An interaction of three resonant modes in a nonlinear lattice

Abstract

The purpose of this paper is to discuss the Hamiltonian H = J 1 + 2J 2 + 3J 3 + αJ 1(2J 2) 1 2 cos(2φ 1 − φ 2) + 2β(J 1J 2J 3 1 2 cos(φ 1 + φ 2 − φ 3 , where the J k 's are canonically conjugate to the φ k 's ( k = 1, 2, 3). In the case β = 0 or α = 0 the corresponding Hamilton equations are integrable. A computer study of the full Hamiltonian was made by Ford and Lunsford ( Phys. Rev. A. 1 (1970) , 59–70). The present paper obtains analytical results that are confronted with the computer study. The results are obtained by expanding the Hamiltonian into a power series about a certain equilibrium point and constructing the corresponding Gustavson normal form up to 4th-order terms. The Gustavson normal form appears as a member of the enveloping algebra of the Lie algebra SO(2, 1). It is shown that the normal form can be used to explain certain features of Figures 5–9 of the above-mentioned computer study. Moreover the Komogloroff-Arnold-Moser theory is invoked to prove that the quasiperiodic solutions of the β = 0 case can be analytically continued to nonzero but sufficiently small β α .