Sufficient conditions for non-equivalence of hyperbolic equations in terms of computer algebra

Startsev Sergey Ya.

This paper devoted to hyperbolic partial differential equations with special properties, which are satisfied, for example, by Darboux integrable equations (i.e., equations with nontrivial kernels of total derivatives by virtue of the equation). The equivalence problem of such equations is considered, that is, the question of whether one of the equations is related to the other by a point change of variables. We can associate an integer r ≥ 0 with any of the above equations. Since the integer r is preserved under point changes of variables, we can formulate a sufficient condition for the non-equivalence of the equations by using this number. We explain how a computer algebra system makes it easy for us to obtain a lower estimate for r and, in some cases, to find the exact value of r in a fully automatic mode. Using a Darboux integrable Moutard equation as an example, we demonstrate the efficiency and usefulness of this approach and show that the Moutard equation is not equivalent to any of the equations in a list of Darboux integrable equations.

Keywords:non-linear hyperbolic partial differential equations, Darboux integrability, equivalence problem, computer algebra.