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

Solitary wave fission and fusion in the (2+1)-dimensional generalized Broer–Kaup system

Via a special Painlevé–Bäcklund transformation and the linear superposition theorem, we derive the general variable separation solution of the (2 + 1)-dimensional generalized Broer–Kaup system. Based on the general variable separation solution and choosing some suitable variable separated functions, new types of V-shaped and A-shaped solitary wave fusion and Y-shaped solitary wave fission phenomena are reported

Nonautonomous solitons in parity-time symmetric potentials

Some analytical solutions of a (1+1)-dimensional nonlinear Schrödinger equation with inhomogeneous diffraction and nonlinearity in the presence of the parity-time symmetric potential are derived. All characteristic parameters such as amplitudes, speeds

Controllable mechanism of breathers in the (2 + 1) -dimensional nonlinear Schrdinger equation with different forms of distributed transverse diffraction

We study the (2+1)-dimensional nonlinear Schrödinger equation with different forms of distributed transverse diffraction in anisotropic graded-index grating waveguides, and obtain an exact two-breather solution for certain functional relations. From thi

Multi-rogue wave and multi-breather solutions in PT-symmetric coupled waveguides

The coupled nonlinear Schrödinger equation in parity-time symmetric coupled waveguides is studied by means of the modified Darboux transformation method. The hierarchies of rational solutions and breather solutions are generated from the plane wave solu

Light bullet in parity-time symmetric potential

We derive light bullet solution of a (3(Formula presented.)1)-dimensional nonlinear Schrödinger equation with inhomogeneous diffraction/dispersion and nonlinearity in presence of the parity-time symmetric potential. All characteristic parameters such a

Stable light-bullet solutions in the harmonic and parity-time-symmetric potentials

Analytical light-bullet solutions of a (3+1)-dimensional nonlinear Schrödinger equation with inhomogeneous diffraction or dispersion and nonlinearity in the presence of the harmonic and parity-time-symmetric potentials are explored. Diffraction or dispe