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

Figure 8

The present figure suggests a possible argument to prove multiplicity of sign changing solutions to the Neumann problem (94). We start with system by considering initial points with (recall that is the positive center of , while is the intersection point of the homoclinic orbit with the positive -axis). We parameterize such initial points as an arc If is sufficiently large (with a lower bound which can be easily estimated from the equation) we find that is a spiral-like curve winding a certain number of times around and with an end point on in the fourth quadrant near the origin. If we fix a (small) positive constant and denote by the region between the homoclinic trajectories of and the level line , we observe that the points of remain on under the action of , while the points of run around the origin along the periodic orbit and will perform a certain number of revolutions if is sufficiently large. More precisely, if we denote by the period of and suppose that , we can find solutions of (94) having precisely -zeros in the interval , for every As in the preceding figures, we have considered and , For graphical reasons a slightly different - and -scaling has been used.