Complex Dynamics in One-Dimensional Nonlinear Schrödinger Equations with Stepwise Potential

Figure 7

The present figure shows some possible rectangular regions which can be considered for the application of Theorem 4. The sets and are the same as in Figure 3 and (as we have already proved in Section 3) they can be used to provide a complex dynamics on positive solutions. If we choose, for instance, the sets and we can prove the presence of a complex dynamics generated by solutions which are negative on the time interval and oscillate in the phase-plane around the point , and then, in the time interval oscillate a certain number of times around the origin. More in detail, given any positive integer , we can produce solutions of (17) which have precisely simple zeros in the interval , provided that is sufficiently large. A lower estimate for can be easily determined by the knowledge of the period of the closed trajectory which bounds “externally” and Similar remarks can be made by selecting other pairs of topological rectangles among those put in evidence with a color. As in the preceding figures, we have considered and , For graphical reasons a slightly different - and -scaling has been used.