Soliton solutions of nonlinear coupled Davey–Stewartson Fokas system using modified auxiliary equation method and extended $$(G'/G^{2})$$-expansion method

The captivating realm of the nonlinear coupled Davey-Stewartson Fokas system is explored in this research paper. As a powerful tool, the proposed system is utilized for the realistic representation of various non-linear dynamical mechanisms in different fields of sciences and engineering including non-linear optical fibers, plasma physics and water waves theory. Two distinct exact methods, namely the modified auxiliary equation method and the extended <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo> <mml:msup><mml:mi>G</mml:mi> <mml:mo>'</mml:mo></mml:msup> <mml:mo>/</mml:mo> <mml:msup><mml:mi>G</mml:mi> <mml:mn>2</mml:mn></mml:msup> <mml:mo>)</mml:mo></mml:mrow> </mml:math> -expansion method, are utilized to acquire the exact soliton solutions of the non-linear coupled Davey-Stewartson Fokas system. A plethora of novel soliton solutions containing anti-kink, kink, bright, dark, dark-bright, bright-dark and some other singular soliton solutions, have been obtained using the employed exact methods. The significance of proposed manuscript lies in the novelty of obtained solutions. Kink, dark and bright solitons have wide applications in optical fiber communications, plasma physics and water waves dynamics. The acquired nontrivial exact solutions contain exponential, trigonometric, rational and hyperbolic functions. Some obtained solutions are visually represented through graphical simulations of 3D, 2D-contour and 2D-line plots, providing a comprehensive visualization of the soliton dynamics.The modulation instability of the proposed nonlinear system has been investigated, which ensures the stability of the system.