Unified structures for solutions of Painlev\'e equation II

Autor:	Zullo, Federico
Rok vydání:	2024
Předmět:	Nonlinear Sciences - Exactly Solvable and Integrable Systems
Druh dokumentu:	Working Paper
Popis:	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 a linear differential equation satisfied by the tau functions and explicit rational expressions linking the solutions of Painlev\'e equation II, Painlev\'e equation XXXIV and the Hamiltonians with the tau functions. Also, recursions for the tau functions and the Hamiltonians will be given. Through the use of suitable parameters inserted into the equations, we consider a degenerate case solvable in terms of Weierstrass elliptic functions. The Yablonskii-Vorob'ev polynomials, the Airy-type solutions and the more general transcendental case are all described in separated Sections. Finally, certain integral relations are given. Comment: 20 pages
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/2412.21018 Zobrazit plný text záznamu View this record from Arxiv