CLASSICAL AND NONCLASSICAL LIE SYMMETRIES OF THE K(m; n) DISPERSION EQUATION Caylah N. Retz A Thesis Submitted to the University of North Carolina Wilmington in Partial Fulfillment Of the Requirements for the Degree of Master of Science Department of Mathematics and Statistics University of North Carolina Wilmington 2012 Approved by Advisory Committee Gabriel Lugo Michael Freeze Russell Herman Chair Accepted by Dean, Graduate School This thesis has been prepared in the style and format Consistent with the journal American Mathematical Monthly. ii TABLE OF CONTENTS ABSTRACT . .v DEDICATION . vi ACKNOWLEDGMENTS . vii LIST OF TABLES . viii LIST OF FIGURES . ix 1 INTRODUCTION . .1 2 ELEMENTARY DIFFERENTIAL GEOMETRY . .6 2.1 Topology . .6 2.2 Manifolds . .7 2.3 Differentiable Maps and Rank . .9 2.4 Submanifolds . 12 2.5 Vector Fields . 12 2.5.1 Flows . 15 2.5.2 Lie Brackets and Lie Algebras . 16 3 LIE GROUPS . 17 3.1 r-Parameter Lie Groups . 17 3.2 Lie Subgroups . 18 3.2.1 Local Lie Groups . 19 3.2.2 Transformation Groups . 19 3.3 One-Parameter Groups of Transformations . 24 3.3.1 Lie Algebras . 24 3.3.2 The Lie Series . 28 3.4 Infinitesimal Transformations . 33 3.4.1 Fundamental Theorem of Lie . 36 4 SYMMETRY GROUPS AND INVARIANCE . 44 iii 4.1 Algebraic Systems . 44 4.1.1 Constructing Invariants . 48 4.2 Prolongation . 50 4.3 Prolongation of Differential Equations . 54 4.3.1 Total Derivatives . 56 4.3.2 The General Prolongation Formula . 57 4.4 Burgers' Equation . 60 4.5 Nonclassical Symmetries . 69 4.5.1 The Nonclassical Method . 71 5 THE K(m; n) DISPERSION EQUATION . 74 5.1 K(2; 2) Equation . 74 5.2 The K(m; n) Equation . 78 5.2.1 Calculations . 79 5.2.2 Generators and Transformation Groups . 81 5.2.3 Invariants and Reductions . 85 5.3 Nonclassical Symmetries of K(m; n)............... 91 6 CONCLUSION . 98 REFERENCES . 103 APPENDIX . 105 A The Coefficient Functions φJ ....................... 105 B Maple Code for K(2; 2).......................... 106 C Maple Code for K(m; n)......................... 107 D Complete Maple Output . 110 E Maple Code for Nonclassical K(m; n).................. 124 iv ABSTRACT The purpose of this thesis is to present applications of Lie groups to solve the K(m; n) dispersion equation. Focus is first placed on discussing the theory behind Lie groups and how they may be applied as a solution technique of a system. Topics of discussion include topology, manifolds, groups, Lie groups, groups of transforma- tions, invariants, and prolongation. We differentiate between what we call classical and nonclassical symmetries and establish methods for calculating each. A sim- ple example of using Lie symmetry methods is thoroughly presented using Burgers' equation to demonstrate the inner calculations behind this technique. Focus is then changed to the K(m; n) equation, where emphasis is placed on finding the sym- metries and summarizing the types of solutions that are produced under both the classical and nonclassical methods. v DEDICATION This thesis is dedicated to my parents, Jeffrey and Tammy Shunk, for their continued support and encouragement in my academic endeavors. vi ACKNOWLEDGMENTS I would first like to express my infinite gratitude and debt to my thesis advisor, Dr. Russell Herman. His guidance and encouragement was key to grasping essential concepts and the backbone of accomplishing this work. He always instructed me by providing reference materials, extensive examples, and lectures well in advance so that I could better comprehend the content as I approached it. His consistent effort to be that prepared and spend so much of his time working with me cannot be thanked enough. This work would have been impossible without him, and I could not have had a better teacher or advisor. I would also like to thank my committee members, Dr. Gabriel Lugo and Dr. Michael Freeze, for their time and feedback regarding this thesis. Both are excellent professors and my appreciation for the talents of each greatly influenced my decision to have them on my committee. All three of the professors mentioned here have greatly impacted my education in their own unique fashion, and they each mean very much to me as a result. Thanks are due to my family for their continued support in my academic adven- tures. I would not have even considered a graduate program without their sugges- tions, so it is certain that I would not be at this point without them. I am especially indebted to my husband, who has helped me through hard times and sacrificed so much for me to get where I am today. vii LIST OF TABLES 1 Complete list of coefficient equations. 61 2 Reduced coefficient list. 62 3 Reduced K(2; 2) coefficient equations. 