The evolution of electromagnetic (EM) solitons due to nonlinear coupling of circularly polarized intense laser pulses with low-frequency electron-acoustic perturbations is studied in relativistic degenerate dense astrophysical plasmas with two groups of electrons: a sparse population of classical relativistic electrons and a dense population of relativistic degenerate electron gas. Different forms of localized stationary solutions are obtained and their properties are analyzed. Using the

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... v-Kolokolov stability criterion, the conditions for the existence and stability of a moving EM soliton are also studied. It is noted that the stable and unstable regions shift around the plane of soliton eigenfrequency and the soliton velocity due to the effects of relativistic degeneracy, the fraction of classical to degenerate electrons and the EM wave frequency. Furthermore, while the standing solitons exhibit stable profiles for a longer time, the moving solitons, however, can be stable or unstable depending on the degree of electron degeneracy, the soliton eigenfrequency and the soliton velocity. The latter with an enhanced value can eventually lead to a soliton collapse. The results should be useful for understanding the formation of solitons in the coupling of highly intense laser pulses with slow response of degenerate dense plasmas in the next generation laser-plasma interaction experiments as well as the novel features of x-ray and γ-ray pulses that originate from compact astrophysical objects.