This paper investigates numerically influence of the external-cavity length on the type of the route-to-chaos of semiconductor lasers under external optical feedback. The study is based on numerical solution of a time-delay model of rate equations, and the solutions are employed to construct bifurcation diagrams and to examine the Fourier frequency spectrum of the laser output. The ratio of the relaxation frequency to the external-cavity resonance frequency is employed to measure the influence of the length of the external cavity. The route-to-chaos is period doubling when this frequency ratio is less than unity. The route is subharmonic when the frequency ratio increases up to 2.25. When the frequency ratio increases further, the transition to chaos becomes quasi-periodic characterized by the compound-cavity frequency and the relaxation frequency as well as their difference.

Influence of the linewidth-enhancement factor on the output and operations of InGaAs/InP pumping lasers emitting at a wavelength of 980nm under strong optical feedback is investigated numerically. The investigations are performed based on intensive numerical integration of an improved time-delay rate equations of semiconductor lasers over wide ranges of the linewidth-enhancement factor and optical feedback strength. The results show that the semiconductor laser operates under strong optical feedback in continuous wave and pulsation at small values of the linewidth-enhancement factor. Under large values of the linewidth-enhancement factor, the laser happens to exhibit chaos and pulsation. We predict that semiconductor laser subjected to strong optical feedback exhibits much more stable pulsing operation under higher values of the linewidth-enhancement factor, which indicates that the laser is locked at the external cavity frequency.

The jump problem and problems with defects on the type change line for model mixed-type equations in the mixed domains are investigated. The explicit solutions of the jump problem are obtained by the method of integral equations and by the Fourier transformation method. The problems with defects are reduced to singular integral equations. Some results for the solution of the equation under consideration are discussed concerning the existence and uniqueness for the solution of the suggested problem.