Hodograph transformations of linearizable partial differential equations. (English) Zbl 0694.35005

In this case, authors’ abstract is a proper review itself and is reproduced below.
In this paper an algorithmic method is developed for transforming quasilinear partial differential equations of the form \[ u_ t=g(u)u_{nx}+f(u,u_ x,...,u_{(n-1)x}),\quad u_{mx}\equiv \partial^ mu/\partial x^ m, \] where dg/du\(\equiv 0\), into semi- linear equations (i.e., equations of the above form with \(g(u)=1)\). This crucially involves the use of hodograph transformation (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 partial differential equations, such as the Harry-Dym equation \(u_ t=(u^{- 1/2})_{xxx}\), typically have movable fractional powers and so do not directly pass the Painlevé tests.
Reviewer: P.N.Bajaj


35A22 Transform methods (e.g., integral transforms) applied to PDEs
35Q99 Partial differential equations of mathematical physics and other areas of application
35K55 Nonlinear parabolic equations
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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