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Computation of Scaling Invariant Lax Pairs with Applications to Conservation Laws

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Computation of Scaling Invariant Lax Pairs with Applications to Conservation Laws COMPUTATION OF SCALING INVARIANT LAX PAIRS WITH APPLICATIONS TO CONSERVATION LAWS by Jacob Rezac A thesis submitted to the Faculty and the Board of Trustees of the Colorado School of Mines in partial fulfillment of the requirements for the degree of Master of Science (Mathematical and Computer Sciences). Golden, Colorado Date Signed: Jacob Rezac Signed: Dr. Willy Hereman Thesis Advisor Golden, Colorado Date Signed: Dr. Willy Hereman Professor and Interim Department Head Department of Applied Mathematics and Statistics ii ABSTRACT There has been a large amount of research on methods for solving nonlinear par- tial differential equations (PDEs) since the 1950s. A completely integrable nonlinear PDE is one which admits solutions, given specific constraints. These integrable non- linear PDEs can be associated with a system of linear PDEs through a compatibility condition. Such a system is called a Lax pair. While Lax pairs are very important in the theory of solving nonlinear PDEs, few methods exist to compute them. This The- sis presents a method for generating Lax pairs for a specific class of nonlinear PDEs. Conservation laws, well-known from physics, are a related concept. Indeed, the ex- istence of an infinite number of conservation laws for a nonlinear PDE also predicts it complete integrability. We discuss a method for the computation of conservation laws, given a PDE's Lax pair. This method is based on work done by Drinfel'd and Sokolov in 1985, which does not seem to have been fully explored in literature. The goal of this Thesis is to demonstrate the efficacy of these two construction methods. As such, we also list a number of Lax pairs and conservation laws computed by the methods presented in the Thesis. iii TABLE OF CONTENTS ABSTRACT . iii LIST OF TABLES . vi ACKNOWLEDGMENTS . vii CHAPTER 1 INTRODUCTION . .1 CHAPTER 2 PRELIMINARIES . .5 2.1 Lax Pairs . .5 2.2 AKNS Scheme . 11 2.3 Gauge Transformations . 16 2.4 Conservation Laws . 21 CHAPTER 3 CONSTRUCTION OF LAX PAIRS . 25 3.1 Scaling Invariance . 25 3.2 Exhaustive Lax Pair Computation Methods . 28 3.3 Uniform Weight Construction . 34 3.4 Weak Lax Pairs and Triviality Concerns . 39 CHAPTER 4 THE CONSTRUCTION OF CONSERVATION LAWS FROM LAX PAIRS . 45 4.1 The Drinfel'd Sokolov Method for Computing Conservation Laws . 45 4.2 Triviality and Simplification . 51 CHAPTER 5 FURTHER EXAMPLES . 55 5.1 Lax Pairs Computed with Exhaustive Methods . 55 iv 5.1.1 Modified Korteweg-de Vries Equation . 55 5.1.2 Kaup-Kuperschmidt Equation . 56 5.1.3 Lax 5th-Order Equation . 57 5.1.4 Sawada-Kotera Equation . 57 5.1.5 Harry-Dym Equation . 58 5.1.6 Drinfel'd-Sokolov System . 59 5.2 Lax Pairs Computed by Uniform Weight Method . 59 5.2.1 Nonlinear Schr¨odingerEquation . 59 5.2.2 Gardner Equation . 60 5.3 Conservation Laws Produced by the Drinfel'd-Sokolov Method . 61 5.3.1 Korteweg-de Vries Equation . 61 5.3.2 Modified Korteweg-de Vries Equation . 61 5.3.3 Sine-Gordon Equation . 62 5.3.4 Sinh-Gordon Equation . 63 5.3.5 Nonlinear Schr¨odingerEquation . 63 CHAPTER 6 CONCLUSIONS AND FURTHER RESEARCH . 67 REFERENCES CITED . 69 APPENDIX - OUTLINE OF DRINFEL'D-SOKOLOV ALGORITHM AND USE OF MATHEMATICA . 73 v LIST OF TABLES Table 5.1 The first four conservation laws for the KdV equation (2.4). 61 Table 5.2 The first four conservation laws for the mKdV equation (2.31). 62 Table 5.3 The first four conservation laws for the sine-Gordon equation (5.4). 63 Table 5.4 The first four conservation laws for the sinh-Gordon equation (5.5). 64 Table 5.5 The first four conservation laws for the NLS equation (5.2). 65 vi ACKNOWLEDGMENTS Thanks to my Thesis advisor, Willy Hereman, for his countless hours of advice (mathematical and otherwise), for his friendship, and for constantly pushing me to be a better mathematician. Thanks to the Department of Applied Math and Statistics at the Colorado School of Mines, including all the faculty and staff who have helped me during my time here. In particular, thanks to my Thesis committee Drs. Paul Martin and Luis Tenorio, both of whom have taught me more math than I care to admit. Also thanks to Dr. Mark Hickman, whose brief sabbatical in Golden put the following work on a surer mathematical footing than could have occurred without him. Further thanks to the many students in my research group the last three years, Terry Bridgman, Jennifer Larue, Tony McCollom, Janeen Neri, Sara Clifton, Oscar Aguilar, Jon Tran, and Allen Voltz. Their work ethic, insightful ideas, and pointed questions all helped push this Thesis to completion. This research was partially funded by NSF research award no. CCF-0830783. Finally, thanks to my family and friends, who both distracted and encouraged me enough to complete this