Hodograph Transformations of Linearizable Partial Differential Equations

Abstract

In this paper an algorithmic method is developed for transforming quasilinear partial differential equations of the form $u_t = g( u )u_{nx} + f( u, u_x , \cdots ,u_{( n - 1)x } ),\, u_{mx} \equiv \partial ^m u/\partial x^m $, where $dg/du \not\equiv 0$, into semilinear equations (i.e., equations of the above form with $g( u ) = 1 )$. This crucially involves the use of hodograph transformations (i.e., transformations involving the interchange of dependent and independent variables). Furthermore, the most general quasilinear equation of the above form is found that can be mapped via a hodograph transformation to a semilinear form. This algorithm provides a method for establishing whether a given quasilinear equation is linearizable, i.e., is solvable in terms of either a linear partial differential equation or of a linear integral equation. In particular, this method is used to show how the Painlevé tests may be applied to quasilinear equations. This appears to resolve the problem that solutions of linearizable quasilinear parital differential equations, such as the Harry–Dym equation $u_1 = ( u^{ - 1/2})_{xxx} $, typically have movable fractional powers and so do not directly pass the Painlevé tests.