Search Results (9)

The objective of the present study is to detect chirped optical solitons of the perturbed Chen–Lee–Liu equation with full nonlinearity. By virtue of the traveling wave hypothesis, the discussed model is reduced to a simple form known as an elliptic equation. The latter equation, which is a second-order ordinary differential equation, is handled by the undetermined coefficient method of two forms expressed in terms of the hyperbolic secant and tangent functions. Additionally, the auxiliary equation method is applied to derive several miscellaneous solutions. Various types of chirped solitons are revealed such as W-shaped, bright, dark, gray, kink and anti-kink waves. Taking into consideration the existence conditions, the dynamical behaviors of optical solitons and their corresponding chirp are illustrated. The modulation instability of the perturbed CLL equation is examined by means of the linear stability analysis. It is found that all solutions are stable against small perturbations. These entirely new results, compared to previous works, can be employed to understand pulse propagation in optical fiber mediums and dynamic characteristics of waves in plasma. Full article

The central objective of this study is to develop some different wave solutions and perform a qualitative analysis on the nonlinear dynamics of the time-fractional chiral nonlinear Schrodinger’s equation (NLSE) in the conformable sense. Combined with the semi-inverse method (SIM) and traveling wave transformation, we establish the variational principle (VP). Based on this, the corresponding Hamiltonian is constructed. Adopting the Galilean transformation, the planar dynamical system is derived. Then, the phase portraits are plotted and the bifurcation analysis is presented to expound the existence conditions of the various wave solutions with the different shapes. Furthermore, the chaotic phenomenon is probed and sensitivity analysis is given in detail. Finally, two powerful tools, namely the variational method (VM) which stems from the VP and Ritz method, as well as the Hamiltonian-based method (HBM) that is based on the energy conservation theory, are adopted to find the abundant wave solutions, which are the bell-shape soliton (bright soliton), W-shape soliton (double-bright solitons or double bell-shaped soliton) and periodic wave solutions. The shapes of the attained new diverse wave solutions are simulated graphically, and the impact of the fractional order $\delta$ on the behaviors of the extracted wave solutions are also elaborated. To the authors’ knowledge, the findings of this research have not been reported elsewhere and can enable us to gain a profound understanding of the dynamics characteristics of the investigative equation. Full article

In this study, we use an analytical method tailored for the in-depth exploration of coupled nonlinear partial differential equations (NLPDEs), with a primary focus on the dynamics of solitons. Traditional methods are quite effective for solving individual nonlinear partial differential equations (NLPDEs). However, their performance diminishes notably when addressing systems of coupled NLPDEs. This decline in effectiveness is mainly due to the complex interaction terms that arise in these coupled systems. Commonly, researchers have attempted to simplify coupled NLPDEs into single equations by imposing proportional relationships between various solutions. Unfortunately, this simplification often leads to a significant deviation from the true physical phenomena that these equations aim to describe. Our approach is distinctively advantageous in its straightforwardness and precision, offering a clearer and more insightful analytical perspective for examining coupled NLPDEs. It is capable of concurrently facilitating the propagation of different soliton types in two distinct systems through a single process. It also supports the spontaneous emergence of similar solitons in both systems with minimal restrictions. It has been extensively used to investigate a wide array of new coupled progressive solitons in birefringent fibers, specifically for complex Ginzburg–Landau Equations (CGLEs) involving Hamiltonian perturbations and Kerr law nonlinearity. The resulting solitons, with comprehensive 2D and 3D visualizations, showcase a variety of coupled soliton configurations, including several that are unprecedented in the field. This innovative approach not only addresses a significant gap in existing methodologies but also broadens the horizons for future research in optical communications and related disciplines. Full article

The main aim of this study is to obtain soliton solutions of the generalized reaction Duffing model, which is a generalization for a collection of prominent models describing various key phenomena in science and engineering. The equation models the motion of a damped oscillator with a more complex potential than in basic harmonic motion. Two effective techniques, the mapping method and Bernoulli sub-ODE technique, are used for the first time to obtain the soliton solutions of the proposed model. Initially, the traveling wave transform, which comes from Lie symmetry infinitesimals, is applied, and a nonlinear ordinary differential equation form is derived. These approaches effectively retrieve a hyperbolic, Jacobi function as well as trigonometric solutions while the appropriate conditions are applied to the parameters. Numerous innovative solutions, including the kink wave, anti-kink wave, bell shape, anti-bell shape, W-shape, bright, dark and singular shape soliton solutions, were produced via the mapping and Bernoulli sub-ODE approaches. The research includes comprehensive 2D and 3D graphical representations of the solutions, facilitating a better understanding of their physical attributes and proving the effectiveness of the proposed methods in solving complex nonlinear equations. It is important to note that the proposed methods are competent, credible and interesting analytical tools for solving nonlinear partial differential equations. Full article

