Differential substitutions and symmetries of hyperbolic equations

Abstract

There are considered differential substitutions of the form $v=P(x,u,u_{x})$ for which there exists a differential operator $H=\sum^{k}_{i=0} α_{i} D^{i}_{x}$ such that the differential substitution maps the equation $u_{t}=H[s(x,P,D_{x}(P),...,D^{k}_{x}(P))]$ into an evolution equation for any function $s$ and any nonnegative integer $k$. All differential substitutions of the form $v=P(x,u,u_{x})$ known to the author have this property. For example, the well-known Miura transformation $v=u_{x}-u^{2}$ maps any equation of the form $$u_{t}=(D^{2}_{x}+2uD_{x}+2u_{x}) [s(x,u_{x}-u^{2},D_{x}(u_{x}-u^{2}),...,D^{k}_{x}(u_{x}-u^{2}))]$$ into the equation $$v_{t}=(D^{3}_{x}+4vD_{x}+2v_{x})[s(x,v,{{\partial v}\over{\partial x }},...,{{\partial^{k} v}\over{\partial x^{k}}})].$$ The complete classification of such differential substitutions is given. An infinite set of the pairwise nonequivalent differential substitutions with the property mentioned above is constructed. Moreover, a general result about symmetries and invariant functions of hyperbolic equations is obtained.