Qualitative Properties of Dispersive PDEs

This book provides a comprehensive collection of contributions by distinguished scholars on the state-of-the-art and future challenges in the field of dispersive partial differential equations, covering long-time behavior, probabilistic methods, dispersive properties, and wave and KdV-type equations.

Format: Hardback
Length: 245 pages
Publication date: 03 December 2022
Publisher: Springer Verlag, Singapore

This book is a treasure trove of insightful contributions from renowned scholars, showcasing the cutting-edge advancements and emerging challenges in the realm of dispersive partial differential equations. Spanning four major areas of exploration, it delves into the intricate behavior of non-linear Schrödinger equations (NLS-type equations), employs probabilistic and nonstandard methods to study NLS equations, explores the dispersive properties of heat, Schrödinger, and Dirac-type flows, and examines wave and KdV-type equations. Within these chapters, a diverse range of crucial mathematical tools are discussed, exhibiting remarkable applicability and versatility that extend far beyond the specific models presented. Furthermore, each contribution is accompanied by an extensive update and comprehensive literature review, ensuring readers have access to the most recent developments in the field. This comprehensive volume serves as a vital resource for researchers, scholars, and students alike, providing a deep understanding of the state-of-the-art and paving the way for future breakthroughs in dispersive partial differential equations.

Introduction:
Dispersive partial differential equations (PDEs) have emerged as a fundamental tool in various fields of science and engineering, offering a powerful framework to describe complex physical phenomena. These equations, which involve the partial differential operators and their derivatives with respect to both space and time, arise in a wide range of applications, including physics, chemistry, biology, and mathematics. The study of PDEs has witnessed significant progress over the years, with researchers developing new methods, theories, and computational techniques to address challenging problems.
State of the Art and Recent Developments:
This book serves as a valuable compilation of contributions by distinguished scholars, presenting the state of the art and some of the most significant latest developments in the field of dispersive PDEs. The material covers a wide range of topics, including long-time behavior of NLS-type equations, probabilistic and nonstandard methods for studying NLS equations, dispersive properties for heat, Schrödinger, and Dirac-type flows, and wave and KdV-type equations. Each chapter provides a comprehensive overview of the current research, highlighting the key findings and open challenges.
Applications:
The applications of dispersive PDEs are diverse and far-reaching. They are used to model a wide range of physical phenomena, such as fluid dynamics, heat transfer, electromagnetic waves, and quantum mechanics. By leveraging the power of PDEs, researchers can gain insights into the behavior of complex systems and develop mathematical models that can accurately describe the underlying physics.
Mathematical Tools:
The study of dispersive PDEs requires a variety of mathematical tools, including differential equations, partial differential equations, probability theory, and numerical methods. These tools are used to solve PDEs, analyze their solutions, and make predictions about the behavior of systems. The development of new and improved mathematical tools has played a crucial role in advancing the field of dispersive PDEs.
Future Challenges:
While dispersive PDEs have made significant strides in recent years, there are still many open challenges that need to be addressed. These challenges include the study of PDEs with complex structures, the development of efficient numerical methods for solving large-scale PDEs, and the integration of PDEs with other fields of science, such as computer science and data science.
Conclusion:
In conclusion, this book provides a comprehensive and up-to-date overview of the state of the art and emerging challenges in the field of dispersive PDEs. It serves as a valuable resource for researchers, scholars, and students alike, offering insights into the latest developments and providing a roadmap for future research. The study of dispersive PDEs is a vibrant and dynamic field, with endless possibilities for discovery and innovation.

Weight: 559g
Dimension: 235 x 155 (mm)
ISBN-13: 9789811964336
Edition number: 1st ed. 2022