Generalized conditional symmetries of evolution equations

On time-dependent symmetries and formal symmetries of evolution equations

We present the explicit formulae, describing the structure of symmetries and formal symmetries of any scalar (1+1)-dimensional evolution equation. Using these results, the formulae for the leading terms of commutators of two symmetries and two formal symmetries are found. The generalization of these results to the case of system of evolution equations is also discussed.

Nonlocal Symmetries of Systems of Evolution Equations

We prove that any potential symmetry of a system of evolution equations reduces to a Lie symmetry through a nonlocal transformation of variables. Based on this fact is our method of group classification of potential symmetries of systems of evolution equations having non-trivial Lie symmetry. Next, we modify the above method to generate more general nonlocal symmetries, which yields a purely al...

Higher Conditional Symmetries and Reduction of Initial Value Problems for Nonlinear Evolution Equations

We prove that the presence of higher conditional symmetry is the necessary and sufficient condition for reduction of an arbitrary evolution equation in two variables to a system of ordinary differential equations. Furthermore, we give the sufficient condition for an initial value problem for an evolution equation to be reducible to a Cauchy problem for a system of ordinary differential equation...

On symmetries of KdV-like evolution equations

It is well known that provided scalar (1+1)-dimensional evolution equation possesses the infinitedimensional commutative Lie algebra of time-independent non-classical symmetries, it is either linearizable or integrable via inverse scattering transform [1, 2]. The standard way to prove the existence of such algebra is to construct the recursion operator [2]. But Fuchssteiner [3] suggested an alt...

Algorithmic computation of generalized symmetries of nonlinear evolution and lattice equations