Many canonical
nonlinear equations describing wave propagation
and solitons exhibit similar integrability properties:
infinite set of symmetries and conserved quantities,
hierarchy of exact solutions, Backlund and Darboux
transformations. Following to an algebraic Sato theory, all
integrable nonlinear equations are considered as reductions
of unifying KP hierarchies, either for a canonical
single-component version or for its
multi-component extension. The KP symbol stands for
Russian physicists, B. Kadomtsev and V. Petviashvili,
who derived a single wave equation in two space and one time
dimensions back to 1970. The abstract Sato theory considers
the KP equation as a tower to infinitely many equations that
depend on infinitely many commuting time variables.
Algebra of the KP hierarchy was found to be extremely rich: it generates various classes of rational and exponential exact solutions. The general KP algebra has reductions to a number of physically meaningful wave equations such as the Korteweg-de Vries, Nonlinear Schrodinger, Sine-Gordon, Yagima-Wadati, Massive Thirring, Boussinesq equations and their modified extensions. Wave propagation, scattering, stationary structures, instabilities can be traced by filtering special solutions of the KP hierarchy.
The formalism of vertex operators acting on algebra of polynomial tau-functions is the powerful method to satisfy the constraints imposed on the KP hierarchy. This method is especially useful in finding the rational and exponential solutions to the constrained nonlinear equations in the form of the Wronskian and Grammian determinants.