Differential invariants of nonlinear equations vtt= f( x, vx) vxx+ g( x, vx)

Abstract

Some 10 years ago, we considered the equations v tt = f( x, v x ) v xx + g( x, v x ) with the intention of their group classification. We found, for the above equations, the equivalence group E generated by an infinite-dimensional Lie algebra L E involving two arbitrary functions of the variable x. We utilized the method of preliminary group classification suggested earlier by one of the authors (NHI), and applied it to a finite-dimensional subalgebra of the equivalence Lie algebra L E . Consequently, we found 33 types of nonlinear wave equations admitting an extension by one of the principal Lie algebra, i.e. of the maximal Lie algebra admitted by our equation with arbitrary functions f( x, v x ) and g( x, v x ). Recently, an infinitesimal technique was developed by NHI that allows one to find invariants of families of differential equations possessing finite or infinite equivalence groups. It is worth noting that the method does not depend on the assumption of linearity of equations. Here, we apply this method for calculation of invariants for the family of nonlinear equations formulated in the title. We show that the infinite-dimensional equivalence Lie algebra L E has three functionally independent differential invariants of the second order. Knowledge of invariants of families of equations is essential for identifying distinctly different equations and therefore for the problem of group classification.