vi §3.7. Numerical Discovery of Solitons 53 §3.8. Hints of Nonlinearity 57 §3.9. Explicit Formulas for n-soliton Solutions 59 §3.10. Soliton Theory and Applications 60 §3.11. Epilogue 62 Problems 63 Suggested Reading 65 Chapter 4. Elliptic Curves and KdV Traveling Waves 67 §4.1. Algebraic Geometry 67 §4.2. Elliptic Curves and Weierstrass ℘-functions 68 §4.3. Traveling Wave Solutions to the KdV Equation 84 Problems 91 Suggested Reading 93 Chapter 5. KdV n-Solitons 95 §5.1. Pure n-soliton Solutions 95 §5.2. A Useful Trick: The τ-function 96 §5.3. Some Experiments 99 §5.4. Understanding the 2-soliton Solution 103 §5.5. General Remarks and Conclusions 109 Problems 109 Suggested Reading 111 Chapter 6. Multiplying and Factoring Differential Operators 113 §6.1. Differential Algebra 113 §6.2. Factoring Differential Operators 121 §6.3. Almost Division 124 §6.4. Application to Solving Differential Equations 125 §6.5. Producing an ODO with a Specified Kernel 127 Problems 130 Suggested Reading 132 Chapter 7. Eigenfunctions and Isospectrality 133

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