partial derivative problem for the generalized Korteweg-de Vries equation

The generalized Korteweg-de Vries equation, which has applications in hydrodynamics, in particular, is essentially the first example of a case in which the partial derivative dressing of a nonlinear equation is constructed by introducing into the dressing operator an arbitrary function of the independent variables of this equation. The proposed algorithm reveals a class of solutions of this equation which are expressed in terms of the solution of algebraic equations. An example of a new type of solution whose derivative with respect to the independent variables has a power-law singularity at some point is presented. (C) 1998 American Institute of Physics. [S0021-3640(98)01221-3].