Klein-Gordon particles in a nonuniform external magnetic field in Bonnor-Melvin rainbow gravity background

We investigate the effect of rainbow gravity on Klein-Gordon (KG) bosons in a quantized nonuniform magnetic field in the background of Bonnor-Melvin (BM) spacetime with a cosmological constant. In the process, we show that the BM spacetime introduces domain walls (i.e., infinitely impenetrable hard walls) at $r=0$ and $r=\pi \mathrm{/}\sqrt{2\mathrm{\Lambda }}$, as a consequence of the effective gravitational potential field generated by such a magnetized BM spacetime. As a result, the motion of KG particles/antiparticles is restricted indefinitely within the range $r\in [0,\pi \mathrm{/}\sqrt{2\mathrm{\Lambda }}]$, and the particles and antiparticles cannot be found elsewhere. Next, we provide a conditionally exact solution in the form of the general Heun function $H_G(a,q,\alpha ,\beta ,\gamma ,\delta ,z)$. Within the BM domain walls and under the condition of exact solvability, we study the effects of rainbow gravity on KG bosonic fields in a quantized nonuniform external magnetic field in the BM spacetime background. We use three pairs of rainbow functions: $f(u)=(1-\tilde{\beta }\mathrm{\mid }E\mathrm{\mid }{)}^{-1},h(u)=1$; and $f(u)=1,h(u)=\sqrt{1-\tilde{\beta }\mathrm{\mid }E{\mathrm{\mid }}^{\upsilon }}$, with $\upsilon =1,2$, where $u=\mathrm{\mid }E\mathrm{\mid }\mathrm{/}{E}_p$, $\tilde{\beta }=\beta \mathrm{/}{E}_p$, and $\beta$ is the rainbow parameter. We find that such pairs of rainbow functions, $(f(u),h(u))$, fully comply with the theory of rainbow gravity, ensuring that $E_p$ is the maximum possible energy for particles and antiparticles alike. Moreover, we show that the corresponding bosonic states form magnetized, rotating vortices, as intriguing consequences of such a magnetized BM spacetime background.