Effective free energy for nonlinear dynamics

Abstract

Nonlinear dissipative systems often become dynamically unstable at critical values of a control parameter h. For h<h_c, all normal modes of the linearized theory decay, whereas, when h exceeds hc, linearized modes with wavenumbers near a nonzero critical value q_ç grow exponentially with time. A nonlinear analysis of the original free energy yields an effective renormalized free energy for the slowly varying envelope function that modulates the plane wave exp(iq_ç·x). For several cases of interest in connection with superfluid 3 He-A, the sign of the resulting quartic coupling produces catastrophic growth, indicating a failure of the small amplitude perturbation theory.