Nonlinear Phase- integral Approximations of stationary waves in nonhomogeneous systems.

Authors: Thylwe, K.-E., Dankowicz, H.
Document Type: Article
Pubstate: Published
Journal: J. Phys
Volume: 30   697-710
Year: 1996


A recent normal-form approximation for dynamical equilibria of one-dimensional Hamiltonian systems is shown to provide a phase-integral (WKB) approximation to solutions of nonlinear differential equations. In the present paper, a restricted class of ordinary differential equations d/sup 2/ Psi /dx/sup 2/+h/sub 2/(x) Psi +h/sub n/(x) Psi /sup n-1/=0, n>or=3, is considered. The integrability of the truncated normal form allows for expressing the solutions as trigonometric expansions in terms of an `amplitude` and `phase`. The method is applied to a Dirichlet boundary value problem on the interval x in [x/sub 0/, x/sub 1/] for n=3 and n=4 where the coefficient functions depend on an additional parameter omega . As in the constant coefficient case, we obtain approximate expressions for eigenvalues omega /sub k/, k=1, 2, ... and eigensolutions near the linear limit. The results show that the interpretative and the predictive power of the linear WKB solutions carry through to the nonlinear regime of small-amplitude, wavelike solutions Psi (x). We further analyse the mechanism by which the `odd-n` nonlinearity in general causes a splitting of the linear eigenvalues. In particular, we discuss the singular threshold behaviour of the doubling mechanism for nonlinearities with n=3. If the coefficient functions become constants, the doubling of eigenvalues corresponding to standing waves of odd numbers of nodes gradually disappears. The method of approximation can be worked out similarly for any `perturbing` polynomial in Psi (x).