Extended multilinear variable separation approach and multivalued localized excitations for some (2+1)-dimensional integrable systems

The multilinear variable separation approach and the related “universal” formula have been applied to many (2+1)-dimensional nonlinear systems. Starting from the universal formula, abundant (2+1)-dimensional localized excitations have been found. In this paper, the universal formula is extended in two different ways. One is obtained for the modified Nizhnik–Novikov–Veselov equation such that two universal terms can be combined linearly and this type of extension is also valid for the (2+1)-dimensional symmetric sine-Gordon system. The other is for the dispersive long wave equation, the Broer–Kaup–Kupershmidt system, the higher order Broer–Kaup–Kupershmidt system, and the Burgers system where arbitrary number of variable separated functions can be involved. Because of the existence of the arbitrary functions in both the original universal formula and its extended forms, the multivalued functions can be used to construct a new type of localized excitations, folded solitary waves (FSWs) and foldons. The FSWs and foldons may be “folded” in quite complicated ways and possess quite rich structures and multiplicate interaction properties.