Inthispaper,weconsideranonlinearsystemofreactiondiffusionequationsarisingfrommathematicalneuroscienceandtwononlinearscalarreactiondiffusionequationsundersomeassumptionsontheircoefficients.Themainpurposeistocoupletogetherlinearizedstabilitycriterion(theequivalenceofthenonlinearstability,thelinearstabilityandthespectralstabilityofthestandingpulsesolutions)andEvansfunctionstoaccomplishtheexistenceandinstabilityofstandingpulsesolutionsofthenonlinearsystemofreactiondiffusionequationsandthenonlinearscalarreactiondiffusionequations.TheEvansfunctionsforthestandingpulsesolutionsareconstructedexplicitly.