Constructing Physical Breather Solutions for the (2 + 1)- and (3 + 1)-Dimensional Extended KdV and KP Systems via the Hirota Bilinear Method

This paper presents an analysis of breather soliton solutions of the integrable extended KdV and KP equations in (2 + 1)- and (3 + 1)-dimensional settings. Employing symbolic computation along with Hirota’s bilinear formalism, we construct positive logarithmic function solutions for the corresponding bilinear equations associated with each model. These solutions serve as a foundation for generating families of breather solutions via appropriate dependent variable transformations tailored to each equation. The resulting breather solutions incorporate multiple free parameters, constrained by specific conditions necessary to ensure their existence. Unlike soliton or lump solutions, breathers capture time-dependent energy localization and modulation effects, making them particularly relevant in modeling transient phenomena such as rogue waves in shallow water, pulse dynamics in nonlinear optics, and localized excitations in plasma and condensed matter systems. Their study enhances the understanding of stability, transverse interactions, and energy transfer in realistic higher-dimensional physical environments. Also, all the obtained results satisfied their corresponding model equation when back-substituted. Furthermore, illustrative graphs of the solutions are provided for suitably selected parameter values.