable performance improvements in the classification and diagnosis of challenging pathologies, like brain tumors. This study proposes employing a handcrafted model based on Weierstrass elliptic function (WEF) and deep feature based on DenseNet-201 to classify brain tumors in MRI images. By calculating the energy of each individual pixel, the Weierstrass coefficients of the WEF are used to capture high frequency image details of the brain image. The WEF mode works to extract the nonlinear patterns in MRI images based on the probability of each pixel. While the dense connectivity of DenseNet-201’s architecture allows to learn features at multiple scales and abstraction levels. These features are passed to support vector machines classifier, which classifies the brain tumor. The results of classification accuracy achieved is 98.55% for combined features of WEF with trained DenseNet-201. Findings on the brain tumor segmentation dataset indicated that the proposed method performed better than alternative techniques for classifying brain tumors.

Dimensionless dynamics: Multipeak and envelope solitons in perturbed nonlinear Schrödinger equation with Kerr law nonlinearity

In this work, the dimensionless form of the improved perturbed nonlinear Schrödinger equation with Kerr law of fiber nonlinearity is solved for distinct exact soliton solutions. We examined the multi-wave solitons and rational solitons of the governing equation using the logarithmic transformation and symbolic computation using an ansatz functions approach. Multi-wave solitons in fluid dynamics describe the situation in which a fluid flow shows several different regions (or peaks) of high concentration or intensity of a particular variable (e.g., velocity, pressure, or vorticity). Multi-wave solitons in turbulent flows might indicate the existence of several coherent structures, like eddies or vortices. These formations are areas of concentrated energy or vorticity in the turbulent flow. Understanding how these peaks interact and change is essential to comprehending the energy cascade and dissipation in turbulent systems. Furthermore, a sub-ordinary differential equation approach is used to create solutions for the Weierstrass elliptic function, periodic function, hyperbolic function, Chirped free, dark-bright (envelope solitons), and rational solitons, as well as the Jacobian elliptic function, periodic function, and rational solitons. Also, as the Jacobian elliptic function's' modulus m approaches values of 1 and 0, we find trigonometric function solutions, solitons-like solutions, and computed chirp free-solitons. Envelope solitons can arise in stratified fluids and spread over the interface between layers, such as layers in the ocean with varying densities. Their research aids in the management and prediction of wave events in artificial and natural fluid settings. In fluids, periodic solitons are persistent, confined wave structures that repeat on a regular basis, retaining their form and velocity over extended distances. These structures occur in a variety of settings, including internal waves in stratified fluids, shallow water waves, and even plasma physics.

Bounds for Weierstrass Elliptic Function and Jacobi Elliptic Integrals of First and Second Kinds

The Weierstrass elliptic function is presented in connection with the Jacobi elliptic integrals of first and second kinds leading to comparing coefficients appearing in the Laurent series expansion with those of Eisenstein series for the cubic polynomial in the meromorphic Weierstrass function. It is unified in the formulation the Weierstrass elliptic function with Jacobi elliptic integral by considering motion of a unit mass particle in a cubic potential in terms of bounded and unbounded velocities and the time of flight with imaginary part in the complex function playing a major role. Numerical tools box used are the Konrad-Gauss quadrature and Runge-Kutta fourth order method.

On the Van der Waals model on granular matters with truncated M-fractional derivative

In this work, exact solutions of the Van der Waals model (vdWm) are investigated with a new algebraic analytical method. The closed-form analysis of the vdW equation arising in the context of the fluidized granular matter is implemented under the effect of time-fractional M-derivative. The vdWm is a challenging problem in the modelling of molecules and materials. Noncovalent Van der Waals or dispersion forces are frequent and have an impact on the structure, dynamics, stability, and function of molecules and materials in biology, chemistry, materials science and physics. The auxiliary equation which is known as a direct analytical method is constructed for the nonlinear fractional equation. The process includes a transformation based on Weierstrass and Jacobi elliptic functions. Wave solutions of the model are analytically verified for the various cases. Then, graphical patterns are presented to show the physical explanation of the model interactions. The achieved solutions will be of high significance in the interaction of quantum-mechanical fluctuations, granular matter and other areas of vdWm applications.

