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The number of polynomial solutions of polynomial Riccati equations
Gasull i Embid, Armengol (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Torregrosa, Joan  (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Zhang, Xiang (Shanghai Jiao Tong University. Department of Mathematics)

Date: 2016
Abstract: Consider real or complex polynomial Riccati differential equations a(x) y=b_0(x) b_1(x)y b_2(x)y^2 with all the involved functions being polynomials of degree at most . We prove that the maximum number of polynomial solutions is 1 (resp. 2) when 1 (resp. =0) and that these bounds are sharp. For real trigonometric polynomial Riccati differential equations with all the functions being trigonometric polynomials of degree at most 1 we prove a similar result. In this case, the maximum number of trigonometric polynomial solutions is 2 (resp. 3) when 2 (resp. =1) and, again, these bounds are sharp. Although the proof of both results has the same starting point, the classical result that asserts that the cross ratio of four different solutions of a Riccati differential equation is constant, the trigonometric case is much more involved. The main reason is that the ring of trigonometric polynomials is not a unique factorization domain.
Note: Número d'acord de subvenció MINECO/MTM2013-40998-P
Note: Número d'acord de subvenció AGAUR/2014/SGR–568
Note: Número d'acord de subvenció EC/FP7/2012/316338
Note: Agraïments: The third author is partially supported by the NNSF of China 11271252 and the Innovation program of Shanghai Municipal Education Commission of China 15ZZ02.
Rights: Tots els drets reservats.
Language: Anglès.
Document: article ; recerca ; submittedVersion
Subject: Explicit solutions ; Number of polynomial solutions ; Polynomial differential equations ; Riccati differential equations ; Trigonometric polynomial differential equations
Published in: Journal of Differential Equations, Vol. 261 (2016) , p. 5071-5093, ISSN 0022-0396

DOI: 10.1016/j.jde.2016.07.019


Preprint
21 p, 377.6 KB

The record appears in these collections:
Research literature > UAB research groups literature > Research Centres and Groups (scientific output) > Experimental sciences > GSD (Dynamical systems)
Articles > Research articles
Articles > Published articles

 Record created 2017-01-23, last modified 2019-07-31



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