Combined multi-scale-variational approach for finding quadratic three-color optical solitons at three-wave cascaded interaction

Multi-color optical solitons have numerous potential applications encouraging researchers to find them by using various approaches based on both exact methods or approximate ones. Among them is widely used a variational approach, whose complexity grows fast with increasing a number of interacting waves. Therefore, a simplification of a mathematical model, describing the multi-waves interaction, without losing the key physical factors of a process under consideration is actual problem. In current paper, such simplification based on combined multi-scale-variational approach is suggested and we demonstrate its applicability for finding temporal three-color optical solitons with equidistant frequencies propagating in a medium with quadratic nonlinear response under large phase mismatching between fundamental wave and its second harmonic. At first, multi-scale method is applied to simplify original set of nonlinear Schrödinger equations. Then, a variational method is used to find approximate soliton-like solutions of the simplified equations. The solution stability criterion is derived based on the Hamiltonian of simplified equations. Computer simulation based on original set of nonlinear Schrödinger equations demonstrates the three-wave structure stabilization after short propagation distance: the pulses propagate without essential energy exchange and changing shapes i.e. some small changes occur). The three-color soliton is even robust to an influence of group velocity mismatching between interacting waves.