INVERSE METHOD FOR THE INVESTIGATION OF NONLINEAR INSTABILITIES ASSOCIATED WITH NEGATIVE-ENERGY MODES

Earlier work on explosively unstable similarity solutions of Hamilton' s equations with Hamiltonians homogeneous of degree N and satisfying r esonance conditions is applied to study the nonlinear stability of lin early stable equilibria with neighboring positive- and negative-energy waves. A multiple-time-scale expansion near equilibrium yields a Hami ltonian system of the assumed structure: In the inverse method an expl osively unstable similarity solution is assumed and one solves for the coefficients of the terms in a Hamiltonian of some given structure. T hrough some general arguments and many examples one concludes that exp losively unstable solutions occur generally for wide ranges of coeffic ient values. Hence the original equilibrium is nonlinearly unstable fo r wide ranges of interaction parameters.