Transformation Methods for Nonlinear Partial Differential Equations

The following sections are included:

• FORWARD

• CONTENTS

The purpose of this chapter is to set an appropriate foundation for the study of partial differential equations (PDE) by transformation-theoretic methods. One of the central issues in such studies hinges on the mode of representation of the PDE. Most of the current literature elects to represent the PDE through the introduction of an appropriate jet bundle. In contrast, this study represents the PDE by introducing a contact manifold of appropriate order. When the number of dependent variables is greater than one, the two methods lead to the same groups of symmetry transformations for a given system of PDE. On the other hand, when there is only one dependent variable, the group of symmetry transformations computed by the jet bundle formalism for certain PDE is only a proper subgroup of the group of symmetry transformations computed on the appropriate contact manifold. In addition, the contact manifold formulation provides a particularly clear means for understanding the limitations imposed by the Backlund theorem, and is suggestive of how to overcome these limitations.

Symmetry methods for partial differential equations have been around since the time of S. Lie, while the classical method of characteristics for a first order PDE has been in use a good deal longer. Recent work from the standpoint of isovector fields of differential ideals has shown that these two methods are actually two facets of the same geometric construct, and that a generalized method of characteristics also exists for second and higher order systems and for systems of nonlinear first order PDE. The purposes of this chapter are to give explicit derivations, examples, and proofs of the results that underlie this order independent method of characteristics. Results of this nature are essential in implementing the symmetry transformation methods presented in the previous chapter because symmetry transformation methods demand that we have at least one solution map of the given system of PDE in order to start the process. The discussion will follow the lines of argument given in [24].

A major obstacle to obtaining solutions of nonlinear PDE by the geometric or transformation-theoretic methods presented in the previous two chapters is the Backlund theorem; that is, the most general transformation (for N > 1) that preserves the contact structure of a contact manifold is a prolongation of a point transformation of the associated graph space. One of the reasons why this theorem holds is the vary complicated nature of the intersections of graphs of different solution maps of the contact ideal, and the attendant requirement that the mappings preserve the graphs of all solution maps of the contact ideal and their complicated intersection structure…

Representations of solutions of PDE by complete systems of primitive integrals have many important ramifications. Some of these are discussed in this chapter. Since complete systems of primitive integrals are associated with completely integrable horizontal ideals of the exterior algebra of a contact manifold, isovector fields of a horizontal ideal will transport graphs of primitive integrals to graphs of primitive integrals. In sharp contrast with isovectors of the contact ideal, isovectors of a horizontal ideal are not necessarily prolongations of vector fields on graph space no matter how many dependent variables are present. Isovectors of a horizontal ideal also form modules, rather than vector spaces over ℝ. Their Lie algebra naturally splits relative to a semidirect product decomposition, and the ideal of the Lie algebra that generates this decomposition is the module of Cauchy characteristic vector fields of the horizontal ideal. Once we have a complete system of primitive integrals, they can be used to generate a resolution (change of coordinates) such that the canonical vector fields {V_i | 1 ≤ i ≤ n} appear as the generators of an n-parameter Abelian group of translations. This resolution also reduces the generators of the completely integrable horizontal ideal to exact 1-forms, and thereby affords significant simplifications in actual calculations. The problem can also be turned around by determining completely integrable horizontal ideals for which the system admits one or more given first integrals.

The resolving coordinate transformations considered in the previous chapter do not preserve the structure of the contact 1-forms, for we have While C’^α = dq’^α. The analogy with classical contact transformations, that obtains for n = 1 or N = 1, suggests that an analysis of transformations that preserve the structure of the generating 1-forms of horizontal ideals will be useful. Indeed, such transformations should provide the means whereby the structure of the set ℌ(K_D) of all completely integrable horizontal ideals can be determined. An explicit determination of this structure is essential because the methods presented in chapter 3 require that every element of ℌ(K_D) must be used in order to obtain all solutions of the given system of PDE under study. In view of the results established previously, such transformations may be expected to contain the pseudogroup of prolongations of point transformations on graph space as a proper subset. There is therefore the possibility of circumventing the Backlund theorem by use of those admissible transformations that are not prolongations.

The following sections are included:

• Appendix A AN ALTERNATIVE TREATMENT OF EXTENDED CANONICAL TRANSFORMATIONS
  • Extended Canonical Transformations
  • Group Properties

• Appendix B MULTIPLE INTEGRAL VARIATIONAL PROBLEMS
  • Formulation of the Problem
  • An Existence Theorem for Euler-Lagrange Equations
  • Aspects Particular to Variational Problems

• REFERENCES

• INDEX