New graphical observations for KdV equation and KdV–Burgers equation using modified auxiliary equation method

This study is made to extract the exact solutions of Korteweg–de Vries–Burgers (KdVB) equation and Korteweg–de Vries (KdV) equation. The original idea of this work is to investigate KdV equation and KdVB equation for possible closed form solutions by employing the modified auxiliary equation (MAE) method. Exact traveling wave solutions of the considered equations are retrieved in the form of trigonometric and hyperbolic functions. Kink, periodic and singular wave patterns are obtained from the constructed solutions. The graphical illustration of the wave solutions is presented using 3D-surface plots to acquire the understanding of physical behavior of the obtained results up to possible extent.

Traveling Wave Solutions and Chaotic Motions for a Perturbed Nonlinear Schrödinger Equation with Power-Law Nonlinearity and Higher-Order Dispersions

This chapter aims to study and solve the perturbed nonlinear Schrödinger (NLS) equation with the power-law nonlinearity in a nano-optical fiber, based upon different methods such as the auxiliary equation method, the Stuart and DiPrima’s stability analysis method, and the bifurcation theory. The existence of the traveling wave solutions is discussed, and their stability properties are investigated through the modulational stability gain spectra. Moreover, the development of the chaotic motions for the systems is pointed out via the bifurcation theory. Taking into account an external periodic perturbation, we have analyzed the chaotic behavior of traveling waves through quasiperiodic route to chaos.

Simulation of bright–dark soliton solutions of the Lonngren wave equation arising the model of transmission lines

In this study, the auxiliary equation method is applied successfully to the Lonngren wave equation. Bright soliton, bright–dark soliton solutions are produced, which play an important role in the distribution and distribution of electric charge. In the conclusion and discussion section, the effect of nonlinearity term on wave behavior in bright soliton traveling wave solution is examined. The advantages and disadvantages of the method are discussed. While graphs representing the stationary wave are obtained, special values are given to the constants in the solutions. These graphs are presented as 3D, 2D and contour.

Cubic-Quartic Optical Solitons in Bragg Gratings Fibers for Perturbed NLSE Having Parabolic-Nonlocal Law of Refractive Index Using Two Integration Schemes

The optical solitons in Bragg gratings fibers for perturbed NLSE having cubic-quartic dispersive reflectivity with parabolic-nonlocal combo law of refractive index are studied. The extended auxiliary equation method and the addendum to Kudryashov's method are introduced. The existence criteria for such solitons are indicated.

Stability and Extraction of New Optical Wave Solutions of Cascaded System in (2 + 1)-dimensions

This paper uses the new modified sub-ODE method, the new extended auxiliary equation method, and the new Jacobi elliptic function expansion method to revisit the (2+1)-dimensional coupled nonlinear Schr¨odinger equation with cascading effect. As a consequence, dark, bright, kinkantikink, singular solitons, Weierstrass elliptic function, doubly periodic, and complex optical soliton solutions are retrieved. All solutions are described, along with the existence criterion on the parameters. As solitons are used for data transfer, the obtained results may be found usage in optical couplers, birefringed fibres, and optoelectronic devices. A comparison of the obtained results with those found in the literature is given. The dynamical behaviour of some of the obtained solutions has been explored for suitable choices of the parameters. Using the property of Hamiltonian systems, the solitons stability is determined.

New and more dual-mode solitary wave solutions for the Kraenkel-Manna-Merle system incorporating fractal effects

This paper introduces the fractal Kraenkel-Manna-Merle (KMM) system, that explains nonlinear short wave propagation with zero conductivity for saturated ferromagnetic materials in an external field. The semi inverse technique and the new auxiliary equation method (NAEM) are used to generate a new set of solutions. The proposed methods are more straightforward, succinct, accurate, and simple to calculate dual mode solitary wave solutions. A collection of exact soliton solutions specifically bright, dark, singular-shaped and singular-periodic are generated. The estimated solutions are obtained using constraint conditions and are displayed through 2D, 3D and contour plots with appropriate parametric values. The arbitrary functions in the solutions are chosen as unique functions to generate some novel soliton structures.