In this work, we obtain exact thick brane models in 4 + 1 dimensions generated by higher order field theory kinks, inspired by specific potentials for φ10 and phi^18 models. We verify that the geodesic equation along the fifth dimension confirms the confining effects of the scalar field on the brane for all of these models. These models provide new solutions with exponential and power-law tails which live in different topological sectors. We show that the resulting branes of specific exponential law models do not possess Z2 -symmetry. Furthermore, we examine the stability of the thick branes, by determining the sign of the w2 term in the expansion of the potential for the resulting Schrödinger-like equation. It turns out that two of the three models of the phi^10 brane are stable, while another contains unstable modes for certain ranges of the model parameters. We also show that the brane solution from the specific phi^18 models are stable, while the others involve neutral equilibrium. The asymptotic behaviour of the brane solutions are also discussed.

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