Multipliers : a general method of analysis for conservation laws of differential equations
Degree GrantorUniversity of Canterbury
Degree NameDoctor of Philosophy
Conservation laws are studied using 'multipliers' - functions which produce divergences when they multiply an equation. Multipliers are found for a number of well-known equations including those of interest in nonlinear physics such as the Korteweg-de Vries and Sine-Gordon equations. It is conjectured that multipliers exist for all conservation laws which are valid for all solutions of an equation. The close links between multipliers and other properties of conservation laws are demonstrated and the identity - at least for Hamiltonian systems - of multipliers with the gradients of conservation laws is shown. By using a formula for the variational derivative of a product of two functions some previously known results are obtained in a simple and direct way. It is also found that the equation ut + un + R = 0, R polynomial, has at most one polynomial conservation law (the equation itself) unless n is odd. The concepts of rank and irreducible terms used by Kruskal et al (J. Math. Phys. 11 952) are generalised and are used to provide a completely new framework for the study of conservation laws. This new framework is used to study the conservation laws of equations such as the Korteweg-de Vries equation and to generalise the result earlier obtained for ut + un + R = 0. Recursion operators are studied and it is found that the concepts used in the framework can be used to give the general form that a recursion operator must take. It is shown that the use of multipliers can produce results for systems of more than one equation by demonstrating that the known integrals for the Henon-Heiles system could be found using multipliers. The framework developed can be incorporated in a computer program and a method of using multipliers by means of such a program is given and illustrated.