Stability problem in nonlinear wave propagation

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Abstract
An explicit expression for the excitation spectrum of the stationary solutions of a nonlinear wave equation is obtained. It is found that all branches of many-valued solutions of a nonlinear wave equation between the (2K+1,2K+2) turning points (branch points in the complex plane of the nonlinearity parameter) are unstable. Some parts of branches between the (2K,2K+1) turning points are also unstable. The instability of the latter is related to the possibility that pairs of complex conjugate eigenvalues cross the real axis in the κ plane.

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