Riemann-Hilbert characterisation of Painlev\'e 5 asymptotics and nonlinear monodromy-Stokes structure

For a generic Painlev\'e 5 equation we characterise all the asymptotics in a right half plane near the point at infinity, that is, we find classified explicit solutions that are, by the Riemann-Hilbert correspondence, labelled with monodromy data filling up the whole monodromy manifold. To do so, in addition to the asymptotics by Andreev and Kitaev along the positive real axis, we require elliptic asymptotics along generic directions and newly provided truncated solutions arising from a general solution along the imaginary axes. To know analytic continuations outside this region we formulate a nonlinear monodromy-Stokes structure, which is observed as changes of monodromy data contained in the explicit expressions of solutions.