SCALING LAWS AT NONLINEAR SCHRODINGER DEFECT SITES

A new family of defect solutions to the nonlinear Schrodinger equation is described. The defects have standing wave dynamics with j concentr ic rings centered at the defect site r=0, and a conical shape as r-->0 with angle of opening phi(j). Using a phase space technique, solution trajectories having a prescribed number (j) of rings are computed alo ng with their corresponding eigenvalue nu(j), and angle of opening phi (j). As in the linear Sturm-Liouville theory, the eigenvalues are orde red so that nu(j-1)<nu(j)<nu(j+1), a fact which is clearly seen from t he phase space structure. The nonlinear eigenfunctions are trajectorie s which lie on the basin boundary between the domains of the attractio n of the two asymptotically stable trajectories in the three dimension al phase space. The asymptotic distribution of the eigenvalues for lar ge j, and the angle of opening at the defect site are both shown to ha ve a power law form, and formulas for the power law exponents are give n.