Fractional calculus with symmetric kernels is a fast-growing field of mathematics with many applications in all branches of science and engineering, notably electromagnetic, biology, optics, viscoelasticity, fluid mechanics, electrochemistry, and signals processing. With the use of the Sardar sub-equation and the Bernoulli sub-ODE methods, new trigonometric and hyperbolic solutions to the time-fractional Caudrey–Dodd–Gibbon–Sawada–Kotera equation have been constructed in this paper. Notably, the definition of our fractional derivative is based on the Jumarie’s modified Riemann–Liouville derivative, which offers a strong basis for our mathematical explorations. This equation is widely utilized to report a variety of fascinating physical events in the domains of classical mechanics, plasma physics, fluid dynamics, heat transfer, and acoustics. It is presumed that the acquired outcomes have not been documented in earlier research. Numerous standard wave profiles, such as kink, smooth bell-shaped and anti-bell-shaped soliton, W-shaped, M-shaped, multi-wave, periodic, bright singular and dark singular soliton, and combined dark and bright soliton, are illustrated in order to thoroughly analyze the wave nature of the solutions. Painlevé analysis of the proposed study is also part of this work. To illustrate how the fractional derivative affects the precise solutions of the equation via 2D and 3D plots. Full article

This paper seeks to find optical soliton solutions for Lakshmanan–Porsezian–Daniel (LPD) model with the parabolic law of nonlinearity. The spatiotemporal dispersion is included to the model, as it can contribute to handling the problem of internet bottleneck. This study was performed analytically using the traveling wave hypothesis to reduce the model to an integrable form. Then, the resulting equation was handled with two approaches, namely, the auxiliary equation method and the Bernoulli subordinary differential equation (sub-ODE) method. With an intentional focus on hyperbolic function solutions, abundant optical soliton waves including W-shaped, bright, dark, kink-dark, singular, kink, and antikink solitons were derived with the existing conditions. Furthermore, the behaviors of some optical solitons are illustrated. The spatiotemporal dispersion was found to significantly affect the pulse propagation dynamics. Finally, the modulation instability (MI) of the LPD model is explained in detail along with the extraction of the expression of MI gain. Full article

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. Full article

This paper aims to seek soliton solutions for the nonlocal generalized Sasa–Satsuma (gSS) equation by constructing the Darboux transformation (DT). We obtain soliton solutions for the nonlocal gSS equation, including double-periodic wave, breather-like, KM-breather solution, dark-soliton, W-shaped soliton, M-shaped soliton, W-shaped periodic wave, M-shaped periodic wave, double-peak dark-breather, double-peak bright-breather, and M-shaped double-peak breather solutions. Furthermore, interaction of these solitons, as well as their dynamical properties and asymptotic analysis, are analyzed. It will be shown that soliton solutions of the nonlocal gSS equation can be reduced into those of the nonlocal Sasa–Satsuma equation. However, several of these properties for the nonlocal Sasa–Satsuma equation are not found in the literature. Full article

This investigation focuses on two novel Kadomtsev–Petviashvili (KP) equations with time-dependent variable coefficients that describe the nonlinear wave propagation of small-amplitude surface waves in narrow channels or large straits with slowly varying width and depth and non-vanishing vorticity. These two variable coefficients, Kadomtsev–Petviashvili (VCKP) equations in (2+1)-dimensions, are the main extensions of the KP equation. Applying the Lie symmetry technique, we carry out infinitesimal generators, potential vector fields, and various similarity reductions of the considered VCKP equations. These VCKP equations are converted into nonlinear ODEs via two similarity reductions. The closed-form analytic solutions are achieved, including in the shape of distinct complex wave structures of solitons, dark and bright soliton shapes, double W-shaped soliton shapes, multi-peakon shapes, curved-shaped multi-wave solitons, and novel solitary wave solitons. All the obtained solutions are verified and validated by using back substitution to the original equation through Wolfram Mathematica. We analyze the dynamical behaviors of these obtained solutions with some three-dimensional graphics via numerical simulation. The obtained variable coefficient solutions are more relevant and useful for understanding the dynamical structures of nonlinear KP equations and shallow water wave models. Full article