Exact solutions and modulation instability analysis of a generalized Kundu-Eckhaus equation with extra-dispersion in optical fibers
Ejaz Hussain, Syed Asif Ali Shah, Muhammad Naveed Rafiq, Adham E. Ragab, Emad A. Az-Zo’bi
Abstract
Abstract This study aims to examine the nonlinear partial differential equation known as the (1+1)-dimensional generalized Kundu-Eckhaus equation with extra-dispersion, which is used to model the transmission of ultra-short femtosecond pulses in an optical fiber. Two versatile techniques, namely the extended <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mfenced close=")" open="("> <mml:mrow> <mml:mstyle displaystyle="false"> <mml:mfrac> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo accent="false">′</mml:mo> </mml:mrow> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>G</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:mfrac> </mml:mstyle> </mml:mrow> </mml:mfenced> </mml:math> -expansion as well as the extended <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>exp</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mo>−</mml:mo> <mml:mi>ϕ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>ξ</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:math> -expansion techniques, are utilized to generate numerous precise answers. Diverse novel collections of exact traveling wave solutions, such as bright solitons, dark solitons, singular solitons, W-shape solutions, M-shape solutions, and rational solutions, are identified as a result. Several of the acquired solutions are interpreted physically through the use of figures. In addition, the modulation instability analysis of the considered equation is performed and presented via 3D and 2D graphs. In the field of nonlinear sciences, the proposed methods have great value and can be applied to other nonlinear evolutionary equations that are used to represent nonlinear physical models.