Soliton dynamics in dispersion-engineered media governed by a power-law higher-order nonlinear schrödinger equation

Abstract This paper investigates exact nonlinear wave dynamics governed by a generalized higher-order nonlinear Schrödinger equation with power-law nonlinearity and mixed spatio-temporal dispersion, a model that arises in dispersion-engineered optical media where conventional chromatic dispersion is negligible. Such regimes are increasingly relevant in photonic crystal fibers, metamaterials, and ultrashort-pulse optical communication systems, where higher-order dispersive and nonlinear effects dominate pulse evolution. To address the analytical challenges posed by this non-integrable higher-order model, the Modified Extended Direct Algebraic Approach (MEDAA) is employed. By reducing the governing partial differential equation to an ordinary differential equation through a traveling-wave transformation and introducing a six-parameter auxiliary equation, MEDAA enables a systematic and unified construction of exact analytical solutions without imposing restrictive trial functions or integrability assumptions. Using this framework, a broad spectrum of exact solutions is derived, including bright, dark, and singular solitons, rational-type waves, periodic solutions, and doubly periodic Jacobi and Weierstrass elliptic function solutions. Explicit parametric constraints ensuring the mathematical consistency and physical feasibility of each solution family are obtained, revealing smooth transitions between localized and periodic wave regimes. Representative two- and three-dimensional profiles are presented to illustrate the distinct morphological characteristics of the solutions. In addition, a detailed modulational instability analysis of the continuous-wave background is carried out to assess the physical relevance of the derived solutions. The instability growth rate and stability conditions are explicitly obtained, identifying parameter regimes that support stable pulse propagation and clarifying the competing roles of higher-order dispersion, self-phase modulation, and derivative nonlinear effects. The novelty of this work lies in the use of a generalized six-parameter auxiliary equation within MEDAA, which significantly enlarges the solution space compared to existing direct algebraic and expansion-based methods. Unlike previous studies that rely on restrictive ansätze or partial integrability, the present approach provides a unified analytical framework capable of generating diverse nonlinear wave structures for realistic higher-order optical models. These results extend earlier analytical treatments and offer new insights into nonlinear wave control in advanced dispersion-engineered media.