Motion of strings, embedding problem and soliton equations

✍ Scribed by Lakshmanan, M. ;Tamizhmani, K. M.

• Publisher: Springer
• Year: 1981
• Tongue: English
• Weight: 641 KB
• Volume: 37
• Category: Article
• ISSN: 0003-6994
• DOI: 10.1007/bf00382623

✦ Synopsis

The motion of a flexible string of constant length in E 3 in interaction, corresponding to a variety of physical situations, is considered. It is pointed out that such a system could be studied in terms of the embedding problem in differential geometry, either as a moving helical space curve in E ~ or by the embedding equations of t w o dimensional surfaces in E 3. The resulting integrability equations are identifiable with standard sofiton equations such as the non-linear Schrödinger, modified K-dV, sine-Gordon, Lund-Regge equations, etc. On appropriate reductions the embedding equations in conjunction wirb suitable local space-time and/or gange symmetries reproduce the AKNS-type eigenvalue equations and Riccati equations associated with soliton equations. The group theoretical properties follow naturally from these studies. Thus the above procedure gives a simple geometric interpretation to a large class of the soliton possessing nonlinear evolution equations and at the same time solves the underlying string equations.