A Steady-state Trio for Bretherton Equation

To investigate if steady-state resonant solution exist for any system of weakly interacting waves in a dispersive medium, a trio is considered in the Bretherton equation based on the homotopy analysis method (HAM). Time-independent spectrum was found when all components were travelling in the same direction. Within the trio, the amplitude of longer component is larger than that of shorter one. As the difference of wave number between components in trio increases or the nonlinearity of whole system increases, the amplitudes of all components tends to increase simultaneously. These findings are helpful to enrich and deepen our understanding about resonant solutions in any dispersive medium, especially for a two-dimensional scenario.