Using Conservation Laws to Solve Toda Field Theories

We investigate the question of how the knowledge of sufficiently many local conservation laws for a model can be utilized to solve the model. We show that for models where the conservation laws can be written in one-sided forms, like $\barpartial Q_s = 0$, the problem can always be reduced to solving a closed system of ordinary differential equations. We investigate the $A_1$, $A_2$, and $B_2$ Toda field theories in considerable detail from this viewpoint. One of our findings is that there is in each case a transformation group intrinsic to the model. This group is built on a specific real form of the Lie algebra used to label the Toda field theory. It is the group of field transformations which leaves the conserved densities invariant.