Elliptic Modulus Research Articles

Exploring the impact of multiplicative white noise on novel soliton solutions with the perturbed Triki–Biswas equation

This study examines the effects of multiplicative white noise on soliton perturbations governed by the Triki–Biswas equation for the first time. Triki–Biswas equation advances research on ultrashort pulse propagation in optical fibers. It modifies the nonlinear Schrödinger equation to describe the behavior of femtosecond pulses more accurately in optical media, becoming a critical tool in the field. The paper employs two innovative methods, the enhanced direct algebraic method and the new projective Riccati equations method to uncover a broad range of soliton solutions, including bright, dark, and singular solitons. The solutions are expressed in terms of Jacobi elliptic functions and exhibit a transition to soliton-type solutions as the elliptic modulus approaches unity. This investigation is the first of its kind to explore the effects of multiplicative white noise within this context, providing new perspectives and methodologies for future research in the field. The study sheds light on previously unexplored aspects of multiplicative white noise and contributes significantly to the body of knowledge in soliton theory and its application to optical fiber technology.

Optical solitons for dispersive concatenation model with power-law of self-phase modulation: a sub-ODE approach

AbstractThis work retrieves a plethora of optical soliton solutions to the dispersive concatenation model with power-law of self-phase modulation. The implementation of the sub-ODE method and its variations and versions yielded such soliton solutions. The intermediary functions were Weierstrass’ elliptic functions as well as Jacobi’s elliptic functions. Their special cases gave way to soliton solutions. In particular, for Jacobi’s elliptic functions, when the modulus of ellipticity approached unity, the soliton solutions have naturally emerged.

Chemical oscillation in ultracold chemistry

We demonstrate the occurrence of oscillatory reactions in the ultra-cold chemistry of atom-molecular Bose-Einstein condensate. Nonlinear oscillations in the mean-field dynamics occur for a specific range of elliptic modulus, giving rise to both in- and out-phase modulations in the atom-molecule population density. The reaction front velocity is found to be controlled by photoassociation, which also regulates the condensate density. Two distinct pairs of in-phase bright localized gap solitons are found as exact solutions, the existence of one of which necessarily requires a background. Cnoidal atomic density waves along with a plane-wave molecular motion are observed in both attractive and repulsive domains. The role of intra- and inter-species interactions on both existence and stability is explicated in the presence of photoassociation.

Classification of Jacobi solutions of double dispersion equation in uniform and inhomogeneous Murnaghan’s rod

The double dispersion equation comprising the Lame coefficient, nonlinear coefficient, and Poisson ratio components is described as the uniform and inhomogeneous Murnaghan’s rod by A. M. Samsonov in Samsonov (2001). In this work, we apply the F expansion method to the double dispersion equation in the uniform and inhomogeneous Murnaghan’s rod, extract the Jacobi elliptic function solution, and classify it into six families of unique solutions. The necessary condition and the degeneration of the Jacobi solutions based upon the elliptic function modulus are given for each solution. The six classifications are formed based on the solutions of the algebraic equations.

The instabilities beyond modulational type in a repulsive Bose–Einstein condensate with a periodic potential

The instabilities of the nontrivial phase elliptic solutions in a repulsive Bose–Einstein condensate (BEC) with a periodic potential are investigated. Based on the defocusing nonlinear Schrödinger (NLS) equation with an elliptic function potential, the well-known modulational instability (MI), the more recently identified high-frequency instability, and an unprecedented – to our knowledge – variant of the MI, the so-called isola instability are identified numerically. Upon varying parameters of the solutions, instability transitions occur through suitable bifurcations, such as the Hamiltonian-Hopf one. Specifically, (i) increasing the elliptic modulus k of the solutions, we find that MI switches to the isola instability and the dominant disturbance has twice the elliptic wave’s period, corresponding to a Floquet exponent μ=π2K(k). The isola instability arises from the collision of spectral elements at the origin of the spectral plane. (ii) Upon varying V0, the transition between MI and the high-frequency instability occurs. Differently from the MI and isola instability where the collisions of eigenvalues happen at the origin, the high-frequency instability arises from pairwise collisions of nonzero, imaginary elements of the stability spectrum; (iii) In the limit of sinusoidal potential, we show that MI occurs from a collision of eigenvalues with μ=π2K(k) at the origin; (iv) we also examine the dynamic byproducts of the instability in chaotic fields generated by its manifestation. An interesting observation is that, in addition to MI, the isola instability could also lead to dark localized events in the scalar defocusing NLS equation.

Optical solitons for the concatenation model: Power-law nonlinearity

The current paper retrieves optical soliton solutions to the concatenation model that is with power-law nonlinearity. The traveling wave hypothesis is the starting point. The detailed mathematical analysis leads to the cnoidal waves and snoidal waves, which eventually lead to the bright and singular soliton solutions when the modulus of ellipticity reaches the appropriate limiting value.

