Folded Solitary Waves and Foldons in 2+1 Dimensions

A general type of localized excitations, folded solitary waves and foldons, are defined and studied both analytically and graphically. The folded solitary waves and foldons may be “folded” in quite complicated ways and possess quite rich structures and abundant interaction properties. The folded phenomenon is quite universal in the real natural world. The folded solitary waves and foldons are obtained from a quite universal formula and the universal formula is valid for some quite universal (2+1)-dimensional physical models. The “universal” formula is also extended to a more general form with many more independent arbitrary functions. PACS.05.45.Yv, 02.30.Jr, 02.30.Ik. In the study of nonlinear science, soliton theory plays a very important role and has been applied in almost all the natural sciences especially in all the physical branches such as the fluid physics, condense matter physics, biophysics, plasma physics, nonlinear optics, quantum field theory and particle physics etc.. Almost all the previous studies of soliton theory especially in high dimensions are restricted in single valued situations. However, the real natural phenomena are very complicated. In various cases, it is even impossible to describe the natural phenomena by single valued functions. For instance, in the real natural world, there exist very complicated folded phenomena such as the folded protein [1], folded brain and skin surfaces and many many other kinds of folded biologic systems[2]. The simplest multi-valued (folded) waves may be the bubbles on (or under) a fluid surface. Various kinds of ocean waves are really folded waves also. To study the complicated folded natural phenomena is very difficult. Similar to the single valued cases, the first important question we should ask is: Are there any stable multivalued (folded) localized excitations? For convenience later, we define the multi-valued localized excitations folded solitary waves. Furthermore, if the interactions among the folded solitary waves are completely elastic, we call them foldons. In (1+1)-dimensional case, the simplest foldons are so-called loop solitons[3] which can be found in many (1+1)-dimensional integrable systems[3] and have been applied in some possible physical fields like the string