Classical M. A. Buhl Problem, Its Pfeiffer–Sato Solutions, and the Classical Lagrange–D’Alembert Principle for the Integrable Heavenly-Type Nonlinear Equations

Abstract

The survey is devoted to old and recent investigations of the classical M. A. Buhl problem of description of the compatible linear vector field equations and their general Pfeiffer and modern Lax–Sato-type special solutions. In particular, we analyze the related Lie-algebraic structures and the properties of integrability for a very interesting class of nonlinear dynamical systems called dispersion-free heavenly-type equations, which were introduced by Plebanski and later analyzed in a series of papers. The AKS-algebraic and related R-structure schemes are used to study the orbits of the corresponding coadjoint actions intimately connected with the classical Lie–Poisson structures on them. It is shown that their compatibility condition coincides with the corresponding heavenly-type equations under consideration. It is also demonstrated that all these equations are originated in the indicated way and can be represented as a Lax compatibility condition for specially constructed loop vector fields on the torus. The infinite hierarchy of conservations laws related to the heavenly equations is described and its analytic structure connected with the Casimir invariants is indicated. In addition, we present typical examples of equations of this kind demonstrating in detail their integrability via the scheme proposed in the paper. The relationship between a very interesting Lagrange–d’Alembert-type mechanical interpretation of the devised integrability scheme and the Lax–Sato equations is also discussed.