Unearthing the existence of intermode soliton-like solutions within integrable quintic Kundu–Eckhaus equation

The main objective of this study is to generate and analyze a wide range of optical soliton and many other traveling wave solutions for the integrable nonlinear complex Quintic Kundu–Eckhaus equation (QKEE), which explains optical solitons’ dynamics and interactions in fiber optic systems. For this purpose, the new extended direct algebraic method (NEDAM) is employed to efficiently discover a new plethora of optical soliton solutions for the aimed QKEE. After implementing a wave transformation, the recommended NEDAM transforms QKEE into a nonlinear ordinary differential equation (NODE), which is then converted into a nonlinear algebraic equation system. Using Maple, the solution of the resultant system facilitates the extraction of the new families of optical soliton solutions and many other physical solutions. Using Mathematica software, some 2D, 3D, and contour graphics are presented that graphically represent obtained optical soliton solutions for various parameter choices to enhance comprehension of the results. These graphics illustrate the presence of intermodal breather solitons (IBSs) within the domain of QKEE and many other physical wave solutions, including bright, dark, and kink solitons. The results not only introduce new soliton dynamics into the model but also demonstrate how these dynamics interact to impact the overall behavior of the optical system.