Nonlinear Dispersive Partial Differential Equations and Inverse Scattering: Exploring Complex Phenomena

Nonlinear dispersive partial differential equations (PDEs) are a class of mathematical equations that describe the evolution of waves propagating through a medium. They arise in a wide range of physical phenomena, including optics, acoustics, and fluid dynamics. The term "nonlinear" refers to the fact that the equations involve nonlinear terms, which describe interactions between waves. The term "dispersive" refers to the fact that the waves travel at different speeds depending on their frequency.

Nonlinear Dispersive Partial Differential Equations and Inverse Scattering (Fields Institute Communications Book 83)

by Peter A. Perry

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Language	:	English
File size	:	10491 KB
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Print length	:	538 pages

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Inverse scattering is a technique for recovering information about a medium from the way that waves scatter from it. This technique is used in a variety of applications, including medical imaging, non-destructive testing, and radar.

Nonlinear Dispersive PDEs

Nonlinear dispersive PDEs are a challenging class of equations to solve. However, in recent years, there has been significant progress in understanding these equations. This progress has been driven by the development of new mathematical techniques, as well as the availability of powerful computers.

One of the most important breakthroughs in the study of nonlinear dispersive PDEs was the discovery of solitons. Solitons are wave-like solutions that are localized in space and time. They can travel long distances without changing their shape. Solitons are found in a variety of physical systems, including optics, acoustics, and fluid dynamics.

Inverse Scattering

Inverse scattering is a technique for recovering information about a medium from the way that waves scatter from it. This technique is based on the fact that the scattering of waves is affected by the properties of the medium. By measuring the scattered waves, it is possible to infer information about the medium's properties.

Inverse scattering is used in a variety of applications, including medical imaging, non-destructive testing, and radar. In medical imaging, inverse scattering is used to create images of the inside of the body. In non-destructive testing, inverse scattering is used to inspect materials for defects. In radar, inverse scattering is used to detect objects in the environment.

Applications

Nonlinear dispersive PDEs and inverse scattering have a wide range of applications in science and engineering. These applications include:

• Optics: Nonlinear dispersive PDEs are used to model the propagation of light in optical fibers. Inverse scattering is used to design optical fibers with specific properties.
• Acoustics: Nonlinear dispersive PDEs are used to model the propagation of sound waves in the atmosphere. Inverse scattering is used to locate underwater objects.
• Fluid dynamics: Nonlinear dispersive PDEs are used to model the flow of fluids. Inverse scattering is used to study turbulence.
• Medical imaging: Inverse scattering is used to create images of the inside of the body.
• Non-destructive testing: Inverse scattering is used to inspect materials for defects.
• Radar: Inverse scattering is used to detect objects in the environment.

Nonlinear dispersive PDEs and inverse scattering are powerful mathematical tools that can be used to describe a wide range of physical phenomena. These tools have a wide range of applications in science and engineering.

As our understanding of nonlinear dispersive PDEs and inverse scattering continues to grow, we can expect to see new applications for these tools in the future.

Nonlinear Dispersive Partial Differential Equations and Inverse Scattering (Fields Institute Communications Book 83)

by Peter A. Perry

4.7 out of 5

Language	:	English
File size	:	10491 KB
Screen Reader	:	Supported
Print length	:	538 pages

FREE