We construct a number of ansatzes that reduce one-dimensional nonlinear
heat equations to systems of ordinary differential equations. Integrating
these, we obtain new exact solution of nonlinear heat equations with
By use of nonlinearization method about spectral problem, a classical
completely integrable system associated with the Harry-Dym (HD) hierarchy is obtained. Furthermore, the involutive solution of each equation
in the HD hierarchy is presented, in particular, the involutive solution
of the well-known...
Ansatzes for the Navier-Stokes field are described. These ansatzes reduce
the Navier-Stokes equations to system of differential equations in three,
two, and one independent variables. The large sets of exact solutions of
the Navier-Stokes equations are constructed.