Linear and nonlinear differential equations as invariants on coset bundles
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.
KeywordsDiffusion Equation Nonlinear Differential Equation Semidirect Product Burger Equation Coset Space
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