Rational Solutions of Riccati-like Partial Differential Equations

When factoring linear partial differential systems with a finite-dimensional solution space or analyzing symmetries of nonlinear ode’s, we need to look for rational solutions of certain nonlinear pde’s. The nonlinear pde’s are called Riccati-like because they arise in a similar way as Riccati ode’s. In this paper we describe the structure of rational solutions of a Riccati-like system, and an algorithm for computing them. The algorithm is also applicable to finding all rational solutions of Lie’s system {∂xu+ u + a1u+a2v+a3, ∂yu+uv+b1u+b2v+b3, ∂xv+uv+c1u+c2v+c3, ∂yv+v+d1u+d2v+d3}, where a1 . . . d3 are rational functions of x and y.