Rational solutions of Riccati-like partial differential equations. (English) Zbl 0985.34007

Li, Ziming; Schwarz, Fritz

J. Symb. Comput. 31, No. 6, 691-716 (2001).

The rational solutions to Riccati-like partial differential equations (for Riccati equations [see W. T. Reid, Riccati differential equations, New York-London: Academic Press (1972; Zbl 0254.34003)]) are considered. These systems arise in a similar way as Riccati ODEs. The structure of rational solutions is obtained, and an algorithm, called RationalSolution (it consists of 5 steps), for finding rational solutions to an associated Riccati-like system is given. As an application of the results obtained, the RationalSolution algorithm is applied to find all rational solutions to Lie’s system \[ \begin{gathered} \partial_xu+u^2+a_1u+a_2v+a_3=0,\quad\partial_xv+uv+c_1u+c_2v+c_3=0,\\ \partial_yu+uv+b_1u+b_2v+b_3=0,\quad\partial_yv+v^2+d_1u+d_2v+d_3=0, \end{gathered} \] where \(a_k,b_k,c_k,d_k\) are rational functions of \(x,y\) and hyperexponential solutions to linear homogeneous differential systems with finite linear dimension in several unknowns.

Reviewer: Valery V.Karachik (Tashkent)

MSC:

34A25	Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
35C05	Solutions to PDEs in closed form
34A05	Explicit solutions, first integrals of ordinary differential equations
65D15	Algorithms for approximation of functions
68W30	Symbolic computation and algebraic computation
35G20	Nonlinear higher-order PDEs

Keywords:

nonlinear PDEs; Riccati equation; coherent system; second-order ordinary differential equations; rational function; algorithm; differential polynomials

Citations:

Zbl 0254.34003

Software:

ALLTYPES

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