Rogue waves in higher-order systems: Lagrangian approach

A Lagrangian approach for finding rogue wave solutions of the extended nonlinear Schrödinger equation system is developed. A list of Lagrangian components for a variety of terms in the extended equation is presented. It turns the analysis of any equation into a simple and straightforward exercise. The fact that these terms can be summed is a major point in achieving this simplicity. Importantly, the technique provides the same result, no matter whether an extension term enters the basic part of the equation or is taken as a perturbation. This significant conclusion is demonstrated by giving several examples. We give several examples of the application of this technique for particular physically-relevant extended equations. As a typical example, the Lagrangian approach is used to evaluate the effects of fourth and sixth order dispersion on rogue waves. The results are compared with solutions found by numerical simulations of the original equation. Good agreements are found, thus confirming the usefulness of the approach. A generalization to even higher-order dispersive terms shows that the phase function for the rogue wave can be written in terms of known functions, e.g. the arcsinh function. When these equations are integrable, the solutions naturally coincide with the exact rogue wave solutions. Obtaining the exact solution this way is demonstrated in the case of Kundu-Echkaus equation.