Power Law Asymptotics for Nonlinear Eigenvalue Problems

Abstract

Nonlinear eigenvalue problems arise in a wide variety of physical settings where oscillation amplitudes are too large to justify linearization. These problems have amplitude dependent frequencies, and one can ask whether scaling laws of the kind that arise in classical Sturm-Liouville theory still pertain. We prove in this paper that in the context of analyzing radially symmetric solutions to a class of nonlinear dispersive wave models, they do. We re-cast the equations as Hamiltonian systems with ‘dissipation’ where the radial variable ‘r’ plays the role of time. We treat the case of a symmetric double-well potential in detail and show that the appropriate nonlinear eigenfunctions are trajectories that start with energies above the center hump and ultimately decay to the peak as r → ∞. The number of crossings over the center line in the double-well (related to the number of times the trajectory ‘bounces’ off the sides) corresponds to the number of zeroes of the nonlinear eigenfunction and the initial energy levels corresponding to these eigenfunctions form a discrete set of values which can be related to the eigenvalues. If u _n (0) ≡ γ _n denotes the value for which the problem has an ‘eigenfunction’ u _n (r) with exactly n zeroes for r ∈ (0, ∞), we prove that the spacings γ _n+1 − γ _n follow power law scaling

$$E_n = \frac{{\gamma _n^{2p + 2} }} {{2p + 2}} - \frac{{\gamma _n^2 }} {2}$$

which we prove follows power law form

$$E_{n + 1} - E_n \sim G_p \cdot n^{1 + (2/p)}$$

Although only symmetric double-well potentials are treated in this paper (both the direct and inverse problem), it is clear that much more general situations can and should be analyzed in the future, including scaling laws for asymmetric potentials and potentials with more than two local minima.