Search Results (11)

This study explores analytical soliton solutions for the cubic–quintic time-fractional nonlinear non-paraxial pulse transmission model. This versatile model finds numerous uses in fiber optic communication, nonlinear optics, and optical signal processing. The strength of the quintic and cubic nonlinear components plays a crucial role in nonlinear processes, such as self-phase modulation, self-focusing, and wave combining. The fractional nonlinear Schrödinger equation (FNLSE) facilitates precise control over the dynamic properties of optical solitons. Exact and methodical solutions include those involving trigonometric functions, Jacobian elliptical functions (JEFs), and the transformation of JEFs into solitary wave (SW) solutions. This study reveals that various soliton solutions, such as periodic, rational, kink, and SW solitons, are identified using the complete discrimination polynomial methods (CDSPM). The concepts of chaos and bifurcation serve as the framework for investigating the system qualitatively. We explore various techniques for detecting chaos, including three-dimensional and two-dimensional graphs, time-series analysis, and Poincarè maps. A sensitivity analysis is performed utilizing a variety of initial conditions. Full article

This article introduces two kinds of processing techniques to solve Jacobian elliptic equations and obtain rich periodic wave solutions. Then, the equation was used as an auxiliary equation to solve the (3+1)-dimensional modified Korteweg de Vries–Zakharov–Kuznetsov (mKDV-ZK) equation. Combined with the mapping method, a large number of new types of exact periodic wave solutions were obtained, many of which were rarely found in previous research. Numerical simulations have demonstrated the evolution of various periodic waves in (3+1)-dimensional mKDV-ZK. The solutions and wave phenomena obtained in this article will help expand our understanding of the equation. Full article

This paper presents an approach to summing a few families of infinite series involving hyperbolic functions, some of which were first studied by Ramanujan. The key idea is based on their contour integral representations and residue computations with the help of some well-known results of Eisenstein series given by Ramanujan, Berndt, et al. As our main results, several series involving hyperbolic functions are evaluated and expressed in terms of $z={}_{2}F_1(1/2,1/2;1;x)$ and $z^{\prime }=dz/dx$. When a certain parameter in these series is equal to $\pi$, the series are expressed in closed forms in terms of some special values of the Gamma function. Moreover, many new illustrative examples are presented. Full article

The Klein–Gordon equation plays an important role in mathematical physics, such as plasma and, condensed matter physics. Exploring its exact solution helps us understand its complex nonlinear wave phenomena. In this paper, we first propose a new extended Jacobian elliptic function expansion method for constructing rich exact periodic wave solutions of the (2+1)-dimensional Klein–Gordon equation. Then, we introduce a novel wave transformation for constructing nonlinear complex waves. To demonstrate the effectiveness of this method, we numerically simulated several sets of complex wave structures, which indicate new types of complex wave phenomena. The results show that this method is simple and effective for constructing rich exact solutions and complex nonlinear wave phenomena to nonlinear equations. Full article

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. Full article

In this paper, exact periodic wave solutions for the perturbed Boussinesq equation with power law nonlinearity are obtained for different nonlinear strengths n. When $n=1$, the periodic traveling wave solutions can be found by the definition of the Jacobian elliptic function. When $n\ge 1$, we construct a transformation to solve for the power law nonlinearity, and the periodic traveling wave solutions can be obtained by applying the extended trial equation method. In addition, we consider the limiting case where the periodicity of the periodic traveling wave solutions vanishes, and we obtain the soliton solution for $n=1$. Numerical simulations show the periodicity of the solution for the perturbed Boussinesq equation. Full article

Cnoidal wave theory perfectly describes nearshore wave characteristics. However, cnoidal wave theory is not widely applied in practical engineering because the formula for the wave profile involves a complex Jacobian elliptic function. In this paper, the approximate cnoidal wave theory is presented. Based on the Biot consolidation theory and the approximate cnoidal wave theory, an analytical solution for the pore water pressure around buried pipelines caused by waves is derived. In addition, based on the principle of effective stress, a theory of soil liquefaction around pipelines is proposed. The theoretical results were virtually identical to the results obtained in a practical flume test. Thus, the analytical method proposed in this paper is feasible. Further, the theory is applied to analyze the instantaneous liquefaction of the seabed around buried pipelines and the stability of the pipeline in the Chengdao oilfield. Full article

