The Institute of Physics (IOP) has written a report that examines the relationship between mathematics and physics.

There is no completely agreed upon definition of mathematical physics, which is closer to mathematics that physics in most respects. The IOP report puts it like this;

Mathematical physics is best described as consisting of two parts: physical research that proceeds primarily through mathematical means and areas of mathematics that work to solve the problems posed by physics.

The Journal of Mathematical Physics define mathematical physics as;

…the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories.

The IOP booklet looks at a wide range of topics in mathematical physics from quantum mechanics, gravity & black holes, random matrix theory, solitons and topological insulators.

Link
Mathematical Physics: What is it and why do we need it?