Factoring linear partial differential operators and the Darboux method for integrating nonlinear partial differential equations (Q1586686)

It is well-known that the methods of Lagrange, Monge, Boole, and Ampère are developed in the process of finding exact solutions of nonlinear partial differential equations. G. Darboux generalized the method of Monge to obtain the most powerful method in those days for explicitly integrating partial differential equations. In [\textit{I. M. Anderson} and \textit{N. Kamran}, Duke Math. J. 87, No. 2, 265-319 (1997; Zbl 0881.35069); \textit{M. Jurás}, ``Generalized Laplace invariants and classical integration methods for second-order scalar hyperbolic partial differential equations in the plane'', in: Differential geometry and applications. Proceedings of the 6th international conference, Brno, Czech Republic, August 28 -- September 1, 1995. Brno: Masaryk University, 275-284 (1996; Zbl 0874.58082); \textit{A. V. Zhiber, V. V. Sokolov} and \textit{S. Ya. Startsev}, Dokl. Math. 52, No. 1, 128-130 (1995; Zbl 0880.35077)] the Darboux method was put in a more precise and efficient form. Following it, the single second-order nonlinear partial differential equation \[ u_{xy}=f(x,y,u,u_x,u_y), \] using the substitution \(u(x,y)\to u(x,y)+\theta v(x,y)\) and cancelling the terms proportional to \(\theta^n\), \(n > 1\), is linearized in the form \[ u_{xy} = Av_x + Bv_y+Cv \] with the coefficients \(A\), \(B\), and \(C\) depending on \(x, y, u,u_x\), and \(u_y\). Studying the equations of the last type, Laplace invented a method for their transformation that is sometimes called the Laplace cascade method. In this paper, using a new definition of the generalized factorization of linear partial differential operators, the author discusses possible generalizations of the Darboux integrability of nonlinear partial differential equations. It is shown that Darboux integrability is intimately related to the (generalized) factorization properties of the corresponding linearized equation.