New self-similar Jacobi rational and polynomial waves in inhomogeneous two-mode fibers: Exact solutions of generalized coupled NLSEs modified

We investigate self-similar periodic and solitary waves in inhomogeneous two-mode fibers governed by a generalized variable-coefficient coupled nonlinear Schrödinger system with ten independent coefficients that account for asymmetric dispersion, Kerr nonlinearities, linear gain/loss, and external potentials in each mode. A constructive similarity transformation maps the variable-coefficient equations to a constant-coefficient pair, yielding explicit relations for the similarity variables, amplitudes, and phases. On the reduced system, an F-expansion framework produces five solvable families of exact solutions classified as rational and polynomial Jacobi-type periodic waves, together with their existence constraints and dispersion relations. The Jacobi families continuously connect to localized states—bright, dark, kink/antikink, and W-shaped solitons—in the elliptic-modulus limit. Mapping back provides chirped self-similar solutions of the original inhomogeneous model. Using prototype dispersion and gain/loss profiles, we quantify pulse energy, width, and chirp to track periodic-to-soliton transitions under parameter modulation. The results substantially enlarge the known solution space for vector NLSEs beyond symmetric, few-coefficient settings and offer analytic control knobs—via the engineered coefficient landscapes—for amplitude, width, and phase, with relevance to pulse shaping and robust transmission in fiber systems.