Time-dependent symmetries of variable-coefficient evolution equations and graded Lie algebras

Polynomial-in-time dependent symmetries are analysed for polynomial-in-time dependent evolution equations. Graded Lie algebras, especially Virasoro algebras, are used to construct nonlinear variable-coefficient evolution equations, both in 1 + 1 dimensions and in 2 + 1 dimensions, which possess higher-degree polynomial-in-time dependent symmetries. The theory also provides a kind of new realization of graded Lie algebras. Some illustrative examples are given.