document, whether they realize it or not. Also to Golden's numerous fine coffee shops (in particular Higher Grounds), which kept me caffeinated and gave me a place to work whenever I needed one. vii CHAPTER 1 INTRODUCTION In the early 1950s, physicists Fermi, Pasta, and Ulam (FPU) investigated [1] an apparent paradox related to the field of statistical mechanics1. The research group was attempting to model wave motion in a nonlinear crystal lattice. Scientific under- standing at the time predicted that energy put into the system would become equally partitioned across the crystal lattice. In an unexpected turn of events, however, nu- merical experiments showed that nearly all of the energy put into the crystal would remain in its original position; e.g., if the energy were initially put into the lowest vibrational mode of a wave, all but 3% of the energy would remain in that same mode [1]. This problem plagued physicists for almost 10 years. In 1965, however, Zabusky and Kruskal [3] explored the problem in its continuous limit. Numerical experiments conducted by Zabusky and Kruskal revealed surprising interactions between solutions of the continuous FPU model: solitary wave solutions passed through each other, emerging with no changes in velocity, amplitude, or shape. Although the problem was nonlinear in nature, wave solutions interacted as though they were linear and obeyed the superposition principle. This discovery hinted at a solution to the FPU paradox. In the continuous limit, these non-interacting waves were analogous to energy not mixing in different wave modes of a nonlinear lattice. Zabusky and Kruskal called these waves solitons2 to emphasize both the solitary nature of the waves and the similarities these waves shared with particles (e.g., the electron or positron). 1In recent years [2] it has been argued that Tsingou should also be credited with these inves- tigations. Most of the numerical computations presented in the paper attributed to FPU were, in fact, done by Tsingou. 2Zabusky originally planned to call these waves solitrons. Solitron, however, was already the name of a manufacturing company (which still exists today) [4]. 1 It was soon noticed that the equation studied by Zabusky and Kruskal was, in fact, a nonlinear partial differential equation (PDE) discovered much earlier in the context of water waves. This so-called Korteweg-de Vries (KdV) equation was derived in 1895, by Dutch mathematicians Korteweg and de Vries, to describe solitary wave behavior observed by naval engineer John Scott Russell in 1834 [5]. Soon, other equations with soliton solutions were found. The modified KdV (mKdV) for example, which is related to the KdV equation by a simple transformation, was investigated by the Zabusky-Kruskal team [3], as well as Miura [6], in the mid-'60s. The sine- Gordon equation, which had been known by plasma physicists for some time, was soon discovered [7] to have soliton solutions as well. As more and more equations of this type were uncovered, the study of exact solutions to nonlinear PDEs exhibiting soliton behavior quickly became a subject important to mathematicians. In a series of papers [6, 8] published in the 1960s and `70s, Gardner, Greene, Kruskal, and Miura (GGKM) found analytic solutions to describe the interaction of waves governed by the KdV equation. Importantly, GGKM developed a general method to find these solutions. The papers also describe other important analytic properties of the KdV equation and its solutions, such as conservation laws. In 1968, Lax proposed a formal technique [9] for finding soliton solutions to non- linear evolution equations based on work done by GGKM. His technique involved relating the original nonlinear PDE to two linear operators via a compatibility condi- tion. These linear operators, called a Lax pair, are the main topic of this Thesis. The work done by Lax was further generalized by Zakharov and Shabat [10] and Ablowitz, Kaup, Newell, and Segur (AKNS) [11]. Soliton solutions were found for more compli- cated nonlinear PDEs, such as the Nonlinear Schr¨odingerequation (NLS), and Lax pairs were again required for the solutions. The solution technique created by these researchers is called the Inverse Scattering Transform (IST). It has been very success- ful over the last 40 years, and has been called one of the most important techniques 2 developed in applied mathematics over that time. During the time-period Lax pairs were first being discovered, mathematicians be- gan to examine conditions under which soliton waves could occur. Many nonlinear waves which had been studied before the discovery of solitons had exhibited \shock wave" solutions, requiring jump boundary conditions and conservation law formu- lations. Applying this analogy to the new nonlinear waves, Miura [6], along with Kruskal, Zabusky, and Whitham [9], discovered nine conservation laws related to the KdV equation in the late 1960s.
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