New patterns of solitons, fractal solitons, soliton molecules and their interactions for the (3 + 1)-dimensional potential-YTSF equation

The significant (3+1)-dimensional potential Yu-Toda-Sasa-Fukuyama equation, broadly applied in many fields such as fluid dynamics, condensed matter physics and statistical mechanics, is solved through the full spatial variable separation approach with a new ansatz which overcomes the deficiency that solutions can only describe non-travelling waves brought by the multilinear full variable separation approach. The new variable separation solutions can model a variety of moving nonlinear waves. Making use of the arbitrary function in these solutions, some interesting examples are delicatedly illustrated, including n-V-branch solitons, n-pulse solitons, 3-ridge solitons, fractal solitons and kinks with spatially–temporally varying backgrounds. The interactions of these nonlinear waves are investigated and their soliton molecules are obtained by means of the velocity resonance. Especially, interactions between fractal solitons, and between fractal solitons and soliton molecules, are explored for the first time. Using a specific example, the mechanism behind the fractal formation is elucidated. The roles of the important parameters are clarified to enhance the understanding of the properties and formation of localized waves. It is revealed that the full spatial variable separation approach is quite effective and promising for solving nonlinear evolution equations and exploring nonlinear wave phenomena.