Inverse Scattering Problems and Their Application to Nonlinear Integrable Systems

Inverse scattering problems are a class of mathematical problems that involve determining the properties of a scattering system from measurements of the scattered waves. These problems arise in a wide variety of applications, including radar, sonar, and medical imaging. In this article, we will discuss the basic principles of inverse scattering problems and their application to the study of nonlinear integrable systems.

Inverse Scattering Problems and Their Application to Nonlinear Integrable Equations (Chapman & Hall/CRC Monographs and Research Notes in Mathematics)

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Background

Scattering problems are concerned with the propagation of waves in a medium. When a wave encounters an obstacle, it will be scattered in all directions. The scattered waves contain information about the properties of the obstacle, such as its size, shape, and composition. Inverse scattering problems involve using measurements of the scattered waves to determine the properties of the obstacle.

Inverse scattering problems are typically solved using a variety of mathematical techniques, including integral equations, Fourier transforms, and Green's functions. The choice of technique depends on the specific problem being solved.

Application to Nonlinear Integrable Systems

Nonlinear integrable systems are a class of dynamical systems that exhibit a remarkable property called integrability. Integrable systems possess a number of special properties, including the existence of an infinite number of conserved quantities. This makes them amenable to a variety of powerful mathematical techniques.

Inverse scattering problems have been used to study a variety of nonlinear integrable systems, including the Korteweg-de Vries equation, the nonlinear Schrödinger equation, and the sine-Gordon equation. These problems have provided valuable insights into the behavior of these systems and have helped to develop new mathematical techniques for their analysis.

Examples

Here are a few examples of how inverse scattering problems have been used to study nonlinear integrable systems:

• The inverse scattering transform has been used to solve the Korteweg-de Vries equation. This equation describes the propagation of waves in a nonlinear medium, such as a fluid or a plasma. The inverse scattering transform provides a way to find exact solutions to the Korteweg-de Vries equation for a variety of initial conditions.
• The inverse scattering method has been used to study the nonlinear Schrödinger equation. This equation describes the propagation of waves in a nonlinear medium with dispersion. The inverse scattering method provides a way to find exact solutions to the nonlinear Schrödinger equation for a variety of initial conditions.
• The inverse scattering method has been used to study the sine-Gordon equation. This equation describes the propagation of waves in a nonlinear medium with a periodic potential. The inverse scattering method provides a way to find exact solutions to the sine-Gordon equation for a variety of initial conditions.

Inverse scattering problems are a powerful tool for studying nonlinear integrable systems. These problems have provided valuable insights into the behavior of these systems and have helped to develop new mathematical techniques for their analysis. As our understanding of inverse scattering problems continues to grow, we can expect to see even more applications of these problems in the future.

References

• Inverse scattering problem on Wikipedia
• What is the inverse scattering problem? on Math Stack Exchange
• Inverse scattering problems for nonlinear integrable systems by Ablowitz, Clarkson, and Fokas

Inverse Scattering Problems and Their Application to Nonlinear Integrable Equations (Chapman & Hall/CRC Monographs and Research Notes in Mathematics)

5 out of 5

Language	:	English
File size	:	10691 KB
Print length	:	414 pages

FREE