Investigating exact solutions, sensitivity, and chaotic behavior of multi-fractional order stochastic Davey–Sewartson equations for hydrodynamics research applications

Our research aims to explore the fractional derivative of the Stochastic Davey–Stewartson equation (FDSDSEs) and make contributions in seven aspects: Firstly, we discovered previously unknown exact solutions for FDSDSEs, including Weierstrass elliptic function solutions. These findings provide new insights for our understanding and analysis of these equations. Secondly, we analyzed the effect of noise on stochastic solutions. We found that higher noise intensity can disrupt patterns and result in flatter surfaces. However, we observed that multiplicative noise can stabilize the solution to some extent. This reveal the role of noise in stochastic dynamics and has important implications for wave behavior in the ocean environment. Thirdly, we emphasized the impact of fractional derivatives on noise dynamics. Our research indicates the importance of considering the order of derivatives α in analysis and modeling. Different order derivatives have different effects on noise, which is significant for ocean researchers in selecting appropriate derivatives to study stochastic systems. Fourthly, we compared the effects of different fractional derivatives on noise in stochastic solutions. We found that choosing appropriate derivative orders can more accurately describe wave behavior in the ocean environment. This discovery has important implications for improving wave models and predicting ocean waves. Fifthly, we studied the influence of tilted wave variations in FDSDSE solutions. We observed spiral-like patterns, providing new insights into wave propagation in the ocean. This is significant for understanding wave behavior in the ocean and relevant ocean engineering applications. Sixthly, we explored the phase diagrams and chaotic behavior of FDSDSE through sensitivity analysis and perturbation factors. This enhances our understanding of practical applications and control strategies in ocean hydrodynamics research and is also important for delving into the dynamic behavior and phase diagrams of these equations. Finally, we compared our research results with well-known wave phenomena in the ocean through visualization, highlighting the practical relevance of our study. These comparisons involve random propagation of ocean solitary waves, twisting propagation of ocean waves, and rotating propagation of solitary waves, providing some inspiration for advancing the fields of ocean science and engineering.

Breathers of Gerdjikov-Ivanov equation under background of elliptic function

As one specific type of local solutions of nonlinear evolution equation, the breathers have the characteristic of envelope oscillation structure. This kind of oscillation is periodic. According to the periodicity of the distribution and evolution directions, there are three kinds of breathers, namely, the Kuznetsov-Ma breather (KMB), the Akhmediev breather (AB), and the general breather (GB). In recent years, the propagation of envelope breathers under the periodic background has been observed in many nonlinear physical fields, including nonlinear optical fibers and hydrodynamics. It is believed that the breathers can arise due to the modulational instability of the periodic waves, and they demonstrate many rich physical properties and dynamic behaviors of interactions. Therefore, recently great attention has been paid to the breathers under the periodic background in nonlinear science. As an important integrable model, the Gerdjikov-Ivanov (GI) equation can be used to describe various nonlinear phenomena in many physical fields such as in the quantum field theory, weak nonlinear dispersive water wave, and nonlinear optics. It is very meaningful to solve various types of solutions of this model to describe the propagation of nonlinear waves. As far as we know, the breather solutions for the GI equation have not been given under the elliptic function background. In this study, firstly, elliptic function solutions of the GI equation are solved by the modified squared wave (MSW) function approach and the traveling wave transformation. Then, we obtain the basic solution of the Lax pair corresponding to the Jacobi elliptic function seed solution. Based on the elliptic function transformation formulas and the integral formulas, the potential function solution can be expressed in terms of the Weierstrass elliptic function. Secondly, by the once iterated Darboux transformation, three types of breather solutions under the elliptic function background are constructed including the GB, the KMB and the AB. In addition, we analyze the dynamic behaviors of these three kinds of breathers, and present their three-dimensional space-time structures. By the twice iterated Darboux transformation, under the dn-periodic background we exhibit three types of interactions between two breathers, i.e. a GB and a KMB, an AB and a KMB, and a GB and an AB. Finally, we also present three types of interactions between two breathers under the general periodic background.

