By Philippe Guyenne, David Nicholls, Catherine Sulem
This ebook is a distinct number of paintings through world-class specialists exploring the newest advancements in Hamiltonian partial differential equations and their functions. issues lined inside of are consultant of the field’s large scope, together with KAM and basic shape theories, perturbation and variational tools, integrable platforms, balance of nonlinear options in addition to functions to cosmology, fluid mechanics and water waves.
The quantity includes either surveys and unique study papers and offers a concise assessment of the above issues, with effects starting from mathematical modeling to rigorous research and numerical simulation. it is going to be of specific curiosity to graduate scholars in addition to researchers in arithmetic and physics, who desire to examine extra in regards to the strong and chic analytical ideas for Hamiltonian partial differential equations.
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Example text
K / ! t; x; //d : D lim a a and u D M ! t; x; / C rx . t; x; a/da D 0; (33) M which is the Benney or V–D–B equation in the Zakharov–Grenier representation. t; x; /d D 0: M (34) Remark 7. The above representation is a variant both of the Madelung transform (where the amplitude a is taken real) and of the WKB method which is a Taylor expansion. t; x; / remains real for x real. This representation already appeared in [7] and [13] . It was used by Grenier [15, 16] for the validation of the semiclassical limit.
Thus, we can analyze these two cases separately. We’ll provide the details for the case of k even below - the equations with k odd behave in a very similar fashion. Remark 8. Note that if we multiply all of the equations in (14) by e and set e D (since we are interested in large times) we get equations that are formally of classical singularly perturbed form. ) Invariant manifold theory has been a powerful tool in the rigorous analysis of singularly perturbed problems and that analogy will guide our use of the center-manifold theory in what follows.
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