Unveiling of Highly Dispersive Dual-Solitons and Modulation Instability Analysis for Dual-Mode Extension of a Non-Linear Schrödinger Equation

The two-mode equations are nonlinear models that describe the behavior of two-way waves moving simultaneously while being affected by confined phase velocity. This article expands a non-linear Schrödinger equation (NLSE) by constructing it as a dual-mode structure. Applying the modified extended direct algebraic method (MEDAM) yields exact and explicit solutions. The results of this investigation have significant implications for the propagation of solitons in nonlinear optics. There are multiple resulted solutions that comprise singular periodic solutions, Weierstrass elliptic doubly periodic solutions, Jacobi elliptic function (JEF), singular soliton, bright soliton, dark soliton, and rational solutions, moreover, hyperbolic wave solutions. We show our acquired traveling wave solutions’ uniqueness and significant addition to current research by contrasting them with the body of existing literature. The method’s effectiveness shows that it may be used to address a wide variety of nonlinear problems across multiple disciplines, particularly in the theory of soliton, as the studied model appears in many applications. Additionally, we display the outlines of some of these discovered solution behaviors in 3D and 2D graphs to help with comprehension. Finally, we analyze modulation instability to examine the stability of the discovered solutions.