  The dynamics of nonlinear waves with controllable dispersion, nonlinearities,
and background continues to be an exciting line of research in recent years. In
this work, we focus to investigate an integrable (3+1)-dimensional nonlinear
model describing the evolution of {water waves} with higher-order temporal
dispersion by characterizing the dynamics of lump and soliton waves on
different spatially-varying backgrounds. We construct explicit lump wave and
kink-soliton solutions for the considered model through its trilinear and
bilinear equations with appropriate forms of the polynomial and
exponential-type initial seed solutions, respectively. Additionally, we obtain
hyperbolic and periodic waves through bilinear B\"acklund transformation. We
explore their propagation and transformation dynamics with spatial backgrounds
through a categorical analysis and clear graphical demonstrations to fully
understand the resulting solutions. Our analysis shows that the lump solution
results into a simultaneous existence of well-localized spike and declining
(coupled bright-dark type wave) structures on a constant background, while the
soliton solution exhibits kink and anti-kink wave patterns along different
spatio-temporal domains with controllable properties. When the arbitrary
spatial background is incorporated appropriately, we can witness the
manifestation of soliton into periodic and interacting dynamical structures. On
the other hand, the lump wave results into the coexistence, interacting wave
and breather formation due to periodic and localized type spatial backgrounds.
The results will be an important addition to the context of nonlinear waves
with controllable backgrounds in higher-dimensional models. Further, the
present study can be extended to investigate several other nonlinear systems to
understand the physical insights of the variable background in their dynamics.

Pattern Formation and Solitons