Darboux Transformations in Integrable Systems

Theory and their Applications to Geometry
Softcover version of original hardcover Auflage 2005
ISBN: 978-90-481-6788-3
Verlag: Springer Netherlands

Buch, Englisch, 308 Seiten, Previously published in hardcover, Format (B × H): 160 mm x 240 mm, Gewicht: 514 g

Reihe: Mathematical Physics Studies

GU Chaohao The soliton theory is an important branch of nonlinear science. On one hand, it describes various kinds of stable motions appearing in - ture, such as solitary water wave, solitary signals in optical ?bre etc., and has many applications in science and technology (like optical signal communication). On the other hand, it gives many e?ective methods ofgetting explicit solutions of nonlinear partial di?erential equations. Therefore, it has attracted much attention from physicists as well as mathematicians. Nonlinearpartialdi?erentialequationsappearinmanyscienti?cpr- lems. Getting explicit solutions is usually a di?cult task. Only in c- tain special cases can the solutions be written down explicitly. However, for many soliton equations, people have found quite a few methods to get explicit solutions. The most famous ones are the inverse scattering method,B¨ acklund transformation etc. The inverse scattering method is based on the spectral theory of ordinary di?erential equations. The Cauchyproblemofmanysolitonequationscanbetransformedtosolving a system of linear integral equations. Explicit solutions can be derived when the kernel of the integral equation is degenerate. The B¨ ac ¨ klund transformation gives a new solution from a known solution by solving a system of completely integrable partial di?erential equations. Some complicated “nonlinear superposition formula” arise to substitute the superposition principlein linear science.

Zielgruppe

Research

Autoren/Hrsg.

Gu, Chaohao

Zhou, Zixiang

Hu, Anning

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Integralrechnungen- und -gleichungen
• Naturwissenschaften Physik Physik Allgemein Theoretische Physik, Mathematische Physik, Computerphysik
• Mathematik | Informatik Mathematik Geometrie Differentialgeometrie

Inhaltsverzeichnis

1+1 Dimensional Integrable Systems.- 2+1 Dimensional Integrable Systems.- N + 1 Dimensional Integrable Systems.- Surfaces of Constant Curvature, Bäcklund Congruences and Darboux Transformation.- Darboux Transformation and Harmonic Map.- Generalized Self-Dual Yang-Mills Equations and Yang-Mills-Higgs Equations.- Two Dimensional Toda Equations and Laplace Sequences of Surfaces in Projective Space.