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

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.

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

Derivation of optical solitons and other solutions for nonlinear Schrödinger equation using modified extended direct algebraic method

By establishing a complete discrimination system for polynomials, the problem of complete root classification for polynomials with complex coefficients is utterly solved, furthermore, the algorithm obtained is made into a general program in Maple, which enables the complete discrimination system and complete root classification of a polynomial to be automatically generated by computer, without any human intervention. Besides, by using the automatic generation of root classification, a method to determine the positive definiteness of a polynomial in one or two indeterminates is automatically presented.

In this study, the stochastic Chen–Lee–Liu equation is considered numerically and analytically which is forced by the multiplicative noise in the Itô sense. The Chen–Lee–Liu equation is a special type of Schrödinger’s equation which has applications in optical fibers and photonic crystal fibers. The stochastic Crank–Nicolson scheme is formed to obtain the computational results. The numerical scheme is analyzed under the mean square sense and Von-Neumann criteria to show consistence and stability, respectively. Meanwhile, stochastic optical soliton solutions are attained by using two techniques, namely, the modified auxiliary equation method and the generalized projective Riccati equation method. These methods provide us with the different types of optical soliton solutions such as hyperbolic, trigonometric, mixed trigonometric, and rational forms. Mainly, the comparison of computational results with newly constructed optical soliton solution is shown graphically. To compare these results, initial conditions and boundary conditions are constructed by selecting some soliton solutions. The 3D and line graphs are drawn by choosing different values of parameters. Additionally, the sensitivity analysis is observed for the different initial values.

The integrable Kuralay equations (IKEs) represent a new class of nonlinear partial differential equations (NLPDEs) that capture the delicate balance between dispersion and nonlinearity in complex physical systems such as nonlinear optical fibers, plasmas, and ferromagnetic spin chains. From a geometric standpoint, these equations correspond to the motion of space curves, where soliton solutions describe the curvature and torsion of evolving wave structures. In this work, we employ the Complete Discrimination System for Polynomial Method (CDSPM) to systematically construct and classify a rich family of optical soliton solutions, including Jacobian elliptic, double-hyperbolic (DH), double-periodic (DP), and trigonometric forms. By exploring the limiting behaviors of elliptic solutions, we also derive solitary wave (SW) structures that exhibit kink-type, periodic, and localized pulse geometries. Furthermore, the bifurcation and stability analyses reveal intricate quasi-periodic and transitional behaviors, highlighting the strong coupling between the system’s geometry and its nonlinear dynamics. The findings not only deepen the theoretical understanding of the Kuralay framework but also offer practical applications in optical communication, plasma energy transport, and nonlinear circuit design. Overall, the study presents a unified analytical and geometrical framework that bridges mathematical theory with real-world physical phenomena.

Optical fiber application of the Improved Generalized Riccati Equation Mapping method to the perturbed nonlinear Chen-Lee-Liu dynamical equation

On the optical solitons and local conservation laws of Chen–Lee–Liu dynamical wave equation

Dynamic properties and optical wave patterns of a high-order nonlinear Schrödinger equation with weak non-local nonlinearity

The purpose of this study is to introduce the wave structures and dynamical features of the third-order nonlinear Schr\"{o}dinger equations (TONLSE). We take the original equation and, using the traveling wave transformation, convert it into the appropriate traveling wave system, from which we create a conserved quantity known as the Hamiltonian. The Jacobian elliptic function solution (JEF), the hyperbolic function solution, and the trigonometric function solution are just a few of the optical soliton solutions to the equation that may be found using the complete discrimination system (CDS) of polynomial method (CDSPM) and also transfer the JEF into solitary wave (SW) soltions. It also includes certain dynamic results, such as bifurcation points and critical conditions for solutions, that might be utilized to explore the dynamic features of the equation employing the CDSPM. This method could also be used for qualitative analysis. The qualitative analysis is used to illustrate the equilibrium points and phase potraits of the equation. Phase portraits are visual representations used in dynamical systems to illustrate a system's behaviour through time. They can provide crucial information about a system's stability, periodic behaviour, and the presence of attractors or repellents.

Optical solitons of the cubic–quartic-nonlinear Schrödinger’s equation having cubic–quintic–septic–nonic form of self-phase modulation

This paper analyzes when a Hopf bifurcation occurs for a class of delay differential systems, and constructs the algorithms for determining the bifurcation's occurrence. By using some algebraic criteria, the complete discrimination system for polynomials and Hurwitz criterion, on-line determining bifurcation's occurrence can be realized. The methods unify and generalize the relevant results in the literature.

A note on exact travelling wave solutions for the modified Camassa–Holm and Degasperis–Procesi equations

Easily testable necessary and sufficient algebraic criteria for delay-independent stability of a class of neutral differential systems

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.