Numerical generation and investigation of rogue waves for discrete nonlinear Schrödinger equations

Abstract

We have numerically generated the rogue waves supported by discrete nonlinear Schrödinger equation (DNLSE). The investigation has been made on pure cubic DNLSE as well as on cubic-quintic discrete nonlinear Schrödinger equation (CQDNLSE). Runge–Kutta fourth (RK4) order method has been employed to obtain these results. For the cubic DNLSE breathers are found to be localized in one of the dimensions and periodic in the other dimensions, whereas rogue waves are found to be appearing from nowhere and disappearing without a trace. For CQDNLSE it is reported that the sign of quintic term possesses a significant effect on the intensity of the rogue waves. Further these waves are appearing from nowhere but instead of disappearing these show periodic reappearance. For larger value of quintic coefficient this phenomenon is visible at shorter length scale. Breathers are reported for the first time for the cubic DNLSE. The obtained results for CQDNLSE are unobserved previously in any constant coefficient discrete nonlinear equation.