Searching for characteristic signatures of a higher order phase transition ͑specifically of the order of three or four͒, we have calculated the spatial profiles and the energies of a spatially varying order parameter in one dimension. In the case of a pth order phase transition to a superconducting ground state, the free energy density depends on temperature as a p , where a = a 0 ͑1−T / T c ͒ is the reduced temperature. The energy of a domain wall between two degenerate ground states is ⑀ p Ӎ a p−1/2 . We have also investigated the effects of a supercurrent in a narrow wire. These effects are limited by a critical current which has a temperature dependence J c ͑T͒Ӎa ͑2p−1͒/2 . The phase slip center profiles and their energies are also calculated. Given the suggestion that the superconducting transition in ͑Ba 1−x K x ͒BiO 3 , for x = 0.4, may be of the order of four, these predictions have relevance for future experiments.