Analytical Methods For Nonlinear Oscillators And Solitary Waves
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Analytical Methods for Nonlinear Oscillators and Solitary Waves
The most well-known analytical method is the perturbation method, which has led to the great discovery of Neptune in 1846, and since then mathematical prediction and empirical observation became two sides of a coin in physics. However, the perturbation method is based on the small parameter assumption, and the obtained solutions are valid only for weakly nonlinear equations, which have greatly limited their applications to modern physical problems. To overcome the shortcomings, many mathematicians and physicists have been extensively developing various technologies for several centuries, however, there is no universal method for all nonlinear problems, and mathematical prediction with remarkably high accuracy is still much needed for modern physics, for example, the solitary waves traveling along an unsmooth boundary, the low-frequency property of a harvesting energy device, the pull-in voltage in a micro-electromechanical system. Now various effective analytical methods have appeared in the open literature, e.g., the homotopy perturbation method and the variational iteration method. An analytical solution provides a fast insight into its physical properties of a practical problem, e.g., frequency-amplitude relation of a nonlinear oscillator, solitary wave in an optical fiber, pull-in instability of a microelectromechanical system, making mathematical prediction even more attractive in modern physics. Nonlinear physics has been developing into a new stage, where the fractal-fractional differential equations have to be adopted to describe more accurately discontinuous problems, and it becomes ever more difficult to find an analytical solution for such nonlinear problems, and the analytical methods for fractal-fractional differential equations have laid the foundations for nonlinear physics.
Partially Integrable Evolution Equations in Physics
Author: R. Conte
language: en
Publisher: Springer Science & Business Media
Release Date: 2012-12-06
In the many physical phenomena ruled by partial differential equations, two extreme fields are currently overcrowded due to recent considerable developments: 1) the field of completely integrable equations, whose recent advances are the inverse spectral transform, the recursion operator, underlying Hamiltonian structures, Lax pairs, etc 2) the field of dynamical systems, often built as models of observed physical phenomena: turbulence, intermittency, Poincare sections, transition to chaos, etc. In between there is a very large region where systems are neither integrable nor nonintegrable, but partially integrable, and people working in the latter domain often know methods from either 1) or 2). Due to the growing interest in partially integrable systems, we decided to organize a meeting for physicists active or about to undertake research in this field, and we thought that an appropriate form would be a school. Indeed, some of the above mentioned methods are often adaptable outside their original domain and therefore worth to be taught in an interdisciplinary school. One of the main concerns was to keep a correct balance between physics and mathematics, and this is reflected in the list of courses.
Dynamics of Lattice Materials
Author: A. Srikantha Phani
language: en
Publisher: John Wiley & Sons
Release Date: 2017-09-25
Provides a comprehensive introduction to the dynamic response of lattice materials, covering the fundamental theory and applications in engineering practice Offers comprehensive treatment of dynamics of lattice materials and periodic materials in general, including phononic crystals and elastic metamaterials Provides an in depth introduction to elastostatics and elastodynamics of lattice materials Covers advanced topics such as damping, nonlinearity, instability, impact and nanoscale systems Introduces contemporary concepts including pentamodes, local resonance and inertial amplification Includes chapters on fast computation and design optimization tools Topics are introduced using simple systems and generalized to more complex structures with a focus on dispersion characteristics