Groups of scalings and invariant sets for higher-order nonlinear evolution equations

Abstract

Consider a nonlinear $m^{\text{th}}$ order evolution PDE \begin{equation} u_t = \mathbf A(u) \equiv A(x,u,u_1, \dots ,u_m), \quad u=u(x,t), \quad (x,t) \in Q=(0,1) \times [0,1], \tag*{(*)} \end{equation} where $A$ is a $C^\infty$ function and $u_t = \partial u/\partial t$, $u_i = \partial ^i u / \partial x^i$. If (*) is invariant under a group of scalings with the infinitesimal generator $ X =x \frac {\partial}{\partial x} + \mu t \frac {\partial}{\partial t} $ ($\mu$ is a constant ``scaling order" of $\mathbf A$), then the PDE admits exact self-similar solutions depending on the single invariant variable $u(x,t) = {\theta } (\xi)$, $ \xi = x/t^{1/\mu}$, where ${\theta }$ solves a nonlinear $m^{\text{th}}$ order ODE associated with the PDE. We prove that when the operator $\mathbf A$ is composed of a finite sum of operators with different scaling orders, $\mathbf A = \sum \mathbf A_i$, and no group of scalings exists, the exact solutions can be constructed via the invariance of the set $S_0 = \{u: u_1 =F(u)/x\}$ of a contact first-order differential structure, where $F$ is a smooth function to be determined. The time-evolution on $S_0$ is shown to be governed by a first-order dynamical system. We thus observe that besides scaling group properties, the invariance of $S_0$ specifies new sets of solutions described by first-order ODEs. The approach applies to a class of nonlinear parabolic equations of the second and of the fourth order.