Exact Traveling Waves and Bifurcation Structure of a Coupled Kudryashov-Type Nonlinear Schrödinger System via the Modified Extended Mapping Method
Abstract
This study obtains exact traveling wave solutions and conducts a bifurcation analysis for the coupled (1+1)-dimensional Kudryashov’s equation. This system of nonlinear Schröodinger-type equations models the simultaneous propagation of optical pulses in a birefringent fiber, incorporating critical higher-order effects like cubic-quartic dispersion and nonlinearities for enhanced physical accuracy. By applying the modified extended mapping method (MEMM), the coupled partial differential equations are systematically reduced to a
manageable ordinary differential equation. This reduction allows for the construction of a broad spectrum of exact solutions such as bright and dark solitons, periodic waves, periodic singular solutions, along with singular solitons, in addition to hyperbolic, exponential, rational, and Weierstrass elliptic function solutions. A key component of the work is a detailed bifurcation analysis of the resulting planar dynamical system. This analysis identifies critical parameter thresholds and classifies all possible qualitative behaviors of the wave solutions, mapping out regions of existence for different solution types. The physical implications of both the derived solutions and the bifurcation structure are discussed in the context of nonlinear optics, particularly for pulse dynamics and stability in dualpolarization optical communication systems. The results demonstrate the efficacy of MEMM and provide new insights that contribute to the understanding and design of advanced optical waveguides.
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