Analysis of the nonlinear conformable Schrödinger equation with Kudryashov's refractive index using the generalized exponential rational function technique

In this paper, we investigate the time-fractional improved (2+1)-dimensional nonlinear Schrödinger equation with power-law nonlinearity, group-velocity dispersion, and spatio-temporal dispersion in nonlinear optics. This equation models the propagation of optical pulses in nonlinear optical fibers. We derive novel optical soliton solutions expressed through exponential and hyperbolic functions, which include bright, bell-shaped, wave, and singular solitons. To illustrate the characteristics of these solutions, we provide two-dimensional, three-dimensional, and contour plots that visualize the magnitude of the conformable improved (2+1)-dimensional nonlinear Schrödinger equation. By selecting suitable values for physical parameters, we demonstrate the diversity of soliton structures and their behaviors. Furthermore, we investigated the influence of the temporal parameter and the conformable fractional-order derivative on the behavior of soliton solutions. The results highlighted the effectiveness and versatility of the modified Kudryashov method in addressing both integer- and fractional-order differential equations, providing analytical solutions that deepen our insight into the dynamics of complex optical systems. These results contribute to the advancement of soliton theory in nonlinear optics and mathematical physics.

Dark solitonic interactions for the (3 + 1)-dimensional coupled nonlinear Schrödinger equations in nonlinear optical fibers

The generalized exponential rational function (GERF) method is used in this work to obtain analytic wave solutions to the Kudryashov-Sinelshchikov (KS) equation. The KS equation depicts the occurrence of pressure waves in mixtures of liquid-gas bubbles while accounting for thermal expansion and viscosity. By applying the GERF method to the KS equation, we obtain analytic solutions in terms of trigonometric, hyperbolic, and exponential functions, among others. These solutions include solitary wave solutions, dark-bright soliton solutions, singular soliton solutions, singular bell-shaped solutions, traveling wave solutions, rational form solutions, and periodic wave solutions. We discuss the two-dimensional and three-dimensional graphics of some obtained solutions under the accurate range space by selecting appropriate values for the involved arbitrary parameters to make this research more praiseworthy. The obtained analytic wave solutions specify the GERF method’s dependability, capability, trustworthiness, and efficiency.

This study presents a mathematical model for understanding wave propagation and soliton behavior in biomechanical tissues, explicitly focusing on the Achilles tendon. Utilizing the Korteweg-de Vries (KdV) equation, the research incorporates the Achilles tendons’ nonlinear elastic and viscoelastic properties to explore how mechanical waves propagate through this complex tissue. The tendon’s nonlinear elasticity leads to wave steepening, while its viscoelasticity introduces dispersive effects that counteract this steepening, resulting in the formation of solitons—stable, localized waves that maintain their shape as they propagate. Key findings from this study reveal that the formation and propagation of solitons are strongly influenced by the tendon’s mechanical properties. Numerical simulations show that stiffer tendons, characterized by a higher elasticity modulus, support faster soliton propagation, with wave speeds ranging from 18.9 m/s in damaged tendons to 28.6 m/s in stiffened tendons. Additionally, soliton amplitude increases with tissue stiffness, with the highest amplitude observed in stiffened tendons (5.1 mm) and the lowest in damaged tendons (3.2 mm). The study also demonstrates that energy dissipation due to the tendon’s viscoelasticity plays a critical role in soliton behavior. Damaged tendons exhibit the highest energy loss (18.6%), leading to shorter soliton propagation distances, while stiffer tendons retain more energy (96.1%) and allow solitons to travel further distances (up to 180 mm). Moreover, the balance between nonlinearity and dispersion is crucial for maintaining soliton stability. Excessive nonlinearity leads to unstable solitons, while higher levels of dispersion contribute to more stable waveforms.

Optical wave solutions of perturbed time-fractional nonlinear Schrödinger equation

Analytic study on the mixed-type solitons for a (2+1)-dimensional N-coupled nonlinear Schrödinger system in nonlinear optical-fiber communication

In this work, the dynamical structure for the extended equation is analyzed through unified Riccati equation expansion (UREE) and the Lie isomorphism method for the (3+1)-dimensional Wazwaz–Benjamin–Bona–Mahony (WBBM) equation. This equation represents the unidimensional propagation of short amplitude long waves on the water’s surface in a medium. These employed techniques are the most powerful and effective ways to obtain different sets of new and more generalized exact soliton solutions of the WBBM equation. Furthermore, what distinguishes this study from other studies is that it not only acquires a variety of analytical wave solutions for the studied models but also, demonstrates the interaction phenomena for these results as they propagate over time. Also, shows various meaningful graphs of the processes that provide valuable wisdom for understanding their behavior. The UREE method directly provides various new exact soliton solutions with some novel dynamical properties. We perform a detailed Lie symmetry analysis to governing equation that leaves the system invariant. The Lie group method explores six Lie isomorphism groups to study the WBBM equation. First, we find infinitesimal transformations employing the one-parameter Lie symmetry method. Second, we solve the infinitesimal generator and reduce the order of the equation. Moreover, we illustrate some two-dimensional (2D), three-dimensional (3D), and contour diagrams of the obtained results and compute the exact analytical solution utilizing the used methods. To find novel solutions, the Adomian method is also used, where the Adomian polynomials are utilized to deal with nonlinear terms. Variety of new analytical solutions with different types of dynamical behavior are analyzed by utilizing the computational software like Mathematica. These new analytical exact wave solutions are demonstrated in various dynamical structures of periodic wave soliton, interaction periodic wave and kink wave soliton, lump wave soliton, doubly soliton, multi-wave soliton, kink periodic, parabolic wave, multisoliton, traveling wave, and standing wave-shaped profiles.

