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The purpose of this study is to introduce the wave structures and dynamical features of the third-order nonlinear Schrödinger equations (TONLSE). We take the original equation and, using the traveling wave transformation, convert it into the appropriate traveling wave system, from which we create a conserved quantity known as the Hamiltonian. The Jacobian elliptic function solution (JEF), the hyperbolic function solution, and the trigonometric function solution are just a few of the optical soliton solutions to the equation that may be found using the complete discrimination system (CDS) of polynomial method (CDSPM) and also transfer the JEF into solitary wave (SW) soltions. It also includes certain dynamic results, such as bifurcation points and critical conditions for solutions, that might be utilized to explore the dynamic features of the equation employing the CDSPM. This method could also be used for qualitative analysis. The qualitative analysis is used to illustrate the equilibrium points and phase potraits of the equation. Phase portraits are visual representations used in dynamical systems to illustrate a system's behaviour through time. They can provide crucial information about a system's stability, periodic behaviour, and the presence of attractors or repellents. [ABSTRACT FROM AUTHOR]

In this paper, we will find the chirped solitons (CS) solutions to the nonlinear schrödinger's equation (NLSE) under polynomial nonlinearity with spatio temporal dispersion (STD) and group velocity dispersion (GVD). We will find CS with Jacobi elliptic functions (JEF). We will also get some new solitary waves (SW) like bright, dark, kink, periodic, hyperbolic, and other solutions. The dynamical behaviour for these waves will also be discussed along with constraint conditions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate resonanat nonlinear Schrödinger equation (RNLSE) with self steeping phenomena to obtain some chirped periodic (CP) and soliton waves. A chirp is a signal in which the frequency increases (up chirp) or decreases (down-chirp) with time. It is commonly used in sonar, radar and laser systems and in other applications, such as in spread-spectrum communications. We obtain chirped periodic waves (CPW) with some Jacobi elliptic functions (JEF). We also obtain some solitary waves (SW) like dark, bright, singular waves of type I and II, hyperbolic, periodic and other solutions. The dynamical behaviour for these waves will also be presented. [ABSTRACT FROM AUTHOR]

In this paper, we study chirp-free solitons (dromions) for the higher-order nonlinear Schrödinger equation (HNLSE) having non-Kerr terms with the aid of sub-ODE scheme. We find bright and singular dromion with domain walls, rational and periodic solitons, Weierstrass elliptic functions (WEFs), Jacobi elliptic solutions (JESs) and other solitary wave solutions for our governing model. The Sub-ODE scheme is very useful scheme, providing so many new solitary wave solutions especially JES and WEF solutions. We will also construct graphs of our results in different dimensions along with constraint conditions. [ABSTRACT FROM AUTHOR]

In this article, we construct multi-wave, homoclinic breather, M-shaped rational and periodic cross kink wave solutions for coupled-Higgs equation (CHE) by using distinct transformations. We obtain these solutions with the aid of logarithmic transformations and symbolic computations. Moreover, we also derive some soliton wave solutions for CHE in polynomial forms. These solutions include solitary wave, soliton wave and Jacobi elliptic function solutions and are found by implementing unified technique. We will also discuss the graphical structure of our newly achieved results. [ABSTRACT FROM AUTHOR]