Complete Discrimination System For Polynomial Research Articles

Exact chirped solutions and Chaotic behaviors of the high-order nonlinear Schrödinger equation with non-Kerr nonlinear terms and cubic-quintic-septic nonlinearities

Abstract In this paper, we obtain the exact solutions and corresponding nonlinear chirps of the high-order nonlinear Schrödinger equation with non-Kerr nonlinear terms and cubic-quintic-septic nonlinearities based on the trial equation method and the complete discrimination system for polynomial. Specifically, we systematically solve all exact chirped solutions of the equation in its general form. Moreover, we discuss the chaotic behaviors of the model while considering the external perturbation terms.

Chaotic Pattern and Solitary Solutions for the (21)-Dimensional Beta-Fractional Double-Chain DNA System

The dynamical behavior of the double-chain deoxyribonucleic acid (DNA) system holds significant implications for advancing the understanding of DNA transmission laws in the realms of biology and medicine. This study delves into the investigation of chaos patterns and solitary wave solutions for the (2+1) Beta-fractional double-chain DNA system, employing the theory of planar dynamical systems and the method of complete discrimination system for polynomials (CDSP). The results demonstrate a diverse spectrum of solitary wave solutions, sensitivity to perturbations, and manifestations of chaotic behavior within the system. Through the utilization of the complete discrimination system for polynomials, a multitude of novel solitary wave solutions, encompassing periodic, solitary wave, and Jacobian elliptic function solutions, were systematically constructed. The influence of Beta derivatives on the solutions was elucidated through parameter comparison analysis, emphasizing the innovative nature of this study. These findings underscore the potential of this system in unraveling various biologically significant DNA transmission mechanisms.

Dynamical properties and travelling wave analysis of Rangwala–Rao equation by complete discrimination system for polynomials

Dynamical properties and travelling wave analysis of Rangwala–Rao equation by complete discrimination system for polynomials

Optical solitons of the (2+1)-dimensional nonlinear Schrödinger equation with spatio-temporal dispersion in quadratic-cubic media

The major objective of this paper is to study the perturbed nonlinear Schrödinger equation with spatio-temporal dispersion in (2+1)-dimensional by using the complete discrimination system for polynomial. This paper discusses the propagation patterns in nonlinear fibers with quadratic-cubic nonlinearity. Additionally, a host of optical wave solutions are obtained, where solitary wave patterns and elliptic functions double periodic patterns are newly received. Besides, some two dimensional figures of solutions are illustrated in parameter space. All these solutions and figures are available to investigate the propagation dynamics of optical solitons in optical fibers.

The Dynamical study and analysis of diverse optical soliton solutions to Chen-Lee-Liu equation by implementing complete discrimination system for polynomial method

Optical solitons are discovered with Chen-Lee-Liu equation (CLLE). CLLE explains the transportation of optical pulses in fibre optics. The goal of the current study is to examine various analytical soliton solutions of CLLE via complete discrimination system for polynomial (CDSP) approach. These consists of elliptic, periodic, irrational, hyperbolic and exponential function solutions. This article also explores the Hamiltonian and topological properties of CLLE. The phase portraits of dynamical system provides the evidence of presence and reliability of topological properties of solutions. CDSP technique is not only suitable for obtaining soliton solutions but for also for conducting qualitative analysis. We also create 3D and 2D graphs by utilising the appropriate values of related parameters to give the physical interpretation of derived chirped solutions.

Exact chirped solutions of perturbed Chen-Lee-Liu equation with refractive index

We study the exact chirped solutions of the perturbed Chen-Lee-Liu equation with a refractive index. Exact chirped solutions and their corresponding chirps are obtained using trial equation method and complete discrimination system for polynomial. Illustrations are presented under certain parameter conditions. Chen-Lee-Liu model describes pulse propagation in optical fibers. Studying chirped solutions to the perturbed Chen-Lee-Liu equation with a refractive index could improve the understanding of physical phenomena and may have applications in the design of optical fibers.

