Linearizable second order Monge-Ampère equations
Second order nonlinear hyperbolic equations generally have solutions that develop singularities in a finite time. However, certain equations in this class are ``completely exceptional'' in that they behave like linear equations. Completely exceptional equations have been classified, and in \((1+1)\) dimensions they are Monge-Ampère equations of the form \[ \kappa_1(u_{tt}u_{xx}-u_{tx}^2) + \kappa_2 u_{tt} + \kappa_3 u_{tx} + \kappa_4 u_{xx} + \kappa_5 = 0, \] where the coefficients \(\kappa_1,\cdots,\kappa_5\) are arbitrary functions of \(x,t,u,u_t\) and \(u_x\). In \((n+1)\) dimensions there is an analogous classification. This suggests that there should be some way of transforming these equations to linear canonical forms, and indeed, some results in this direction have been obtained by \textit{A. Donato, U. Ramgulam} and \textit{C. Rogers} [Meccanica 27, 257-262 (1992; Zbl 0764.35066)]. Here the author considers the problem of linearizing the first order quasilinear systems which are equivalent to second order Monge-Ampère equations. It is shown that this can always be done in \((1+1)\) dimensions if the coefficients \(\kappa_1,\cdots,\kappa_5\) are constants, or depend only on the first derivatives of the solution. In addition, in \((2+1)\) and \((3+1)\) dimensions some conditions on the coefficents are found which guarantee that the first order system can be linearized.
linearization of first order quasilinear systems
scientific article; zbMATH DE number 1130501
scientific article