Nonlinear Phenomena pp 439-451 | Cite as

# Linear and nonlinear differential equations as invariants on coset bundles

## Abstract

Among the ways in which group theory adapts its methods to study nonlinear systems, we have come upon a rather natural construction -natural for a group theorist- whereby certain families of differential equations become identified with invariants built within a group. The equations embedded in this way thus far are tensor generalizations of Burgers equation, third- or higher-order-derivative Korteweg-de Vries equations, and similar generalizations of the diffusion and Hírota equations. The groups which harbour these are subgroups of inhomogeneous linear groups and some of their normal extensions. The construction is natural for a group theorist because any space *C*, homogeneous under the action of a group *G* may be identified with a *coset* space *C* = *H/G* by some subgroup *H* ≈ *G*. We are able to find the appropriate coordinates of this space so that they become the set of independent and dependent variables satisfying the above differential equations.

## Keywords

Diffusion Equation Nonlinear Differential Equation Semidirect Product Burger Equation Coset Space## Preview

Unable to display preview. Download preview PDF.

## References

- [1]G. W. Bluman and J. D. Cole,
*Similarity Methods for Differential Equations*. Applied Mathematical Sciences #13, Springer Verlag, 1974.Google Scholar - [2]A. O. Barut and R. Rączka,
*Theory of Group Representations and Applications*. Polish Scientific Publishers, Warszaw, 1977.Google Scholar - [3]S. Helgason,
*Differential Geometry and Symmetric Spaces*. Academic Press, New York, 1962.Google Scholar - [4]R. Hermann,
*The Geometry of Non-Linear Differential Equations, Bäcklund Transformations and Solitons. Part A*. Math-Sci Press, Interdisciplinary Mathematics, Vol. XII, Brookline Ma., 1976.Google Scholar - [5]K. B. Wolf, L. Hlavatý, and S. Steinberg, Nonlinear differential equations as invariants under group action on coset bundles. I. Burgers and Korteweg-de Vries equation families.
*preprint*Comunicaciones Técnicas IIMAS #319 (1982).Google Scholar - [6]J. M. Burgers, A mathematical model illustrating the theory of turbulence.
*Adv. Appl. Mech.***1**, 171–199 (1948).Google Scholar - [7]G. B. Whittham,
*Linear and Nonlinear Waves*John Wiley & Sons, New York, 1974.Google Scholar - [8]K. B. Wolf,
*Integral Transforms in Science and Engineering*. Plenum Press, New York, 1979.Google Scholar - [9]R. Hirota, Exact solutions of the Korteweg-de Vries equation for multiple collisions of solitons.
*Phys. Rev. Lett.***27**, 1192–1194 (1971).CrossRefGoogle Scholar - [10]E. Hopf, The partial differential equation
*u*_{t}+*u u*_{x}=*μu*_{xx}.*Commun. Pure Appl. Math.***3**, 201–230 (1950);J. D. Cole, On a quasilinear parabolic equation occurring in thermodynamics.*Q. Appl. Math.***9**, 225–236 (1951).Google Scholar - [11]K. B. Wolf, Nonlinear group action and differential equations.
*Talk delivered at the topical AMS meeting, Albuquerque NM, November 1976*.Google Scholar - [12]G. Rosen and G. W. Ullrich, Invariance group of the equation ∂u/∂ = −
**u**·∇u.*SIAM J. Appl. Math.***24**, 286–288 (1973).CrossRefGoogle Scholar - [13]C. P. Boyer and M. Peñafiel, Conformal symmetry of the Hamilton-Jacobi equation.
*Nuovo Cimento***31B**, 195–210 (1976).Google Scholar