The two-singular-manifold method: I. Modified Korteweg-de Vries and sine-Gordon equations

Abstract

We address the problem of finding, by Painleve analysis only, the Backlund transformation of partial-differential equations (PDEs) having two families of movable singularities with opposite principal parts, such as the modified Korteweg-de Vries (MKdV), sine-Gordon or nonlinear Schrodinger equations. This first paper gives an almost algorithmic method which extends the singular-manifold method of Weiss(1986) that is unable to handle these equations. First, with only one singular manifold at a time, we obtain the Darboux transformation. Second, we assume that the ratio of two functions defining the singular manifolds satisfies the most general projective Riccati system with undetermined coefficients; the Darboux transformation then generates a very small number of determining equations, admitting a unique solution. Equivalent to the Lax pair of the Zakharov-Shabat-Ablowitz-Kaup-Newell-Segur scheme by the canonical linearization of the Riccati system. The method is here applied to the MKdV and sine-Gordon equations.