Extended rotation and scaling groups for nonlinear evolution equations

Abstract

A (1+1)-dimensional nonlinear evolution equation is invariant under the rotation group if it is invariant under the infinitesimal generator V = x∂ u − u∂ 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 = xF(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 $$\tilde S_0 $$ that depends on two constants ∈ and n ≠ 1. When ∈ = 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.