Be careful with variable separation solutions via the extended tanh-function method and periodic wave structures

We analyze the extended tanh-function method to realize variable separation, however, we find that various "different" solutions obtained by this method are seriously equivalent to the general solution derived by the multilinear variable separation approach. In order to illustrate this point, we take a general (2 + 1)-dimensional Korteweg–de Vries system in water for example. Eight kind of variable separation solutions for a general (2 + 1)-dimensional Korteweg–de Vries system are derived by means of the extended tanh-function method and the improved tanh-function method. By detailed investigation, we find that these seemly independent variable separation solutions actually depend on each other. It is verified that many of so-called "new" solutions are equivalent to one another. Based on the uniform variable separation solution, abundant localized coherent structures can be constructed. However, we must pay our attention to the solution expression of all components to avoid the appearance of some un-physical related and divergent structures: seemly abundant structures for a special component are obtained while the divergence of the corresponding other component for the same equation appears

Non-Linear Lattice

The development of mathematical techniques, combined with new possibilities of computational simulation, have greatly broadened the study of non-linear lattices, a theme among the most refined and interdisciplinary-oriented in the field of mathematical physics. This Special Issue mainly focuses on state-of-the-art advancements concerning the many facets of non-linear lattices, from the theoretical ones to more applied ones. The non-linear and discrete systems play a key role in all ranges of physical experience, from macrophenomena to condensed matter, up to some models of space discrete space-time