Date of Award:

12-2021

Document Type:

Dissertation

Degree Name:

Doctor of Philosophy (PhD)

Department:

Mathematics and Statistics

Committee Chair(s)

Ian M. Anderson

Committee

Ian M. Anderson

Committee

Mark E. Fels

Committee

Nathan Geer

Committee

Andreas Malmendier

Committee

Oscar Varela

Abstract

In the study of partial differential equations (PDE), one is often concerned as to whether or not explicit solutions can be obtained via various integration techniques. One such technique, known as the method of Darboux, has had particular success in solving nonlinear problems as demonstrated by the classical works of Goursat. Recently, Anderson, Fels, and Vassiliou provided a far-reaching generalization of Vessiot’s group-theoretic interpretation of the method of Darboux. This generalization allows for the characterization of Darboux integrable systems in terms of fundamental geometric invariants as well as the construction of Darboux integrable systems in general.

In this work, we refine the theory of Anderson, Fels, and Vassiliou by providing conditions for which their construction gives rise to various classes of second-order PDE in the plane of the form

F(x,y,u,ux,uy,uxx,uxy,uyy) = 0.

We use this refinement to completely characterize all linear Darboux integrable PDE in the plane and provide a simple proof concerning the classification of all PDE equivalent to the wave equation uxy = 0. We then study the fundamental invariants associated to several classes of Darboux integrable equations, in particular, f-Gordon equations of the form

uxy= f(x,y,u,ux,uy).

In doing so, we construct several new examples of Darboux integrable f-Gordon equations with interesting geometric structure.

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