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The aim of this textbook is to introduce the theory of nonlinear dispersive equations to graduate students in a constructive way. The first three chapters are dedicated to preliminary material, such as Fourier transform, interpolation theory and Sobolev spaces. The authors then proceed to use the linear Schrodinger equation to describe properties enjoyed by general dispersive equations. This information is then used to treat local and global well-posedness for the semi-linear Schrodinger equations. The end of each chapter contains recent developments and open problems, as well as exercises.

Differential equations, Partial. --- Nonlinear wave equations. --- Nonlinear wave equations --- Differential equations, Partial --- Engineering & Applied Sciences --- Applied Mathematics --- Partial differential equations --- Mathematics. --- Partial differential equations. --- Partial Differential Equations. --- Math --- Science --- Wave equation --- Differential equations, partial.

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This textbook introduces the well-posedness theory for initial-value problems of nonlinear, dispersive partial differential equations, with special focus on two key models, the Korteweg–de Vries equation and the nonlinear Schrödinger equation. A concise and self-contained treatment of background material (the Fourier transform, interpolation theory, Sobolev spaces, and the linear Schrödinger equation) prepares the reader to understand the main topics covered: the initial-value problem for the nonlinear Schrödinger equation and the generalized Korteweg–de Vries equation, properties of their solutions, and a survey of general classes of nonlinear dispersive equations of physical and mathematical significance. Each chapter ends with an expert account of recent developments and open problems, as well as exercises. The final chapter gives a detailed exposition of local well-posedness for the nonlinear Schrödinger equation, taking the reader to the forefront of recent research. The second edition of Introduction to Nonlinear Dispersive Equations builds upon the success of the first edition by the addition of updated material on the main topics, an expanded bibliography, and new exercises. Assuming only basic knowledge of complex analysis and integration theory, this book will enable graduate students and researchers to enter this actively developing field.

Partial differential equations --- differentiaalvergelijkingen

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Mother infant interactions are know to play an important role in the formation of maternal bonding in humans and in mice. Mouse studies can help to elucidate on the underlying mechanisms that influence maternal bonding in humans, for this reason the development of protocols capable to assess maternal bonding in mice has arisen. This study has aimed to assess whether a adapted radial-arm maze is able to reliably measure infant-to-mother bonding in juvenile mice and if expression of maternal preference changes between two crucial stages , namely, p21 and p28. Furthermore, to assess whether changes in mother-pup interactions lead to differences in neonatal and juvenile infant-to-mother interaction, two different C57BL/6 groups of mice have been tested. One group underwent prenatal VPA while the other group was comprised of control mice, prenatal VPA exposure has been shown to disrupt early mother-infant interaction. Lastly, to explore if evidence of changes in mother-infant interactions can be predictive of changes in adult functioning behaviors. A series of adult functioning tests were performed Our results could not validate the adapted radial-arm maze as a potential protocol for the assessment of infant-to-mother bonding in mice. Furthermore, no noticeable differences were found in neonatal USVs between VPA and control mice or in the adapted radial-arm maze at p21 or p28. Lastly, spatial memory was found to be impaired in VPA mice, similarly, repetitive behaviors were found to impair normal social functioning in adult VPA mice but no general social or communication functioning impairments were found. However, due to lack of evidence of changes in early mother-infant interaction, these results could not be directly attributed to impaired maternal bonding.