Superposed periodic kink and pulse solutions of coupled nonlinear equations

We present novel previously unexplored periodic solutions, expressed in terms of Jacobi elliptic functions, for both a coupled ϕ4 model and a coupled nonlinear Schrödinger equation (NLS) model. Remarkably, these solutions can be elegantly reformulated as a linear combination of periodic kinks and antikinks, or as a combination of two periodic kinks or two periodic pulse solutions. However, we also find that for m=0 and a specific value of the periodicity (or at a nonzero value of the elliptic modulus m) this superposition does not hold. These results demonstrate that the notion of superposed solutions extends to the coupled nonlinear equations as well.

Accretion of modified Chaplygin–Jacobi gas and modified Chaplygin–Abel gas onto Schwarzschild black hole

Herein, we have discussed about the accretion of two extremely interesting forms of dark energies, namely “Modified Chaplygin–Jacobi Gas” (MCJG) and “Modified Chaplygin–Abel Gas” (MCAG) onto a Schwarzschild black hole of dimension 4. First, considering the above-mentioned dark energies, we have deduced the rate of change of mass and then evolved the required mass from it in terms of the redshift [Formula: see text]. We have also plotted the mass versus redshift graphs for different values of the elliptic modulus [Formula: see text] in the case of MCJG and different values of [Formula: see text] in the case of MCAG, respectively. Also, mass versus redshift graphs have been portrayed in the different stages of the FRW universe. We have found that in the case of both MCJG and MCAG accretion, the mass of the black holes follow an increasing pattern. But as expected in the case of the phantom energy-filled universe, the mass of the black hole shows decreasing nature. Again in the case of the [Formula: see text]CDM stage of the universe, accretion of MCAG primarily leads to an increase in mass, and then it gradually diminishes.

Rogue wave solutions on different periodic backgrounds for the (2 + 1)-dimensional Heisenberg ferromagnetic spin chain equation

In this paper, we will present exact rogue wave solutions on different periodic backgrounds for the (2+1)-dimensional Heisenberg ferromagnetic spin chain equation, which can govern nonlinear spin dynamics of the (2+1)-dimensional Heisenberg ferromagnetic spin chains with bilinear and anisotropic interactions in the semiclassical limit. We will derive different spectral parameters and corresponding periodic eigenfunctions of Lax pair according to the nonlinearization of spectral problem. Via Darboux transformation method, first- and second-order rogue wave solutions will be derived under two different periodic seed solutions, which can be expressed by the Jacobian elliptic functions dn and cn. Meanwhile, dynamic features of these various solutions with different parameters on two kinds periodic backgrounds taking different elliptic modulus will be analytically and graphically analyzed.

Rogue wave solutions over double-periodic wave background of a fifth-order nonlinear Schrödinger equation with quintic terms

Studies on rogue waves (RWs) over constant and periodic wave background are of contemporary interest in several areas of physics. These RWs arise not only on such backgrounds but also appear on double-periodic wave background. In this paper, we derive RW solutions over a double-periodic wave background of a fifth-order nonlinear Schrödinger equation. We create the double-periodic wave background with elliptic functions (in combinations of cn, sn and dn) as seed solutions in the first iteration of Darboux transformation. We then calculate the growth rate of instability for these double-periodic solutions under different values of elliptic modulus parameter. On the top of these background waves, we generate RWs with the help of second independent solution to the eigenvalue problem. Further, we analyze the differences that occur in the appearance of RWs with reference to the lower-order and higher-order dispersions terms. In addition to the above, we examine the derived solutions in detail for certain system and elliptic modulus parameter values and highlight some interesting features that we obtain from our studies.

W-Shaped Bright Soliton of the (2 + 1)-Dimension Nonlinear Electrical Transmission Line

In this paper, we investigate solitary wave solutions of the nonlinear electrical transmission line by using the Jacobi elliptic function and the auxiliary equation methods. We obtain Jacobi elliptic function solutions as well as kink, bright, dark, and W-shaped solitons as a result. For specific values of the Jacobi elliptic modulus, we depict bright, dark, and W-shaped soliton solutions as suitable parameters of the structure. Using the auxiliary equation method gives the combined bright–bright and dark–dark optical solitons in optical fibers. One result emerges from this analysis: the potential parameters and free parameters of the method can be employed to degenerate W-shaped bright and dark solitons. The acquired results are general and can be used for many applications in nonlinear dynamic systems.

Coupled sn-cn, sn-dn, cn-dn Jacobi elliptic functions and solitons solutions in magneto-optic waveguides with quadratic-cubic nonlinearity

The current paper is dedicated to the study of the propagation of progressive solitons in magneto-optical waveguides that carry the quartic-cubic nonlinearity described by a coupled system of of nonlinear Schrödinger equations. In this direction, we take the help of a modified F-expansion method which allows us to widen the class of solutions to these coupled equations and permit the propagation of different waves in the two channels. First of all, we constructed the solutions in terms of Jacobi elliptic functions in the forms of and After that, we obtained the progressive soliton solutions when these functions approached their limiting values with regards to the modulus of ellipticity. The progressive solutions include bright-dark soliton, bright-bright soliton, dark-dark soliton, bright-front soliton, dark-front soliton. In addition, we present the figures 1–6 which graphically exhibit the representative structures of each explicit solution for some special parameter values.

