Stability of lie groups of nonlinear hyperbolic equations

Abstract

In this thesis the stability of the Lie group invariance of classical solutions of large classes of nonlinear partial differential equations is studied. We give a theoretical framework for the construction of approximate groups for nonlinear partial differential equations. In particular, stability symmetries for the perturbed nonlinear wave equation սtt+ eut = (ƒ (x, u) ux]x are presented here for the first time. This research is a particularly important stability study, since it applies to large classes of - earlier unknown - classical solutions of nonlinear partial differential equations as well as to their symmetries. These equations and solutions model, amongst others, important laws of nature. Chapter 1 is devoted to the general concepts of Lie group theory. A detailed account is given of the applications of Lie groups to both ordinary and partial differential equations. To date the only known method of obtaining particular solutions to complicated systems of differential equations is by Lie group symmetry analysis. This is now well known in the literature. The Lie group symmetry analysis, however, has some limitations. Any small perturbation of an equation disturbs the group admitted by it and this reduces the practical value of group theoretic methods in general. The theory of stability analysis presented in chapter 2 overcomes this problem. This technique, originated by N .Kh. Ibragimov around 1988, generates groups that are stable under small, or even classes of more arbitrary, perturbations of the differential equations involved. The exact Lie groups admitted by the nonlinear wave equation սtt = (ƒ (x, u) ux]x and the corresponding perturbed equation are discussed in chapter 3. Finally, in chapter 4, the construction of stability groups admitted by the perturbed nonlinear wave equation are set out in detail.