Superposed nonlinear waves and transitions in a (3+1)-dimensional variable-coefficient eight-order nonintegrable Kac–Wakimoto equation

In this work, we investigate the Kac–Wakimoto equation associated with Lie algebra e16, which is one of the highest-order Hirota bilinear models in soliton theory. As the present model is nonintegrable, it does not admit the Painlevé integrability test, Lax pairs, or N-soliton solution, the focus of the present work is to construct explicit soliton solutions and study the influence of variable coefficients and arbitrary background by identifying a simple yet effective mathematical method. Especially, we wish to unearth the dynamical features of one- and two-solitons (solitary waves) by exploring the spatially-varying dispersion-nonlinearity coefficients with an additional temporally-modulated background in e16 equation. Particularly, the impact of hyperbolic and solitary waves varying in one spatial direction modulates the kink solitons by inducing different nonlinear transitions, revealing phenomena like bending, reflection, fusion, oscillation, and V-type soliton formations. Additionally, we observe the superposition of kink-soliton with V-type soliton, localized bell-type bright solitons, well and double-well type dark solitons, and periodic waves with interesting dynamics. Our results are well explained through extensive analysis and categorical graphical illustrations that ensure better understanding. Though the adapted methodology looks simple, it is much more efficient and systematic in exploring several classes of variable-coefficient nonlinear models, which cannot be possible with many other sophisticated techniques. The presented results will be a good addition of new knowledge to the existing literature on nonlinear waves (solitons).