76 4 Groups, Solutions, and Invariants of K(2; 2) equation. 78 5 Reduced K(m; n) coefficient equations. 80 6 Further reduced K(m; n) coefficient equations . 80 7 K(m; n) generator results. 82 8 Generators, Groups, and Solutions of K(m; n). 83 9 Invariants of K(m; n) equation. 88 10 Nonclassical generators of K(m; n) equation. 97 viii LIST OF FIGURES 1 Charts on a manifold M .........................8 2 Tangent Vector . 13 3 Path, or orbit, from (x; y) to (x1; y1): .................. 20 4 Composition Ψ("; Ψ(δ; x)) = Ψ(" + δ; x). 22 5 Orbit: the path translated level curves take . 23 6 Tangent plane housing x∗ = X(x; "). 34 7 It's all connected! . 43 8 Invariant Surface. 51 9 Solution process. 59 ix 1 INTRODUCTION By definition, a differential equation is an equation relating one or more deriva- tives of an unknown function. Solutions of simple differential equations may be found using one, or more, of many techniques developed in an elementary differen- tial equations course. Most differential equations, however, require more rigorous solution techniques because of their non-linearity and higher orders. As a result, thought must be placed on which solution techniques would be the most fruitful for solving the equation(s) of interest. In this thesis, emphasis is placed on demonstrat- ing one class of solution methods in particular, known as Lie symmetry methods. One of the earliest solution techniques of differential equations we learn is sep- aration of variables. The question that we now wish to present is whether a PDE that is not separable can be made so by a change of variables. The idea is that once the appropriate change of variables, or transformation, is found, the system of differential equations reduces to a system of ordinary differential equations that have known solutions. We will spend a considerable portion of this thesis describing the theory behind Lie symmetry methods with this idea in mind. We will then turn focus to applying these methods to a differential equation of particular interest: the K(m; n) dispersion equation, which models the dispersion patterns of liquid drops [11]. Finding a change of variables that makes a system of differential equations sep- arable is not always simple task. Obviously, the more challenging the system, the more difficult it would be to have insight on what change of variables would work. The key to this proposed method, then, is being able to calculate what change of variables would simplify any given system. Fortunately, we have a brilliant process developed by Sophus Lie, called Lie symmetry methods, that we will use to put 1 this idea in motion [7]. The pivotal piece of the theory that Lie developed is the discovery of transformation groups, called Lie groups, that continually map curves into other curves. One basic principle in the theory behind Lie symmetry methods is that solutions of differential equations can be represented by specific functions that we will call invariants. The level curves associated with these functions are known as solution curves. When looking at the level curves, there is a direction that allows each curve to be \slid" into a nearby level curve. This is known as mapping a solution curve into another. If we can identify a \direction" that allows the level curves to be mapped into each other, we can use it as criterion that any set of new coordinates have to uphold. As long as the solution curves \slide" into each other, we know the new coordinates have not altered the solutions in any way. When we find these mapping coordinates, we say that we have found a symmetry of that function under that mapping. So far, we have recognized that a change of variables may make an equation solvable and that as long as the new variables do not change the solution curves, then the variable change is valid. Thought must now be placed on how we are to distinguish what mappings will work. First, we define an orbit as the path that the mapped solution curves will take. Once this function is found, we have a set condition that all of our mappings must satisfy. If the invariant functions which describe the solution curves follow the path of the orbit, then we know that the solution curves mapped along its path will slide into each other. To use this, we must first describe the environment where everything lives and how it all behaves, as we will do in Chapter 2. In the beginning of Chapter 2, we venture into the discussion of topology and manifolds. Before we can find the invariants or orbits of a system, we must describe the region where the solution curves live. We would like to have a space that makes 2 it easier to find and use these variables and symmetries.