Effects of Wiener process on analytical wave solutions for (3+1) dimensional nonlinear Schrödinger equation using modified extended mapping method

In this study, we investigate the soliton solutions for stochastic (3+1) dimensional nonlinear Schrödinger equation (NLSE). Our Study mainly depends on applying the modified extended mapping method to construct various and novel stochastic solutions for the proposed model. These solutions including (dark, bright, and singular) solitons. Moreover, singular periodic solutions, Weierstrass elliptic function solutions, and Jacobi elliptic function (JEF) solutions are raised. For a variety of nonlinear stochastic partial differential equations, this method offers a practical and effective method for determining exact stochastic solutions. The stochastic noise effects forcing the wave to behave differently from deterministic cases prognosticated. To illustrate the physical characteristics of the established stochastic solutions, some selected solutions are presented graphically. We used Mathematica (11.3) packages to find the coefficients and Matlab (R2015a) packages to plot the graphs.

Nonlinear dissipative wave trains in a system of self-propelled particles

The paper addresses the existence of modulated nonlinear periodic wave trains in a system of self-propelled particles (SPPs). The reductive perturbation method reduces the model hydrodynamics equations to a one-dimensional (1D) complex Ginzburg-Landau (CGL) equation. The modulational instability (MI) phenomenon is studied, where an expression for the instability growth rate is proposed. The latter is used to discuss regions of parameters where trains of solitonic waves are likely to be obtained. This is highly influenced by the values of the variances of Gaussian noise in self-diffusion and binary collision. Solutions for the CGL equations are also studied via the Porubov technique, using a combination of Jacobi and Weierstrass elliptic functions. Wave propagation in the self-propelled particles flock includes modulated nonlinear wave trains, nonlinear spatially localized periodic patterns, and continuous waves.

Semi-analytical estimates for the chaotic diffusion in the Second Fundamental Model of Resonance. Application to Earth’s navigation satellites

We discuss the applicability of the Melnikov and Landau–Teller theories in obtaining semi-analytical estimates of the speed of chaotic diffusion in systems driven by the separatrix-like stochastic layers of a resonance belonging to the ‘second fundamental model’ (SFM) (Henrard and Lemaitre, 1983). Stemming from the analytic solution for the SFM in terms of Weierstrass elliptic functions, we introduce stochastic Melnikov and Landau–Teller models allowing to locally approximate chaotic diffusion as a sequence of uncorrelated ‘jumps’ observed in the time series yielding the slow evolution of an ensemble of trajectories in the space of the adiabatic actions of the system. Such jumps occur in steps of one per homoclinic loop. We show how a semi-analytical determination of the probability distribution of the size of the jumps can be arrived at by the Melnikov and Landau–Teller approximate theories. Computing also the mean time required per homoclinic loop, we arrive at estimates of the chaotic diffusion coefficient in such systems. As a concrete example, we refer to the long-term diffusion of a small object (e.g. Earth navigation satellite or space debris) within the chaotic layers of the so-called 2g+h lunisolar resonance, which is of the SFM type. After a suitable normal form reduction of the Hamiltonian, we compute estimates of the speed of diffusion of these objects, which compare well with the results of numerical experiments.

Extraction of solitons in magneto–optic waveguides for coupled NLSEs with Kudryashov's law of nonlinearity via modified extended direct algebraic method

In this paper, the modified extended direct algebraic technique is used for determining new and various traveling wave solutions to the coupled nonlinear Schrödinger equation (NLSE) with Kudryashov's law of nonlinearity. This model describes the propagation of optical solitons through magneto-optic wave-guides. Various solutions such as dark solitons, singular solitons, hyperbolic solutions, singular periodic solutions, rational solutions, Jacobi elliptic solutions and Weierstrass elliptic function solution are raised. To demonstrate the physical behavior of the obtained solutions, 3D and 2D graphs are introduced.

