Soliton solutions of the third-order perturbed nonlinear Schrödinger equation having the Kudryashov’s law of self-modulation form and modulation instability analysis

In this paper, coupled resonant Davey-Stewartson (CRDS) system is studied. The resonant concept is quite important in fluid dynamics, magneto-acoustic waves and plasma physics. CRDS system models the two-wave propagation with periodic wave patterns and short-long wave propagation. Our primary aim is obtaining soliton solutions of this important CRDS system via generalized F-expansion method (GFEM) and auxiliary equation method (AEM). As a result of the application of the aforementioned methods to the model, soliton solutions both known in the literature and a rare type have been obtained. We obtained dark, bright, periodic-singular, two-dark and two-bright soliton solutions. Also, two-dark and two-bright soliton solutions are quite rare soliton types according to the literature research. The 3D and contour graphics of the obtained soliton solutions were drawn. On the other hand, we did modulation instability (MI) analysis on obtained solutions and according to the MI analysis, obtained results are clearly stable. As far as we know, the relevant methods were applied for the first time to this model. Again, modulation instability analysis was performed on the model for the first time. Therefore, the study includes innovative reviews and conclusions.

The bright, dark and kink soliton solutions of the coupled Zakharov-Kuznetsov (ZK) equation are obtained by using the solitary wave ansatz method, variational approximation (VA), variational iteration method (VIM) and Adomian's decomposition method (ADM). The bright and dark soliton solutions are multiple soliton solutions at time t = 0 which reduce into stationary soliton through single soliton for sufficiently large time t . The approximate solutions by the VA, VIM and ADM are compared with the exact solution obtained by the ansatz method. The VIM gives better approximate solution of the coupled ZK equation than the VA. The absolute error and convergence analysis of the approximate solutions by the VIM and ADM show that the approximate solutions converge to the exact solution. The modulation instability is used to discuss the stability of the steady state solution of the coupled ZK equation and it demonstrates that the nonlinear term in the equation decides the modulational stability of the soliton solution.

The generalized derivative higher order non-linear Schrödinger (DNLS) equation describes pluses propagation in optical fibers and can be regarded as a special case of the generalized higher order non-linear Schrödinger equation. We derive a Lagrangian and the invariant variational principle for DNLS equation. Using the amplitude ansatz method, we obtain the different cases of the exact bright, dark and bright–dark solitary wave soliton solutions of the generalized higher order DNLS equation. By implementing the modulation instability analysis and stability analysis solutions, the stability analysis of the obtained solutions and the movement role of the waves are analyzed. All solutions are analytic and stable.

A simple but efficient approach for studying on nonlinear differential-difference equations

By using the extended hyperbolic function method, we havestudied a quintic discrete nonlinear Schrödinger equation and obtainednew exact localized solutions, including the discrete bright solitonsolution, dark soliton solution, bright and dark soliton solution,alternating phase bright soliton solution, alternating phase dark solitonsolution, and alternating phase bright and dark soliton solution, if aspecial relation is bound on the coefficients of the equation.

This study explores novel optical soliton solutions for the generalized derivative nonlinear conformable Schrödinger equation under the influence of multiplicative white noise. Using the new Kudryashov method, various solutions are derived, including solitary waves, bright, dark, singular, and W-shaped soliton solutions. The study investigates their dynamic behavior and physical characteristics, emphasizing the role of the conformable order derivative and temporal parameters through three-dimensional, two-dimensional, and contour plots. Incorporating multiplicative white noise into soliton analysis presents an innovative approach, advancing the understanding of nonlinear optical phenomena. Noise management techniques modeled in this study help simulate real-world scenarios where fibers face stochastic disturbances, aiding in the design of robust communication systems. Further, understanding noise’s impact on soliton stability offers insights for minimizing errors in signal processing and enhancing the reliability of optical fiber communication networks.

On symmetry preserving and symmetry broken bright, dark and antidark soliton solutions of nonlocal nonlinear Schrödinger equation

For isotropic and homogeneous nonlinear left-handed materials, for which the effective medium approximation is valid, Maxwell's equations for electric and magnetic fields lead naturally, within the slowly varying envelope approximation, to a system of coupled nonlinear Schrodinger equations. This system is equivalent to the well-known Manakov model that under certain conditions, is completely integrable, and admits bright and dark soliton solutions. It is demonstrated that left- and right-handed (normal) nonlinear media may have compound dark and bright soliton solutions, respectively [corrected] These results are also supported by numerical calculations.

