Extended Rotation and Scaling Groups for Nonlinear Evolution Equations

A $(1+1)$-dimensional nonlinear evolution equation is invariant under the rotation group if it is invariant under the infinitesimal generator $V=x\partial_u-u\partial_x$. Then the solution satisfies the condition $u_x=-x/u$. For equations that do not admit the rotation group, we provide an extension of the rotation group. The corresponding exact solution can be constructed via the invariant set $R_0=\{u:u_x=x F(u)\}$ of a contact first-order differential structure, where $F$ is a smooth function to be determined. The time evolution on $R_0$ is shown to be governed by a first-order dynamical system. We introduce an extension of the scaling groups characterized by an invariant set that depends on two constants $\epsilon$ and $n\ne1$. When $\epsilon=0$, it reduces to the invariant set $S_0$ introduced by Galaktionov. We also introduce a generalization of both the scaling and rotation groups, which is described by an invariant set $E_0$ with parameters $a$ and $b$. When $a=0$ or $b=0$, it respectively reduces to $R_0$ or $S_0$. These approaches are used to obtain exact solutions and reductions of dynamical systems of nonlinear evolution equations.