New soliton waves and modulation instability analysis for a metamaterials model via the integration schemes

Abstract In this paper, the exact analytical solutions to the generalized Schrödinger equation are investigated. The Schrodinger type equations bearing nonlinearity are the important models that flourished with the wide-ranging arena concerning plasma physics, nonlinear optics, fluid-flow, and the theory of deep-water waves, etc. In this exploration, the soliton and other traveling wave solutions in an appropriate form to the generalized nonlinear Schrodinger equation by means of the extended sinh-Gordon equation expansion method, tan(Γ(ϖ))-expansion method, and the improved cos(Γ(ϖ)) function method are obtained. The suggested model of the nonlinear Schrodinger equation is turned into a differential ordinary equation of a single variable through executing some operations. One soliton, periodic, and singular wave solutions to this important equation in physics are reached. The periodic solutions are expressed in terms of the rational functions. Soliton solutions are obtained from them as a particular case. The obtained solutions are figured out in the profiles of 2D, density, and 3D plots by assigning suitable values of the involved unknown constants. Modulation instability (MI) is employed to discuss the stability of got solutions. These various graphical appearances enable the researchers to understand the underlying mechanisms of intricate phenomena of the leading equation. The individual performances of the employed methods are praiseworthy which deserves further application to unravel any other nonlinear partial differential equations (NLPDEs) arising in various branches of sciences. The proposed methodologies for resolving NLPDEs have been designed to be effectual, unpretentious, expedient, and manageable.