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Rogue waves, also known as freak waves, monster waves, killer waves, extreme waves, and abnormal waves, are unusually large and suddenly appearing surface waves in the sea (Dysthe et al. 2008; Kharif et al. 2009). Since they appear and disappear without warning, they can be dangerous to ships, even to large ones. The famous Japanese print, The Great Wave Of Kanagawa, by Katsushika Hokusai (see cover of this book) can be viewed as an artistic rendering of a rogue wave. In the old days, sailors often talked about rogue waves that they saw in the sea, but such stories were often dismissed as myths or folklore. The first verified measurement of a rogue wave was on January 1, 1995, on the Draupner platform in the North Sea, where a 25.6 m wave was observed—much larger than the background wave field. Subsequently, rogue wave events have been observed in water tanks, thus allowing their physical mechanisms to be better understood. While there could be multiple mechanisms for the appearance of rogue waves, one of them is modulation instability, also called Benjamin-Feir instability, where a uniform nonlinear wavetrain (Stokes wavetrain) is unstable. When this Stokes wavetrain breaks up, it could generate rogue waves if the instability perturbation is seeded properly. The study of rogue waves has spread from oceanography to other branches of physics such as nonlinear optics, plasma and Bose-Einstein condensates (BEC) because the mathematical model for one-dimensional surface waves in the ocean— the nonlinear Schrödinger (NLS) equation, also governs wave phenomena in those other systems. The NLS equation admits explicit rational solutions that exhibit rogue-wave-like behaviors, which is the underlying mathematical reason for the appearance of rogue water waves in laboratory experiments and possibly in the ocean as well. Since the NLS equation also governs those other physical systems, optical rogue waves, plasma rogue waves, and BEC rogue waves would be expected. Indeed, such rogue waves have been observed in laboratories of optics, plasma, and BEC, thus widening rogue occurrences in the natural world. It should be recognized that there are also other physical settings where wave behaviors are governed not by the NLS equation but by other model equations such as the Manakov system and the three-wave resonant interaction system. Such model equations turn out to admit rogue wave solutions as well, due to modulation instability of a uniform background wave. Then, rogue events can be expected in even more diverse physical systems. Rogue waves can be studied both experimentally and theoretically. In the theoretical approach, one can derive and analyze rogue wave solutions in physically relevant mathematical models in order to gain insight into properties of rogue events. What makes this theoretical treatment possible is that, many physically relevant mathematical models, such as the NLS equation, the Manakov system, the three-wave resonant interaction system, and the long-wave-short-wave resonant interaction system, are all integrable, meaning that they can be solved analytically. Due to their integrability, we are able to derive explicit rogue wave solutions in those systems, which provide us with detailed quantitative information on rogue wave dynamics. In addition, due to the explicitness of these rogue wave solutions, we are able to perform various asymptotic analysis on these solutions, which results in asymptotic predictions of very fascinating rogue wave patterns. These rogue patterns turn out to be closely related to root structures of certain special polynomials, such as the Yablonskii-Vorob’ev polynomial hierarchy, the Okamoto polynomial hierarchies, and Adler-Moser polynomials. This beautiful connection between rogue patterns and special polynomials is a testament of the rich structure of rogue waves in integrable systems. This book summarizes the current state of knowledge on rogue waves in physically important integrable systems. The first chapter derives many of these integrable systems in physical settings such as water waves, optics, and plasma. This chapter provides physical motivations for our mathematical studies in later chapters. The second chapter derives rogue wave solutions in a wide array of integrable systems, including those obtained in Chap. 1 and much beyond. In the literature, rogue waves in many of those integrable systems were originally derived by generalized Darboux transformation. We will derive these rogue waves almost exclusively by the bilinear method, since rogue wave expressions by the bilinear method are much more explicit than those by Darboux transformation. The third chapter analyzes patterns of rogue waves in certain asymptotic limits such as large internal parameters. Connections between rogue patterns and root structures of special polynomials will be revealed, and universality of these rogue patterns in integrable systems will be established. The fourth chapter describes laboratory experiments on rogue waves in physical settings such as optical fibers, water tanks, plasma, and BEC. The last chapter covers topics that are closely related to rogue waves of the earlier chapters, such as rogue waves arising from a nonuniform background, robustness of rogue waves, partial-rogue waves, and lump patterns in the Kadomtsev-Petviashvili I equation. This book is intended as a monograph on the theoretical treatments of rogue waves. Its intended readership is researchers and graduate students in diverse mathematical and physical fields, where rogue waves are an interest of study. Most derivations are self-contained, and the reader should be able to follow them without much help from other sources.