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Über dieses Buch

This book focuses on the theory of the Zakharov system in the context of plasma physics. It has been over 40 years since the system was first derived by V. E. Zakharov – and in the course of those decades, many innovative achievements with major impacts on other research fields have been made.

The book represents a first attempt to highlight the mathematical theories that are most important to researchers, including the existence and unique problems, blow-up, low regularity, large time behavior and the singular limit. Rather than attempting to examine every aspect of the Zakharov system in detail, it provides an effective road map to help readers access the frontier of studies on this system.



Chapter 1. Physical Background of Zakharov Equations and Its Soliton Solutions

There are many monographs on plasma physics, such as Braginskii [26], Dendy [46].
Boling Guo, Zaihui Gan, Linghai Kong, Jingjun Zhang

Chapter 2. On the Existence, Blowup and Large Time Behavior of the Zakharov System

We present in this book a wide-range survey of important topics on mathematical theories of Zakharov system Zakharov (Sov Phys JETP 35:908-914, 1972, [196]) established in 1972, with particular emphasis on various modern developments. It was Sulem and Sulem (C R Acad Sci Paris 289:173-176, 1979, [180]) who originally issued in the existence and uniqueness of global solutions to one dimensional Zakharov system. Global existence of the initial-boundary value problem for 1D Zakharov system was first obtained by Guo and Shen (Acta Math Appl Sin 3(5):310–324, 1982, [85].
Boling Guo, Zaihui Gan, Linghai Kong, Jingjun Zhang

Chapter 3. Studies on Generalized Zakharov System

In this chapter, we are concerned with the studies for some generalized Zakharov systems, including Zakharov system in nonhomogeneous medium, Klein–Gordon–Zakharov system, ion-acoustic Zakharov system, quantum Zakharov system and magnetic Zakharov system.
Boling Guo, Zaihui Gan, Linghai Kong, Jingjun Zhang

Chapter 4. Low Regularity Theories of Zakharov System

In the last two decades, low regularity theory has certainly been one of the fastest growing areas for the study of dispersive equation(s) owing to the application of modern analysis tools in partial differential equations. In this theory, we are asked for whether the equation possesses a unique solution or at least a local solution, which continuously depends on the given initial data belonging to some spaces with lower regularity.
Boling Guo, Zaihui Gan, Linghai Kong, Jingjun Zhang

Chapter 5. Singular Limit of Klein–Gordon–Zakharov System with Infinite Propagation Speed

Problem of singular limit for Zakharov type system concerns on the convergence of the system with infinite propagation speed.
Boling Guo, Zaihui Gan, Linghai Kong, Jingjun Zhang


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