Unified structures for solutions of Painlevé equation II and Somos-4 like relations for the tau functions

We present certain general structures related to the solutions of Painlevé equation II and to the solutions of the differential equation satisfied by the corresponding Hamiltonian equations, together with the tau functions. By taking advantage of the Bäcklund transformations we find different explicit rational expressions linking the solutions of Painlevé equation II, Painlevé equation XXXIV and the Hamiltonians with the tau functions. Wronskians among different tau functions and the derivatives of the tau functions themselves will be expressed in terms of rational functions of tau functions too. A non-autonomous Somos-4 type relation solved by these functions is given. For the Somos-4 type relation we consider degenerate cases through the use of suitable parameters inserted into the equations: the autonomous case solvable in terms of Weierstrass elliptic functions, the case corresponding to the Yablonskii-Vorob’ev polynomials, the Airy-type solutions and the more general transcendental case.

💡 Research Summary

The paper investigates a unified framework that links the Painlevé II differential equation, its associated Hamiltonian system, and the corresponding τ‑functions. Starting from a parametrised Hamiltonian
(H_n = \frac{p_n^2}{2m} - a p_n (q_n^2 + b z + c) - e_n q_n - \frac{m g^2}{2 a^2})
with parameters (a,b,c,m,e_n,g) (where (e_n = e + 2 a b m n)), the authors write the non‑autonomous Hamilton equations for ((q_n,p_n)). By scaling (H_n = 2 b m,h_n) they obtain a central second‑order nonlinear ODE for (h_n) (equation 7), which simultaneously encodes Painlevé II (for (q_n)) and Painlevé XXXIV (for (p_n)). This ODE admits rational, elliptic, Airy‑type, and generic transcendental solutions, depending on the choice of the parameters.

The core of the analysis is the use of Okamoto’s Bäcklund transformations, written as a discrete map ((q_n,p_n)\mapsto (q_{n+1},p_{n+1})). These transformations preserve the Hamiltonian and induce a discrete dynamics for (h_n): \