Modern geometry of non-linear PDE's

Thus mathematical insufficiency of the existing quantum theory and first of all of the quantum fields is directly coupled with the fact that the fundamentals of the general theories of non-linear PDEs have not been developed and worked out in the due systematic and complete way.

Indeed it is a true paradox that even though the whole of Theoretical Physics, Mechanics of Continua Media, Control Theory, Mathematical Modeling in Economy and Biology, etc., is based on the non-linear PDEs—not to mention those areas of Pure Mathematics as Differential Geometry, for instance—

**past century's mathematicians did not pay enough attention to them**. Therefore insulated cases could be found of important to applications equations that were studied by experts specialized in very narrow sectors by using "artisanal" methods. The idea shared by almost all mathematicians of that epoch was expressed by Richard Courant, who wrote that

**the problems associated with differential equations of order greater that one are so diversified that the construction of an unified theory can not be possible**.

This viewpoint though is sharply in opposition to the remarkable results of the classical period of the development of the theories of non-linear PDEs, which can be represented on the historical axis as the interval

**Gaspard Monge—Sophus Lie**. In those times such a theory was interpreted as a sector of Differential Geometry and Monge and Lie themselves are known as geometers. In the 800s mathematicians like Riemann, Hamilton, Jacobi, Poisson, Frobenius, Doarboux, Backlund, Riquijer, and many others, set the first stones to the foundations of the general theory of PDEs.

**This periods end with the epic work of Sophus Lie, which his contemporaries and their followers would not fully appreciate**. Frobenius Theorem, Darboux Lemma, Tensor Analysis, Sophus Lie's Contact Geometry and the based on it complete theory of first order non-linear PDEs, Hamiltonian formalism, Lie Groups and Algebras, and Differential Invariants are some results and construction from the classical period of the theory of non-linear PDEs that are commonly used by contemporary mathematics. In the 900s for mathematics it begins the age of the sets theory's dominion. Too bad, in such a time the results from the classical period were partially ignored but mostly they were almost completely forgotten. Similar doom awaited works of those few—like Luigi Bianchi, Eugene Vessiot, Emile Goursat, Tullio Levi-Civita, Emily Noether, A. Tresse— who struggled to carry on the classical tradition and obtained results of the highest value. Among them the most lucky one was

**the celebrated Noether's theorem**which for a long time was the unique source of conserved quantities for many physical theories. Only the works of Elie Cartan—among the first ones who realized the crucial importance of an invariant language without using local coordinates for Differential Geometry—succeeded in casting a bridge between the Classical and the Modern period.

Only in the 70s of the past century some results or works by Lie, Backlund, Darboux, and other classical one—that have been resting in libraries under the dust for nearly a century—were brought back to life during the recent boom period linked with the discovery of Integrable Systems. Noteworthily

**this revival of classical mathematics was due to specialists of Mathematical and Theoretical Physics**—not to pure mathematicians. In other words, Nature always has the last word. The study of the Integrable Systems revealed that the study of non-linear PDEs is not so impossible like it was assumed earlier. Like it has been proved not very later, many structures appearing within the theories of the equations of Integrable Systems are common to all non-linear PDEs. So on the basis of a synthesis of classical theory, theory of formal integrability, Spencer's and Goldshmidt's cohomological theory of formal integrability, and several ideas borrowed from the theory of integrable systems, begins the modern era of the geometrical theory of non-linear PDEs. But what is this theory?