162 research outputs found
Decomposition Method for Kdv Boussinesq and Coupled Kdv Boussinesq Equations
This paper obtains the solitary wave solutions of two different forms of Boussinesq equations that model the study of shallow water waves in lakes and ocean beaches. The decomposition method using He’s polynomials is applied to solve the governing equations. The travelling wave hypothesis is also utilized to solve the generalized case of coupled Boussinesq equations, and, thus, an exact soliton solution is obtained. The results are also supported by numerical simulations. Keywords: Decomposition Method, He’s polynomials, cubic Boussinesq equation, Coupled Boussinesq equation
Grassmannian flows and applications to nonlinear partial differential equations
We show how solutions to a large class of partial differential equations with
nonlocal Riccati-type nonlinearities can be generated from the corresponding
linearized equations, from arbitrary initial data. It is well known that
evolutionary matrix Riccati equations can be generated by projecting linear
evolutionary flows on a Stiefel manifold onto a coordinate chart of the
underlying Grassmann manifold. Our method relies on extending this idea to the
infinite dimensional case. The key is an integral equation analogous to the
Marchenko equation in integrable systems, that represents the coodinate chart
map. We show explicitly how to generate such solutions to scalar partial
differential equations of arbitrary order with nonlocal quadratic
nonlinearities using our approach. We provide numerical simulations that
demonstrate the generation of solutions to
Fisher--Kolmogorov--Petrovskii--Piskunov equations with nonlocal
nonlinearities. We also indicate how the method might extend to more general
classes of nonlinear partial differential systems.Comment: 26 pages, 2 figure
Nonlinear Waves and Dispersive Equations
Nonlinear dispersive equations are models for nonlinear waves in a wide range of physical contexts. Mathematically they display an interplay between linear dispersion and nonlinear interactions, which can result in a wide range of outcomes from finite time blow-up to scattering. They are linked to many areas of mathematics and physics, ranging from integrable systems and harmonic analysis to fluid dynamics and general relativity. The conference did focus on the analytic aspects and PDE aspects
Mathematical Theory of Water Waves
Water waves, that is waves on the surface of a fluid (or the interface between different fluids) are omnipresent phenomena.
However, as Feynman wrote in his lecture, water waves that are easily seen by everyone, and which are usually used as an example of waves in elementary courses, are the worst possible example; they have all the complications that waves can have. These complications make mathematical investigations particularly challenging and the physics particularly rich.
Indeed, expertise gained in modelling,
mathematical analysis and numerical simulation of water waves can be expected to lead to progress in issues of high societal impact
(renewable energies in marine environments, vorticity generation and wave breaking, macro-vortices and coastal erosion, ocean
shipping and near-shore navigation, tsunamis and hurricane-generated waves, floating airports, ice-sea interactions,
ferrofluids in high-technology applications, ...).
The workshop was mostly devoted to rigorous mathematical theory for the exact hydrodynamic
equations; numerical simulations, modelling and experimental issues were included insofar as they
had an evident synergy effect
Book of Abstracts
USPCAPESFAPESPCNPqINCTMatICMC Summer Meeting on Differentail Equations.\ud
São Carlos, Brasil. 3-7 february 2014
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