H. Holden, N.H. Risebro: Front Tracking for Hyperbolic Conservation Laws

Springer-Verlag, Berlin Heidelberg, 2015, pp. XIV+517

This is the second edition of a well-received book providing the fundamentals of the theory hyperbolic conservation laws. Several chapters have been rewritten, new material has been added, in particular, a chapter on space dependent flux functions and the detailed solution of the Riemann problem for the Euler equations.

Hyperbolic conservation laws are central in the theory of nonlinear partial differential equations and in science and technology. The reader is given a self-contained presentation using front tracking, which is also a numerical method. The multidimensional scalar case and the case of systems on the line are treated in detail. A chapter on finite differences is included.

From the reviews of the first edition:

"It is already one of the few best digests on this topic. The present book is an excellent compromise between theory and practice. Students will appreciate the lively and accurate style." D. Serre, MathSciNet

"I have read the book with great pleasure, and I can recommend it to experts as well as students. It can also be used for reliable and very exciting basis for a one-semester graduate course." S. Noelle, Book review, German Math. Soc.

"Making it an ideal first book for the theory of nonlinear partial differential equations...an excellent reference for a graduate course on nonlinear conservation laws." M. Laforest, Comp. Phys. Comm.

Springer: https://www.springer.com/gp/book/9783662475065

Springer eBook: https://link.springer.com/book/10.1007%2F978-3-662-47507-2

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Mathematical Reviews

H. Holden, K. H. Karlsen, K.-A. Lie, N. H. Risebro: Operator Splitting for Nonlinear Partial Differential Equations with Rough Solutions. Analysis and Matlab Programs

EMS Series of Lectures in Mathematics, EMS Publishing House, Zurich, 2010, pp. viii+226



F. Gesztesy, H. Holden, J. Michor, and G. Teschl: Soliton Equations and Their Algebro-Geometric Solutions. Volume II: (1 + 1)-Dimensional Discrete Models

Cambridge Studies in Advanced Mathematics, volume 114. Cambridge University Press, Cambridge, 2008, pp. x+452

As a partner to Volume 1: Dimensional Continuous Models, this monograph provides a self-contained introduction to algebro-geometric solutions of completely integrable, nonlinear, partial differential-difference equations, also known as soliton equations. The systems studied in this volume include the Toda lattice hierarchy, the Kac-van Moerbeke hierarchy, and the Ablowitz-Ladik hierarchy. An extensive treatment of the class of algebro-geometric solutions in the stationary as well as time-dependent contexts is provided. The theory presented includes trace formulas, algebro-geometric initial value problems, Baker-Akhiezer functions, and theta function representations of all relevant quantities involved. The book uses basic techniques from the theory of difference equations and spectral analysis, some elements of algebraic geometry and especially, the theory of compact Riemann surfaces. The presentation is constructive and rigorous, with ample background material provided in various appendices. Detailed notes for each chapter, together with an exhaustive bibliography, enhance understanding of the main results.

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F. Gesztesy, H. Holden: Soliton Equations and Their Algebro-Geometric Solutions. Volume I: (1 + 1)-Dimensional Continuous Models

Cambridge Studies in Advanced Mathematics, volume 79. Cambridge University Press, Cambridge, 2003, pp. x+530.

The focus of this book is on algebro-geometric solutions of completely integrable nonlinear partial differential equations in (1+1)-dimensions, also known as soliton equations. Explicitly treated integrable models include the KdV, AKNS, sine-Gordon, and Camassa-Holm hierarchies as well as the classical massive Thirring system. An extensive treatment of the class of algebro-geometric solutions in the stationary as well as time-dependent contexts is provided. The formalism presented includes trace formulas, Dubrovin-type initial value problems, Baker-Akhiezer functions, and theta function representations of all relevant quantities involved. The book uses techniques from the theory of differential equations, spectral analysis, and elements of algebraic geometry (most notably, the theory of compact Riemann surfaces). The presentation is rigorous, detailed, and self-contained, with ample background material provided in various appendices. Detailed notes for each chapter together with an exhaustive bibliography enhance the presentation offered in the main text.

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H. Holden, B. Øksendal, J. Ubøe, T. Zhang: Stochastic Partial Differential Equations. A Modeling, White Noise Functional Approach.


Second edition,
Universitext, Springer-Verlag, New York, 2010, pp. xv+305.

The first edition of Stochastic Partial Differential Equations: A Modeling, White Noise Functional Approach, gave a comprehensive introduction to SPDEs driven by space-time Brownian motion noise. In this, the second edition, the authors extend the theory to include SPDEs driven by space-time Lévy process noise, and introduce new applications of the field.