Dispersive optical soliton solutions in birefringent fibers with stochastic Kaup–Newell equation having multiplicative white noise

In this article, we study for the first time the stochastic Kaup–Newell equation in birefringent fibers. Three integration algorithms are applied, namely, the unified Riccati equation expansion method, the addendum to Kudryashov's method, and the addendum to modified sub‐ODE method. Many types of optical soliton solutions such as Jacobi‐elliptic function solutions, Weierstrass elliptic function solutions, bright solitons, dark solitons, singular solitons, and periodic function solutions are obtained. Numerical schemes give a visual perspective to the solutions derived analytically.

Weierstrass elliptic function solutions and degenerate solutions of a variable coefficient higher-order Schrödinger equation

In this paper, the auxiliary equation method is used to study the Weierstrass elliptic function solutions and degenerate solutions of the variable coefficient higher order Schrödinger equation, including Jacobian elliptic function solutions, trigonometric function solutions and hyperbolic function solutions. The types of solutions of the variable coefficient higher-order Schrödinger equation are enriched, and the method of seeking precise and accurate solutions is extended. It is concluded that the types of degenerate solutions are related to the coefficients of the equation itself when the degenerate solutions are obtained from the solutions of the Weierstrass elliptic functions. In addition, the solutions form of the equation is extended from the power series expansion form to the Laurent series expansion form, and the corresponding solutions are obtained. After the conversion formula between the Weierstrass elliptic function solutions and the Jacobian elliptic function solutions is constructed, the Jacobian elliptic function solutions of the higher order Schrödinger equation with variable coefficients are also obtained. These have not been previously studied.

Dynamics of newly created soliton solutions via Atangana–Baleanu Fractional (ABF) for system of (ISALWs) equations

In this work, newly created soliton solutions for ion sound and Langmuir waves with an Atangana–Baleanu fractional (ABF) are given. Symbolic software is used to perform the unified solver method (USM) and Weierstrass elliptic function method (WEFM) in order to solve this model. In terms of hyperbolic functions, extended trigonometric functions, and so forth. Single-wave solutions that were entirely novel and universal were attained. The way the soliton solutions behaved in connection with the two-dimensional (2D) and three-dimensional (3D) graphics was also investigated. The dynamic investigation of the newly created soliton solutions reveals that they have several soliton forms like single-soliton, bell-shaped and mixed-form soliton profiles. This strategy is viewed as promising for handling a range of ABF evolution systems.

Analytical solutions and dynamic behaviors to synchronous oscillation of same bubbles at vertices of cuboid and rectangle

The present work focuses on the nonlinear dynamics of the synchronous oscillating multiple bubbles in two typical spatial locations, namely, cuboid and rectangle arrangements. The governing equation for such synchronous oscillating multiple bubbles is derived from a modified Rayleigh–Plesset equation. Theoretical results including the collapse time and analytical solution (in three forms) for multiple vapor bubbles, as well as the maximum/minimum radii, oscillation period, and analytical solution in the form of Weierstrass elliptic function for multiple gas-filled ones, are provided. On the basis of these results, we not only study the dynamic characteristics of multi-bubbles straightforwardly but also carefully observe a series of evolution behaviors of bubbles when the number of bubbles decreases gradually on the order of 8→4→2→1. It should be pointed out that we also compare the multi-bubble behaviors between the general cuboid/rectangle arrangements and the corresponding cube/square arrangements under two reasonable restrictions, respectively. Furthermore, the limiting behaviors of the synchronous oscillating multiple gas-filled bubbles are discussed as the initial pressure of the gas in bubble approaches to zero.