Optical wave propagation for the sixth-order nonlinear partial differential equation in optical fiber

In this paper, a sixth-order nonlinear partial differential equation which can be used to describe the propagation pulse in optical fibers is studied by using the methods of trial equation and complete discrimination system for polynomial. We have obtained some exact optical wave solutions, including singular solutions, solitary wave solutions, and elliptic function double periodic solutions, which can be completely classified in parameter space.

Traveling wave solutions of a coupled Schrödinger-Korteweg-de Vries equation by the generalized coupled trial equation method

The coupled Schrödinger-Korteweg-de Vries equation is a critical system of in nonlinear evolution equations. It describes various processes in dusty plasma, such as Langmuir waves, dust-acoustic waves, and electromagnetic waves. This paper uses the generalized coupled trial equation method to solve the equation. By the complete discrimination system for polynomial, a series of exact traveling wave solutions are obtained, including discontinuous periodic solutions, solitary wave solutions, and Jacobian elliptical function solutions. In addition, to determine the existence of the solutions and understand their properties, we draw three-dimensional images of the modules of the solutions with Mathematica. We obtain more comprehensive and accurate solutions than previous studies, and the results give the system more profound physical significance.

Optical wave solutions of the nonlinear Schrödinger equation with an anti-cubic nonlinear in presence of Hamiltonian perturbation terms

In this paper, we study the nonlinear Schrödinger equation with an anti-cubic nonlinear with Hamiltonian perturbation terms by using the complete discrimination system for polynomial. The main result is a complete classification of traveling wave patterns in parameters space from which we can recognize the form of optical wave only based on the physical parameters condition.

Optical wave propagation for the resonant nonlinear Schrödinger equation with arbitrary refractive index in optical fiber

In this paper, we study the resonant nonlinear Schrödinger equation which describes the propagation of optical solitons in optical fibers. Through the trial equation method and the complete discrimination system for polynomial we obtain abundant optical wave propagation patterns for the model and divide them into three categories. Besides, we analyze the topological stability of these patterns. Ultimately, we give physical representations of optical wave patterns and draw some figures to show their temporal-space structures.

Optical solitons of Sasa–Satsuma equation in birefringent fibers

This paper studies the propagation of optical solitons of the Sasa–Satsuma model in birefringent fibers. By the complete discrimination system for polynomial, we get three types of optical wave patterns for the model, where singular rational patterns and elliptic functions double periodic patterns are new propagation patterns. Besides, we show the temporal-space structures and propagation properties for some patterns with specific parameter values by drawing graphs. These results are beneficial to study the nonlinear dynamics of optical wave propagation in optics.

Propagation of optical solitons for Kudryashov’s law with dual form of generalized non-local nonlinearity

In this paper, we study the optical solitons for Kudryashov’s law with dual form of generalized non-local nonlinearity in optical fibers. Through the trial equation method and the complete discrimination system for polynomial, we get various optical wave patterns for the model. Among those, the new patterns including rational function pattern, trigonometric function pattern and elliptic function pattern. In addition, we draw some graphs to show the propagation of optical wave patterns and the temporal-space structures.

The Classification of the Exact Single Travelling Wave Solutions to the Constant Coefficient KP-mKP Equation Employing Complete Discrimination System for Polynomial Method

The purpose of this article is to explore different types of solutions for the Kadomtsev-Petviashvili-modified Kadomtsev-Petviashvili (KP-mKP) equation which is termed as KP-Gardner equation, extensively used to model strong nonlinear internal waves in ( 1 + 2 )-dimensions on the stratified ocean shelf. This evolution equation is also used to describe weakly nonlinear shallow-water wave and dispersive interracial waves traveling in a mildly rotating channel with slowly varying topography. Introducing Liu’s approach regarding the complete discrimination system for polynomial and the trial equation technique, a set of new solutions to the KP-mKP equation containing Jacobi elliptic function have been derived. It is found that these analytical solutions numerically exhibit different nonlinear structures such as solitary waves, shock waves, and periodic wave profiles. The reliability and effectiveness are confirmed from the numerical graphs of the solutions. Finally, the existence and validity of the various topological structures of the solutions are confirmed from the phase portrait of the dynamical system. Based on this investigation, it is confirmed that the method is not only suited for obtaining the classification of the solutions but also for qualitative analysis, which means that it can also be extended to other fields of application.

