Chaohao Gu, Anning Hu, Zixiang Zhou

Darboux Transformations in Integrable Systems (eBook, PDF)

Theory and their Applications to Geometry

• Format: PDF

Produktbeschreibung

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.

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Produktdetails

• Verlag: Springer Netherlands
• Seitenzahl: 308
• Erscheinungstermin: 9. Juli 2006
• Englisch
• ISBN-13: 9781402030888
• Artikelnr.: 37357222

Autorenporträt

Chaohao Gu, Fudan University, Shanghai, China / Hesheng Hu, Fudan University, Shanghai, China / Zixiang Zhou, Fudan University, Shanghai, China

Inhaltsangabe

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.

Rezensionen

From the reviews: "The book is concerned with mutual relations between the differential geometry of surfaces and the theory of integrable nonlinear systems of partial differential equations. It concentrates on the Darboux matrix method for constructing explicit solutions to various integrable nonlinear PDEs. ... This book can be recommended for students and researchers who are interested in a differential-geometric approach to integrable nonlinear PDE's." (Jun-ichi Inoguchi, Mathematical Reviews, Issue 2006 i)