Optical Solitons in Fiber Bragg Gratings with Dispersive Reflectivity Having Five Nonlinear Forms of Refractive Index

This paper implements the trial equation approach to retrieve cubic–quartic optical solitons in fiber Bragg gratings with the aid of the trial equation methodology. Five forms of nonlinear refractive index structures are considered. They are the Kerr law, the parabolic law, the polynomial law, the quadratic–cubic law, and the parabolic nonlocal law. Dark and singular soliton solutions are recovered along with Jacobi’s elliptic functions with an appropriate modulus of ellipticity.

Cnoidal waves and dark solitons with linear third-order dispersion and self-steepening effect

The present paper investigates the conditions needed for the generation and propagation of cnoidal waves in the form of dark solitons under the influence of self-steepening effect and linear third-order dispersion. This kind of optical pulses could be obtained in single-mode waveguides due to the precise balance between higher orders of nonlinear and dispersive phenomena. As a basic equation, it is used the nonlinear amplitude equation (NAE). It is found a new analytical solution of this equation in the form of a cnoidal wave (Jacobi elliptic sine function). In the article it is presented that the solution can be reduced to dark soliton at specific values of the modulus of ellipticity κ.

Rogue waves on the double-periodic background for a nonlinear Schrödinger equation with higher-order effects

In this paper, we study the dynamics of rogue waves on the double-periodic background for a nonlinear Schrödinger equation with higher-order effects. First, we consider two types of double-periodic solutions in terms of the Jacobi elliptic functions. When the elliptic modulus approaches 1, such both double-periodic wave solutions can reduce to the Akhmediev breather solution. Second, with both double-periodic waves as seed solutions, we derive rogue wave solutions from the Darboux transformation method. Further, we demonstrate localized structures of rogue waves formed on double-periodic waves for two sets of different eigenvalues. In addition, we discuss how the higher-order effect affects double-periodic background waves and rogue waves. We expect that our obtained results may help understand rogue waves manifestations on the double-periodic background occurring in hydrodynamics and nonlinear optics with higher-order effects.

Singular and Bright soliton solutions for the coupled system corresponding to Gerdjikov-Ivanov equation with four-wave mixing terms

The extended trial function method is utilized in birefringent fibers to generate optical soliton solutions to the coupled system that correspond to the Gerdjikov-Ivanov equation with four-wave mixing terms. The approach yields singular and bright soliton solutions, and in addition extremely important singular and singular periodic soliton solutions are extracted by taken the limit of the modulus of ellipticity of the Jacobi’s elliptic function. The conditions under which the soliton solution exist are also provided.

Dispersive optical solitons with Schrödinger–Hirota model having multiplicative white noise via Itô Calculus

The current work is on the retrieval of dispersive stochastic optical solitons that are modeled by the Schrödiner–Hirota equation with the fiber maintaining power law of nonlinearity. The stochasticity, from the multiplicative white noise effect, leads to the implementation of the Itô Calculus. The ϕ6–expansion scheme yields soliton solutions with the limiting approach being applied to Jacobi's elliptic functions with respect to the modulus of ellipticity.

Cubic–quartic optical solitons in fiber Bragg gratings with anti-cubic nonlinearity using the modified extended direct algebraic method

Objective:This paper obtains cubic–quartic solitons in fiber Bragg gratings with anti-cubic nonlinearity. Methods:The extended direct algebraic integration algorithm is adopted. Results:Dark, bright and singular solitons are recovered along with their respective parameter constraints for their existence. Conclusion:The integration tool yields solutions in terms of Jacobi’s elliptic functions which led to the emergence of optical solitons from the limiting operation applied to the modulus of ellipticity. The results are thus extremely promising to move ahead with this approach to study additional forms of self-phase modulation coupled with other forms of dispersive effects.

Stationary optical solitons with complex Ginzburg–Landau equation having nonlinear chromatic dispersion and Kudryashov's refractive index structures

This work obtains stationary optical solitons for complex Ginzburg–Landau equation that is structured with Kudryashov's self–phase modulation structures. The chromatic dispersion is also taken to be nonlinear. Six forms of nonlinear refractive index are considered. The adopted integration scheme is the generalized G′/G-expansion approach that yields solutions in terms of Jacobi's elliptic functions. The limiting approach, when applied with the modulus of ellipticity leads to the stationary optical solitons that finally emerge from the model.

Sequel to “stationary optical solitons with Kudryashov’s laws of refractive index” (generalized temporal evolution)

This work is a sequel to the previously reported results on stationary optical solitons with several forms of Kudryashov’s law of self-phase modulation. The study is conducted with generalized temporal evolution by the aid of extended trial function approach. The solutions that naturally emerged from this scheme are in terms of Jacobi’s elliptic function which finally led to stationary solitons with the modulus of ellipticity approaching its appropriate limit.