This investigation focuses on solitary wave solutions and dynamic analysis of the complex coupled Maccari system. We employ the new extended hyperbolic function method to establish bright-wave and dark-wave profiles of the model. The resulting solutions include hyperbolic, trigonometric, and exponential-type functions. Furthermore, we explore the model's dynamical characteristics via multiple perspectives, including phase portrait analysis, quasi-periodic and chaotic patterns, sensitivity analysis and Lyapunov exponent. The analysis validates the robustness of the new extended hyperbolic function method on one hand and extends the understanding of complex wave structures in Maccari's system on the other hand.

In this paper, we study the boundedness of the commutator of the rough fractional Hausdorff operator on grand-variable-Herz-Morrey spaces when the symbol functions belong to bounded mean oscillations (BMO) space.