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 $r=\pi /\sqrt{2\Lambda }$, 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 $r\in [0,\pi /\sqrt{2\Lambda }]$, 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: $f\left(u\right)={(1-\tilde{\beta }|E\left|\right)}^{-1},h\left(u\right)=1$; and $f\left(u\right)=1,h\left(u\right)=\sqrt{1-\tilde{\beta }|E{|}^{\upsilon }}$, with $\upsilon =1,2$, where $u=\left|E\right|/E_p$, $\tilde{\beta }=\beta /E_p$, and β is the rainbow parameter. We find that such pairs of rainbow functions, $\left(f\right(u),h(u\left)\right)$, 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.