In this paper, we use a new, extended Jacobian elliptic function expansion method to explore the exact solutions of the (2+1)-dimensional asymmetric Nizhnik–Novikov–Veselov (aNNV) equation, which is a nonlinear physical model to describe an incompressible fluid. Combined with the mapping method, many new types of exact Jacobian elliptic function solutions are obtained. As we use two new forms of transformation, most of the solutions obtained are not found in previous studies. To show the complex nonlinear wave phenomena, we also provide various wave structures of a group of solutions, including periodic wave and solitary wave structures of ordinary traveling wave solutions, horseshoe-type wave, s-type wave and breaker-wave structures superposed by two kinds of waves: chaotic wave structures with chaotic behavior and spiral wave structures. The results show that this method is effective and powerful and can be used to construct various exact solutions for a wide range of nonlinear models and complex nonlinear wave phenomena in mathematical and physical research. Full article

In this investigation, some analytical solutions to both conserved and non-conserved rotational pendulum systems are reported. The exact solution to the conserved oscillator (unforced, undamped rotational pendulum oscillator), is derived in the form of a Jacobi elliptical function. Moreover, an approximate solution for the conserved case is obtained in the form of a trigonometric function. A comparison between both exact and approximate solutions to the conserved oscillator is examined. Moreover, the analytical approximations to the non-conserved oscillators including the unforced, damped rotational pendulum oscillator and forced, damped rotational pendulum oscillator are obtained. Furthermore, all mentioned oscillators (conserved and non-conserved oscillators) are linearized, and their exact solutions are derived. In addition, all obtained approximations are compared with the four-order Runge–Kutta (RK4) numerical approximations and with the exact solutions to the linearized oscillators. The obtained results can help several authors for discussing and interpreting their results. Full article

We show that the results we had previously obtained on diagonals of 9- and 10-parameter families of rational functions in three variables $x$, $y$, and $z$, using creative telescoping, yielding modular forms expressed as pullbacked ${}_2{F}_1$ hypergeometric functions, can be obtained much more efficiently by calculating the $j$-invariant of an elliptic curve canonically associated with the denominator of the rational functions. These results can be drastically generalized by changing the parameters into arbitrary rational functions of the product $p=xyz$. In other cases where creative telescoping yields pullbacked ${}_2{F}_1$ hypergeometric functions, we extend this algebraic geometry approach to other families of rational functions in three or more variables. In particular, we generalize this approach to rational functions in more than three variables when the denominator can be associated to an algebraic variety corresponding to products of elliptic curves, or foliations in elliptic curves. We also extend this approach to rational functions in three variables when the denominator is associated with a genus-two curve such that its Jacobian is a split Jacobian, corresponding to the product of two elliptic curves. We sketch the situation where the denominator of the rational function is associated with algebraic varieties that are not of the general type, having an infinite set of birational automorphisms. We finally provide some examples of rational functions in more than three variables, where the telescopers have pullbacked ${}_2{F}_1$ hypergeometric solutions, because the denominator corresponds to an algebraic variety that has a selected elliptic curve. Full article

An updated Power Index Method is presented for nonlinear differential equations (NLPDEs) with the aim of reducing them to solutions by algebraic equations. The Lie symmetry, translation invariance of independent variables, allows for traveling waves. In addition discrete symmetries, reflection, or $180°$ rotation symmetry, are possible. The method tests whether certain hyperbolic or Jacobian elliptic functions are analytic solutions. The method consists of eight steps. The first six steps are quickly applied; conditions for algebraic equations are more complicated. A few exceptions to the Power Index Method are discussed. The method realizes an aim of Sophus Lie to find analytic solutions of nonlinear differential equations. Full article