Explicit position-space profiles for Y-Hamiltonian -5/2 cascade solutions

Derive explicit analytical formulas for the position-space profile |u(t,θ)|^2 = |Σ_{n=0}^{∞} α_n(t) e^{i n θ}|^2 during the -5/2 energy cascade in the Y-Hamiltonian systems defined by couplings in Equation (C_nmij_equation_Y), characterizing the large-scale θ-dependence beyond approximations via polylogarithmic expansions of g_n.

Background

For the Y-Hamiltonian family, the -5/2 cascade leads to θ-dependent large-scale structures and cusp formation at small scales in the spatial profile u(t,θ). While the qualitative behavior is described and cusp formation is justified via asymptotics of g_n, the paper does not provide closed-form expressions for the full θ-dependent profiles.

Explicit formulas would enable precise characterization of scaling behavior, amplitude distributions, and singularity formation, facilitating comparison with numerical studies and other analytical models.