Thomas J. Bridges – författare
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6 produkter
6 produkter
Del 31 - Cambridge Monographs on Applied and Computational Mathematics
Symmetry, Phase Modulation and Nonlinear Waves
Inbunden, Engelska, 2017
1 054 kr
Skickas inom 7-10 vardagar
Nonlinear waves are pervasive in nature, but are often elusive when they are modelled and analysed. This book develops a natural approach to the problem based on phase modulation. It is both an elaboration of the use of phase modulation for the study of nonlinear waves and a compendium of background results in mathematics, such as Hamiltonian systems, symplectic geometry, conservation laws, Noether theory, Lagrangian field theory and analysis, all of which combine to generate the new theory of phase modulation. While the build-up of theory can be intensive, the resulting emergent partial differential equations are relatively simple. A key outcome of the theory is that the coefficients in the emergent modulation equations are universal and easy to calculate. This book gives several examples of the implications in the theory of fluid mechanics and points to a wide range of new applications.
Del 426 - London Mathematical Society Lecture Note Series
Lectures on the Theory of Water Waves
Häftad, Engelska, 2016
845 kr
Skickas inom 7-10 vardagar
In the summer of 2014 leading experts in the theory of water waves gathered at the Newton Institute for Mathematical Sciences in Cambridge for four weeks of research interaction. A cross-section of those experts was invited to give introductory-level talks on active topics. This book is a compilation of those talks and illustrates the diversity, intensity, and progress of current research in this area. The key themes that emerge are numerical methods for analysis, stability and simulation of water waves, transform methods, rigorous analysis of model equations, three-dimensionality of water waves, variational principles, shallow water hydrodynamics, the role of deterministic and random bottom topography, and modulation equations. This book is an ideal introduction for PhD students and researchers looking for a research project. It may also be used as a supplementary text for advanced courses in mathematics or fluid dynamics.
596 kr
Skickas inom 5-8 vardagar
355 kr
Skickas inom 5-8 vardagar
Del 1558 - Lecture Notes in Mathematics
Singularity Theory and Equivariant Symplectic Maps
Häftad, Engelska, 1993
487 kr
Skickas inom 10-15 vardagar
This monograph is a study of the local bifurcations of multiparameter symplectic maps of arbitrary dimension in the neighbourhood of a fixed point. The problem is reduced to a study of critical points of an equivariant gradient bifurcation problem, using the correspondence between orbits of a symplectic map and critical points of an action functional. New results on singularity theory for equivariant gradient bifurcation problems are obtained and then used to classify singularities of bifurcating period-q points. Of particular interest is the presentation of a general framework for analyzing group-theoretic aspects and singularities of symplectic maps (particularly period-q points). Topics include bifurcations when the symplectic map has spatial symmetry and a theory for the collision of multipliers near rational points with and without spatial symmetry. The monograph also includes 11 self-contained appendices each with a basic result on symplectic maps.
Transverse Instability of Solitary Waves
Multisymplectic Dirac Operators and the Evans Function
Inbunden, Engelska, 2025
1 492 kr
Skickas inom 5-8 vardagar
This book presents a wide-ranging geometric approach to the stability of solitary wave solutions of Hamiltonian partial differential equations (PDEs). It blends original research with background material and a review of the literature. The overarching aim is to integrate geometry, algebra, and analysis into a theoretical framework for the spectral problem associated with the transverse instability of line solitary wave solutions—waves that travel uniformly in a horizontal plane and are embedded in two spatial dimensions. Rather than focusing on individual PDEs, the book develops an abstract class of Hamiltonian PDEs in two spatial dimensions and time, based on multisymplectic Dirac operators and their generalizations. This class models a broad range of nonlinear wave equations and benefits from a distinct symplectic structure associated with each spatial dimension and time. These structures inform both the existence theory (via variational principles, the Maslov index, and transversality conditions) and the linear stability analysis (through a multisymplectic partition of the Evans function). The spectral problem arising from linearization about a solitary wave is formulated as a dynamical system, with three symplectic structures contributing to the analysis. A two-parameter Evans function—depending on the spectral parameter and transverse wavenumber—is constructed from this system. This structure enables new results concerning the Evans function and the linear transverse instability of solitary waves. A key result is an abstract derivative formula for the Evans function in the regime of small stability exponents and transverse wavenumbers. To illustrate the theory, the book introduces a class of vector-valued nonlinear wave equations in 2+1 dimensions that are multisymplectic and admit explicit solitary wave solutions. In this example, the stable and unstable subspaces involved in the Evans function construction are each four-dimensional and can be explicitly computed. The example is used to demonstrate the geometric instability condition and to explore the inner workings of the theory in detail.