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Special symmetries to standard Riccati equations and applications. (English) Zbl 1202.34007

The general Riccati equation

dϕ(ξ) dξ=p(ξ)ϕ 2 (ξ)+q(ξ)ϕ(ξ)+r(ξ)(1)

is studied. Here, p,q and r are continuous functions, defined on some interval [a,b]. Using Lie group symmetry, the authors obtain new integrability conditions for the generalized Riccati equation. Using this condition, 7 families of Riccati equations in standard form (1) are obtained which are integrable by quadratures. The obtained results are applied to construct travelling wave solutions for nonlinear evolution equations.

MSC:
34A05Methods of solution of ODE
34C14Symmetries, invariants (ODE)
34A34Nonlinear ODE and systems, general
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