Analytic Studies on the Helmholtz Spatial Solitons in Power-Law Optical Materials

This study investigates soliton solutions and dynamic wave behaviors in the complex Ginzburg-Landau equation, a model that plays a central role in describing diverse physical phenomena such as superconductivity, nonlinear optical fibers, liquid crystals, second-order phase transitions, and field theory strings. To derive closed-form solutions, we employ two advanced analytical techniques: the new rational extended sinh-Gordon equation expansion method (ShGEEM) and the modified generalized exponential rational function method (mGERFM). These methods yield a wide range of solitonic structures, such as complex and singular solitons, oscillatory periodic waves, bright, dark, and multi-wave profiles. In this work, new families of exact solitary wave solutions with ShGEEM and several hyperbolic, trigonometric, and exponential solutions with mGERFM are presented. Further, the obtained solutions are checked for accuracy by substituting them back into Mathematica. For the dynamics of solutions, 2D plots, 3D surfaces, and contour graphs have been constructed for some values of parameters in the presence of the [Formula: see text]-fractional derivative to understand wave structures and their evolution. In general, the present study not only consolidates the aspects of nonlinear wave dynamics in the field of chemical and physical oceanography but also provides pathways for further research on nonlinear fractional-order models. The originality of the present study lies in the point that the complex Ginzburg-Landau equation has not been studied within the ShGEEM and mGERFM frameworks.

This paper investigates the resonant nonlinear Schrödinger equation (RNSE) with parabolic law nonlinearity, modeling optical pulse propagation in nonlinear optical fibers. By employing the Kumar–Malik approach, we have derived some analytical soliton solutions for the considered equation. These solutions are in the form of Jacobi elliptic, hyperbolic, trigonometric, exponential functions are obtained by this analytical approach. Dark, bright, singular, and periodic wave solitons are created by selecting proper values for the parameters. The new results are compared with previously obtained results. In addition, the physical properties of the presented solutions are represented by 2d, contour and 3d graphs created by selecting appropriate constant parameters. The findings of this study are novel. The acquired results highlight the simplicity, efficacy, and dependability of this method in the analysis of various nonlinear models encountered in the fields of mathematical physics and engineering.

This paper investigates the nonlinear conformable Schrödinger equation with nonlocal nonlinearities and Kudryashov’s arbitrary refractive index. The study employs the Uniform Method (UM) and the Extended Hyperbolic Function Method (EHFM) to derive various optical soliton solutions for this present conformable equation. To demonstrate the importance of the novel solutions, the paper presents two-dimensional, three-dimensional, and contour plots, illustrating kink-type, wave, bright, bell-shaped, and mixed dark–bright soliton solutions. Further, the impact of the fractional order parameter, temporal parameter, and nonlinearity coefficient on these optical solutions is analyzed, providing valuable insights into the conformable nonlinear Schrödinger model. The behavior of these optical solutions is analyzed through illustrative graphs that account for variations in the conformable order derivative, temporal parameter, and nonlinearity coefficient. The methodologies applied in this research show potential for broader application to various nonlinear Schrödinger equations in fields such as applied mathematics and nonlinear optics. The research suggests that these methods could be applied in the future to investigate other differential equations with fractional and integer orders across different fields of applied sciences. This study expands our understanding of nonlinear optics and has potential practical implications in various fields like optical signal processing, laser technology, and telecommunications.

Exact analytical soliton solutions play an important role in soliton fields. Soliton solutions were obtained with some special constraints on the nonlinear parameters in nonlinear coupled systems, but they usually do not hold in real physical systems. We successfully release all usual constrain conditions on nonlinear parameters for exact analytical vector soliton solutions in N-component coupled nonlinear Schrödinger equations. The exact soliton solutions and their existence condition are given explicitly. Applications of these results are discussed in several present experimental parameter regimes. The results would motivate experiments to observe more novel vector solitons in nonlinear optical fibers, Bose-Einstein condensates, and other nonlinear coupled systems.

The soliton waves’ physical behavior on the pseudo spherical surfaces is studied through the analytical solutions of the nonlinear (1+1)–dimensional Kaup–Kupershmidt (KK) equation. This model is named after Boris Abram Kupershmidt and David J. Kaup. This model has been used in various branches such as fluid dynamics, nonlinear optics, and plasma physics. The model’s computational solutions are obtained by employing two recent analytical methods. Additionally, the solutions’ accuracy is checked by comparing the analytical and approximate solutions. The soliton waves’ characterizations are illustrated by some sketches such as polar, spherical, contour, two, and three-dimensional plots. The paper’s novelty is shown by comparing our obtained solutions with those previously published of the considered model.

Analytical soliton solutions and wave profiles of the (3+1)-dimensional modified Korteweg–de Vries–Zakharov–Kuznetsov equation

A class of analytical traveling wave and soliton solutions to generalized nonautonomous nonlinear Schrodinger equation with an external potential are constructed by using homogeneous balance principle and F-expansion technique. Constraint conditions for analytical solutions are obtained at the same time which admit different types of external potentials. Various types of analytical traveling waves and soliton solutions are studied in detail. Stability analysis of the solutions is discussed numerically. Results show that stable propagation of solitons can be maintained in nonuniform distributed nonlinear media.