Unified structures for solutions of Painlev\'e equation II and Somos-4 like relations for the tau functions

TL;DR

This paper explores the interconnected structures of Painlevé II solutions, tau functions, and Somos-4 relations, revealing explicit rational formulas, special solutions, and degenerations including elliptic, polynomial, and transcendental cases.

Contribution

It introduces a unified framework linking Painlevé II, tau functions, and Somos-4 relations, with explicit formulas and analysis of special solutions and degenerations.

Findings

Explicit rational expressions linking Painlevé II and tau functions.

A non-autonomous Somos-4 type relation solved by these functions.

Degenerate cases include elliptic, polynomial, and transcendental solutions.

Abstract

We present certain general structures related to the solutions of Painlev\'e 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 B\"acklund transformations we find different explicit rational expressions linking the solutions of Painlev\'e equation II, Painlev\'e 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.