Some new types of truncated M-fractional exact soliton solutions of the two important quantum plasma physics models, extended quantum Zakharov–Kuznetsov and extended quantum nonlinear Zakharov–Kuznetsov, are successfully achieved by applying the expa function technique, the improved (G′/G)-expansion technique, and the Sardar sub-equation technique. These two models have many useful applications when explaining the waves in the quantum electron-positron-ion magnetoplasmas as well as weakly nonlinear ion-acoustic waves in plasma. The obtained results are in the form of dark, bright, periodic, and other soliton solutions. The results are verified and represented by two-dimensional, three-dimensional, and contour graphs. The results are newer than the existing results in the literature due to the use of fractional derivatives. Hence, the solutions will be fruitful in future studies on these models. The solutions obtained are useful in the areas of applied physics, applied mathematics, dynamical systems, and nonlinear waves in plasmas and in dense space plasma. The applied techniques are simple, fruitful, and reliable for solving other models in mathematical physics.

In this paper we explore the new analytical soliton solutions of the truncated M-fractional nonlinear (1+1)-dimensional Akbota equation by applying the expa function technique, Sardar sub-equation and generalized kudryashov techniques. Akbota is an integrable equation which is Heisenberg ferromagnetic type equation and have much importance for the analysis of curve as well as surface geometry, in optics and in magnets. The obtained results are in the form of dark, bright, periodic and other soliton solutions. The gained results are verified as well as represented by two-dimensional, three-dimensional and contour graphs. The gained results are newer than the existing results in the literature due to the use of fractional derivative. The obtained results are very helpful in optical fibers, optics, telecommunications and other fields. Hence, the gained solutions are fruitful in the future study for these models. The used techniques provide the different variety of solutions. At the end, the applied techniques are simple, fruitful and reliable to solve the other models in mathematical physics.

Clout of fractional time order and magnetic coupling coefficients on the soliton and modulation instability gain in the Heisenberg ferromagnetic spin chain

<p>We have introduced various novel soliton waves and other analytic wave solutions for nonlinear Schrödinger equation with cubic, quintic, septic, and nonic nonlinearities. The modified extended direct algebraic method governs the transmission of various solitons with different effects. The combination of this system enables the obtaining of analytical soliton solutions with some unique behaviors, including bright, dark, and mixed dark-bright soliton solutions; singular soliton solutions; singular periodic, exponential, rational wave solutions; and Jacobi elliptic function solutions. These results realize the stability of the nonlinear waves' propagation in a high-nonlinear-dispersion medium that is illustrated using 2D and 3D visuals and contour graphical diagrams of the output solutions. This research focused on determining exact soliton solutions under certain parameter conditions and evaluating the stability and reliability of the soliton solutions based on the used modified extended direct algebraic method. This will be useful for many various domains in technology and physics, such as biology, optics, and plasma physical science. At the end, we use modulation instability analysis to assess the stability of the wave solutions obtained.</p>

Temporal behavior of bright and dark spatial solitons in photorefractive crystals having both the linear and quadratic electro-optic effects based on low amplitude approximations

This work employs the sub-ODE method to compute new soliton solutions for the component generalization of [Formula: see text]-dimensional Schwarz–Kortweg–de Vries (SKdV) equation in shallow water waves. The rational soliton solutions (RSS), Jacobian elliptic function (JEF), periodic solutions (PS), hyperbolic function solutions (HFS), positive solitons, dark soliton solutions (DSS) and bright soliton solutions (BSS) are constructed with this technique. Solitons-like solutions, HFS and trigonometric-function solutions (TFS) are also found as the JEF’s parameter m approaches values of 1 and 0, respectively. The suggested approach for the nonlinear equations in mathematical physics is concise, simple, effective, and promising.

In this article, four powerful techniques namely, the three wave method, double exponential, homoclinic breather, M-Shaped rational, M-Shaped with one kink, and M-Shaped with two kink are putting into practice to bolster the new optical solutions of the coupled form of the nonlinear (2+1)-dimensional Kundu-Mukherjee-Naskar equation in Fiber Bragg Gratings (FBGs). Implementation of the appropriate functions of the solutions results in a different form of multi-wave solutions with their interaction phenomena such as dark soliton solutions, singular soliton solutions, bright soliton solutions, kink soliton solutions, Multi-Peak soliton solutions, multi dark-bright soliton solutions, multi bright soliton solutions, multi dark soliton solutions with different structure, multi M-Shaped soliton solutions, multi M-Shaped soliton solutions, M-Shaped soliton solutions with different structure, kink soliton solutions with different structure, and periodical M-Shaped soliton solutions. Three-dimensional graphics demonstrating the dependability and productivity of the suggested methodologies are used to describe the dynamics of the produced solutions.