The subject of physical mathematics is concerned with physically motivated mathematics and is different from mathematical physics. The Journal of Physical Mathematics is an important journal in the field.
String theorist Greg Moore said this about physical mathematics in his vision talk at Strings 2014.
"The use of the term “Physical Mathematics” in contrast to the more traditional “Mathematical Physics” by myself and others is not meant to detract from the venerable subject of Mathematical Physics but rather to delineate a smaller subfield characterized by questions and goals that are often motivated, on the physics side, by quantum gravity, string theory, and supersymmetry, (and more recently by the notion of topological phases in condensed matter physics), and, on the mathematics side, often involve deep relations to infinite-dimensional Lie algebras (and groups), topology, geometry, and even analytic number theory, in addition to the more traditional relations of physics to algebra, group theory, and analysis."