  Electromagnetic Star-Planet Interaction (SPI) describes the phenomenon, when
a planet couples to its host star via electromagnetic forces. Alfv\'en waves
can establish such a coupling by forming Alfv\'en wings. SPI allows phenomena
that we do not know from the Solar System. Wing-wing interaction is such an
example, where the Alfv\'en wings of two planets merge and interact
non-linearly. In this paper we focus on the effects that SPI has on other
planets and the stellar wind. First, we analyse the different wave structures
connected to SPI. The second part then investigates wing-wing interaction. Our
study applies a magnetohydrodynamic model to describe a stellar system with
multiple possible planets. As an example, we chose TRAPPIST-1 and its two
innermost planets. We extended the PLUTO code to simulate collisions between
atmospheric neutral particles and plasma ions. Neutral gas clouds imitate the
planets and move through the simulation domain. That allows the simulation of
fully time-dependent stellar systems. We analysed the wave structures, which
result from the interaction between stellar wind and TRAPPIST-1 b. The inward
going wave structure is an Alfv\'en wing. The outward going part of the
interaction consists of an Alfv\'en wing, slow mode waves, the planetary wake
and a slow shock. We quantified the strength of the respective wave
perturbations at the outer planets to be on the order of 10\% to 40\% of the
local background values of thermal, magnetic and dynamic pressure. Wing-wing
interaction occurs due to the relative position of two planets during their
conjunction and shows three phases. First there is an initial, non-linear
intensification of the Poynting flux by $20\%$, an intermediate phase with
reduced Poynting flux and a third phase when the Alfv\'en wing of planet c goes
through planet b's wave structures with another intensification of the Poynting
flux.