Because the authors allow the noise to be in both space and time, the solutions to SPDEs are usually of the distribution type, rather than a classical random field. To make this study rigorous and as general as possible, the discussion of SPDEs is therefore placed in the context of Hida white noise theory. The key connection between white noise theory and SPDEs is that integration with respect to Brownian random fields can be expressed as integration with respect to the Lebesgue measure of the Wick product of the integrand with Brownian white noise, and similarly with Lévy processes.

The first part of the book deals with the classical Brownian motion case. The second extends it to the Lévy white noise case. For SPDEs of the Wick type, a general solution method is given by means of the Hermite transform, which turns a given SPDE into a parameterized family of deterministic PDEs. Applications of this theory are emphasized throughout. The stochastic pressure equation for fluid flow in porous media is treated, as are applications to finance.

Graduate students in pure and applied mathematics as well as researchers in SPDEs, physics, and engineering will find this introduction indispensible. Useful exercises are collected at the end of each chapter.

From the reviews of the first edition:

"The authors have made significant contributions to each of the areas. As a whole, the book is well organized and very carefully written and the details of the proofs are basically spelled out... This is a rich and demanding book… It will be of great value for students of probability theory or SPDEs with an interest in the subject, and also for professional probabilists." —Mathematical Reviews

"...a comprehensive introduction to stochastic partial differential equations." —Zentralblatt MATH

Springer: https://www.springer.com/gp/book/9780387894874

Springer (eBook): https://link.springer.com/book/10.1007%2F978-0-387-89488-1

S. Albeverio, F. Gesztesy, R. Høegh-Krohn: Solvable Models in Quantum Mechanics

Second edition with an Appendix by P. Exner
AMS Chelsea Publishing, volume 350. Chelsea Publishing, American Mathematical Society, Providence, 2005. pp. xiv+452.

[from Springer edition] Next to the harmonic oscillator and the Coulomb potential the class of two-body models with point interactions is the only one where complete solutions are available. All mathematical and physical quantities can be calculated explicitly which makes this field of research important also for more complicated and realistic models in quantum mechanics. The detailed results allow their implementation in numerical codes to analyse properties of alloys, impurities, crystals and other features in solid state quantum physics. This monograph presents in a systematic way the mathematical approach and unifies results obtained in recent years. The student with a sound background in mathematics will get a deeper understanding of Schrödinger Operators and will see many examples which may eventually be used with profit in courses on quantum mechanics and solid state physics. The book has textbook potential in mathematical physics and is suitable for additional reading in various fields of theoretical quantum physics.

AMS: https://bookstore.ams.org/chel-350-h/

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H. Holden, B. Øksendal, J. Ubøe, T. Zhang: Stochastic Partial Differential Equations. A Modeling, White Noise Functional Approach.

First edition, Birkhauser, Basel–Boston. 1996, pp. xv+305.


S. Albeverio, F. Gesztesy, R. Høegh-Krohn: Solvable Models in Quantum Mechanics

Texts and Monographs in Physics. Springer-Verlag, Berlin-Heidelberg-New York-London-Paris-Tokyo, 1988, pp. xiv+452.
Translation into the Russian, Mir, Moscow 1991
(Translated by Yu. A. Kuperin, K. A. Makarov, V. A. Geiler)

Next to the harmonic oscillator and the Coulomb potential the class of two-body models with point interactions is the only one where complete solutions are available. All mathematical and physical quantities can be calculated explicitly which makes this field of research important also for more complicated and realistic models in quantum mechanics. The detailed results allow their implementation in numerical codes to analyse properties of alloys, impurities, crystals and other features in solid state quantum physics. This monograph presents in a systematic way the mathematical approach and unifies results obtained in recent years. The student with a sound background in mathematics will get a deeper understanding of Schrödinger Operators and will see many examples which may eventually be used with profit in courses on quantum mechanics and solid state physics. The book has textbook potential in mathematical physics and is suitable for additional reading in various fields of theoretical quantum physics.

Springer: https://www.springer.com/gp/book/9783642882036

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Mathematical Reviews

H. Holden, N.H. Risebro: Front Tracking for Hyperbolic Conservation Laws

Springer-Verlag, Berlin Heidelberg, 2002, pp. XIV+517

H. Holden: Sturm–Liouville Operators and Hilbert Spaces: A Brief Introduction

Tapir, Trondheim, 2001. pp. vi+82.