June 9, 1999 9:11 WSPC/147-MPLB 0135 Modern Physics Letters B, Vol. 13, No. 8 (1999) 225–232 c  World Scientific Publishing Company SELF-TRAPPING ON A GENERALIZED NONLINEAR TETRAHEDRON M. I. MOLINA * Facultad de Ciencias, Departamento de F´ ısica, Universidad de Chile, Casilla 653, Santiago, Chile. Received 31 March 1999 We analyze the dynamical self-trapping of an excitation propagating on a generalized n-sites tetrahedron, characterized by having every site at equal distance from each other. The evolution equation is given by the Discrete Nonlinear Schr¨ odinger (DNLS) equa- tion. For completely localized initial conditions, we find an exact solution for the critical nonlinearity strength (χ/V )c as a function of the number of sites n of the generalized tetrahedron. This critical nonlinearity, that marks the onset of the self-trapping tran- sition, is always negative for n ≥ 3 and its magnitude increases monotonically with n, always remaining inside the sector delimited by (|χ|/V )= n and (|χ|/V )=2n. 1. Introduction The Discrete Nonlinear Schr¨ odinger (DNLS) equation i dC n dt = V  n·n ′ C m − χ |C n | 2 C n , (1) is a paradigmatic equation describing the propagation of an excitation on a discrete nonlinear medium. It can describe a set of coupled, anharmonic oscillators, 1 energy transport in molecular crystals 2,3 or power switching in nonlinear optical devices. 4 In a solid state context, (1) describes the propagation of an electron or “excitation” propagating in a discrete medium, with strong electron–phonon coupling. 5 In (1), C n (t) is the probability amplitude for the excitation to be at site n at time t, V is the hopping between nearest neighbor sites and χ is the nonlinearity parameter, proportional to the square of the electron–phonon coupling. The prime in the sum in (1) restricts the summation to nearest-neighbors only. Exact solutions of (1) are known only for the special cases of a dimer (two sites) and a trimer (three sites). For the case of a dimer, Eilbeck, Lomdahl and Scott 1 found a bifurcation in the stationary states at χ/V = 2. Later, Kenkre and Campbell 6 studied the time-dependent problem and showed that the probability difference p = |C 1 | 2 −|C 2 | 2 obeyed the same equation as a classical particle in a * E-mail: mmolina@abello.dic.uchile.cl 225