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When superconducting (SC) and charge-density wave (CDW) orders compete, novellow temperature behaviors can result. From an analysis of theLandau-Ginzberg-Wilson theory of competing orders, we demonstrate the genericoccurrence of a `fragile' SC phase at low temperatures and high fields in thepresence of weak disorder. Here, the SC order is largely concentrated in thevicinity of dilute dislocations in the CDW order, leading to transitiontemperatures and critical currents that are parametrically smaller than thosecharacterizing the zero field SC phase. This may provide the outline of anexplanation of the recently discovered `resilient' superconducting phase athigh fields in underdoped YBa$_2$Cu$_3$O$_{6+\delta}$.

Fragile superconductivity in the presence of weakly disordered charge density waves