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

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.


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


Zbl 0254.34003


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