Integrability, and stability aspects for the non-autonomous perturbed Gardner KP equation: Solitons, breathers, Y-type resonance and soliton interactions

This article examines the soliton-type solutions, their interactions, and the integrable properties of a non-autonomous perturbed Gardner KP (NPGKP) equation. For the NPGKP equation under consideration, a bilinear structure, and a Bäcklund transformation are designed explicitly, which claim the integrability of the system under some constraints. The stability of the obtained solutions is discussed using modulation instability. The bilinear form demonstrates the dynamic characteristics of multiple solitons, breathers, and their interactions in response to external impulses. Furthermore, it allows for determining the solitons’ amplitudes and velocities. The two-soliton solution yields a first-order breather solution. At the same time, the analytical investigation focuses on the interaction between a single breather and a single-soliton within the multi-soliton solution. This investigation also identifies the resonance of $Y$-shaped solitons and examines the dynamical characteristics of the interaction between resonant $Y$-shaped solitons and $M$-fissionable pulses. The multi-shock solutions and the collisions between shocks are analyzed in the presence of external influences. Graphical representations of the relationships between different sorts of achieved solutions are provided explicitly.