New coupled rogue waves propagating backward and forward and modulation instability in a composite nonlinear right- and left-handed transmission line

The aim of this study is to investigate new exact coupled rogue wave solutions of the coupled nonlinear Schrödinger equation system in a nonlinear left- and right-handed composite transmission line by the semi-discrete approximation. By means of this approximation, we found coupled type I and II rogue waves of the above-mentioned equation system. The solutions obtained are expressed in the form of new coupled rogue wave solutions of type I and II. This approximation used is efficient, powerful and can be considered as an alternative to establish new rogue waves of different types of the Schrödinger equation system applied in mathematical physics. In addition, in order to display the underlying dynamics of the coupled type I and II rogue wave solutions obtained, 3D plots are drawn. The computational results obtained show not only the efficiency and robustness of this approximation, but also the potential applicability of this technique to other significant nonlinear Schrödinger equation systems. These results obtained show that coupled rogue waves well exist in the nonlinear left- and right-handed composite transmission line and that the zones of instability could also gradually disappear when this line operates mainly at low frequencies.