A cusp, bright breathers, dark breathers, kink, bright Rogue waves and some soliton waves solutions are obtained by using the exp(− ϕ (ξ ))-expansion method for the fourth order Benjamin-Ono equation and BBM equations. The obtained solutions might be indicated and meaningful for narrating the physical phenomena in the realworld. For compatible values of the arbitrary parameter included in the solution, we plot the 3D surface of the all obtained solutions which are shown in Figure 1 to Figure 10.

The rouge waves in an optically injected semiconductor lasers are receiving a lot of interest. In this work, the generating and controlling of the Rogue waves in semiconductor lasers have been investigated. For this purpose, the laser’s rate equations are solved numerically in a specified time interval, then the field amplitude and the intensity versus time are calculated and analyzed. To solve the rate equations, the famous finite deference method (FDM) is used. Also the rouge waves are counted using the standard definition that is mentioned in the context. Furthermore, the effects of the injection strength and the detuning frequency on the Rogue wave’s occurrence are studied. Results show that by increasing the injection amplitude, the number of Rogue waves decreases significantly, so that rouge waves vanish at the large values of the injection amplitude. Also increasing the detuning frequency, reduces the number of Rogue waves and this reduction is more sensitive at the large injection amplitude.

In this article, one gives a classification of the solutions to the one dimensional nonlinear focusing Schrodinger equation (NLS) by considering the modulus of the solutions in the (x, t) plan in the cases of orders 3 and 4. For this, we use a representation of solutions to NLS equation as a quotient of two determinants by an exponential depending ont. This formulation gives in the case of the order 3 and 4, solutions with, respectively 4 and 6 parameters. With this method, beside Peregrine breathers, we construct all characteristic patterns for the modulus of solutions, like triangular configurations, ring and others.