Kerr geodesics in terms of Weierstrass elliptic functions

Kerr geodesics in terms of Weierstrass elliptic functions

Solution Structures of an Electrical Transmission Line Model with Bifurcation and Chaos in Hamiltonian Dynamics

Employing the Riccati–Bernoulli sub-ODE method (RBSM) and improved Weierstrass elliptic function method, we handle the proposed [Formula: see text]-dimensional nonlinear fractional electrical transmission line equation (NFETLE) system in this paper. An infinite sequence of solutions and Weierstrass elliptic function solutions may be obtained through solving the NFETLE. These new exact and solitary wave solutions are derived in the forms of trigonometric function, Weierstrass elliptic function and hyperbolic function. The graphs of soliton solutions of the NFETLE describe the dynamics of the solutions in the figures. We also discuss elaborately the effects of fraction and arbitrary parameters on a part of obtained soliton solutions which are presented graphically. Moreover, we also draw meaningful conclusions via a comparison of our partially explored areas with other different fractional derivatives. From our perspectives, by rewriting the equation as Hamiltonian system, we study the phase portrait and bifurcation of the system about NFETLE and we also for the first time discuss sensitivity of the system and chaotic behaviors. To our best knowledge, we discover a variety of new solutions that have not been reported in existing articles [Formula: see text], [Formula: see text]. The most important thing is that there are iterative ideas in finding the solution process, which have not been seen before from relevant articles such as [Tala-Tebue et al., 2014; Fendzi-Donfack et al., 2018; Ashraf et al., 2022; Ndzana et al., 2022; Halidou et al., 2022] in seeking for exact solutions about NFETLE.

On the propagation of cnoidal wave and overtaking collision of slow shear Alfvén solitons in low β magnetized plasmas

The overtaking collisional phenomenon of slow shear Alfvén solitons are studied in a low beta (β = kinetic pressure/magnetic pressure) collisionless, magnetized plasma consisting of electron and ion fluids. By employing a reductive perturbation technique, the Korteweg–de Vries (KdV) equation is deduced for investigating the nonlinear slow shear Alfvén wave. Before embarking on the study of the overtaking collisions, the stability analysis of the KdV equation is studied using the bifurcation theory. Also, a nonlinear periodic solution of the KdV equation is derived for the first time in the Weierstrass elliptic function formula. Moreover, the condition for converting the Weierstrass elliptic function expression to soliton is discussed. Furthermore, it is found that only density dip (rarefactive) solitons are formed in the super-Alfvénic regime. The next step includes the use of the Hirota bilinear method, which results in two and three shear Alfvén soliton solutions and their subsequent phase shifts. The influence of the plasma parameters on the amplitude as well as width of the slow shear Alfvén wave solitons are examined analytically and numerically. We also find out the profiles of overtaking interaction of slow shear Alfvén dip solitons having different amplitudes and speeds numerically. This study is important for understanding the phenomena of nonlinear slow shear Alfvén wave structures both in space and in laboratory plasmas.

Cutting the Coon amplitude

The Coon amplitude is a q-deformed generalization of the Veneziano amplitude exhibiting a semi-infinite sequence of poles that converge on an accumulation point, from which a branch cut emerges. A number of recent papers have provided compelling evidence that the residues of this amplitude satisfy the positivity requirements imposed by unitarity. This paper investigates whether positivity is also satisfied along the branch cut. It is demonstrated for a wide range of q-values that positivity violations occur in a region of the branch cut exponentially close to the accumulation point according to a scale set by q. The closing section of the paper discusses possible interpretations of this fact and strategies for excising negativity from the partial wave coefficients.An appendix presents derivations of instrumental identities relating the q-gamma and q-polygamma functions to the Weierstrass elliptic and quasiperiodic functions.

Operations for D-Algebraic Functions

A function is differentially algebraic (or simply D-algebraic) if there is a polynomial relationship between some of its derivatives and the indeterminate variable. Many functions in the sciences, such as Mathieu functions, the Weierstrass elliptic functions, and holonomic or D-finite functions are D-algebraic. These functions form a field, and are closed under composition, taking functional inverse, and derivation. We present implementation for each underlying operation. We also give a systematic way for computing an algebraic differential equation from a linear differential equation with D-finite function coefficients. Each command is a feature of our Maple package NLDE available at https://mathrepo.mis.mpg.de/DAlgebraicFunctions/NLDEpackage.