Optical solitons in optical metamaterials with anti-cubic nonlinearity

What the paper will show is how we attempt to study optical solitons of the nonlinear Schrödinger (NLS) equation with anti-cubic (AC) nonlinearity which describes the propagation of pulses in optical metamaterials. By using trial equation method and the complete discrimination system for polynomial, we obtain all the possible forms of optical propagation patterns for the model. Finally, we draw the graphs for several typical patterns and show their temporal-space structures.

The traveling wave mode for nonlinear Biswas–Milovic equation in magneto-optical wave guide coupling system with Kudryashov’s law of refractive index

This paper studies the traveling wave modes for nonlinear Biswas–Milovic equation in magneto-optical wave guide coupling system with Kudryashov’s law of refractive index. By the complete discrimination system for polynomial and trial equation method, we derive a series of traveling wave solutions of the coupling system. These results show the abundant propagation patterns. In particular, by analyzing the topological stability and dynamic behavior we get four stable modes of this system. Under some special parameters, we give the concrete representations of solutions.

Thirty traveling wave solutions to the systems of ion sound and Langmuir waves

Nine kinds of general solutions of certain popular first-order ordinary differential equation are obtained by direct calculations. Compared with complete discrimination system for polynomials, our classifications of solutions straightly depend on the coefficients of the ordinary differential equation and hence are easier to be employed. According to the nine kinds of solutions, we establish thirty traveling wave solutions to the coupled systems of ion sound and Langmuir waves, including rational function solution, exponential function solution, trigonometric function solutions, hyperbolic function solutions and Jacobi elliptic function solutions. To the best of our knowledge, many of them are new. Solutions discussed in two recent articles are showed to be equivalent to some special cases of our thirty traveling wave solutions. Our solutions include the kink and anti-kink soliton solutions, dark and bright soliton solutions as well as periodic soliton solutions and solitary wave solutions. We believe that these solutions will be of great use to researchers concerning with nonlinear physical phenomena. In addition, the famous (1+1)-dimensional dispersive long wave equations is taken to illustrate the applicability of these thirty solutions.

Exact Solutions for the (2 + 1)-Dimensional Nizhnik-Novikov-Veselov System

Finding the exact solutions of nonlinear partial differential equations plays an important part in nonlinear problems. We use the complete discrimination system for polynomials [1-6] to get exact solutions for the (2 + 1)-dimensional Nizhnik-Novikov-Veselov system in this paper. We divide different situations to discuss the solutions and get three kinds of solutions.

Exact Solution of Yu-Toda-Sasa-Fukugama Potential Equation In (3+1) Dimension

This article solves the Yu-Toda-Sasa-Fukugama potential equation in (3+1) dimension [1-4]. First of all, we carry out the traveling wave transformation on the equation. Then, the equation can be transformed into integral form. At last, we use the complete discrimination system for polynomial [5-12] to obtain all exact solutions of the equation.

Classifying Exact Traveling Wave Solutions to the Coupled-Higgs Equation

By the complete discrimination system for polynomials, we classify exact traveling wave solutions to the Coupled-Higgs Equation.

Analysis of Critical Velocities for an Infinite Timoshenko Beam Resting on an Elastic Foundation Subjected to a Harmonic Moving Load

Critical velocities are investigated for an infinite Timoshenko beam resting on a Winkler-type elastic foundation subjected to a harmonic moving load. The determination of critical velocities ultimately comes down to discrimination of the existence of multiple real roots of an algebraic equation with real coefficients of the 4th degree, which can be solved by employing Descartes sign method and complete discrimination system for polynomials. Numerical calculations for the European high-speed rail show that there are at most four critical velocities for an infinite Timoshenko beam, which is very different from those gained by others. Furthermore, the shear wave velocity must be the critical velocity, but the longitudinal wave velocity is not possible under certain conditions. Further numerical simulations indicate that all critical velocities are limited to be less than the longitudinal wave velocity no matter how large the foundation stiffness is or how high the loading frequency is. Additionally, our study suggests that the maximum value of one group velocity of waves in Timoshenko beam should be one “dangerous” velocity for the moving load in launching process, which has never been referred to in previous work.