Proceedings of the Conference on Nonlinear Coherent Structures in Physics and Biology,
Heriot-Watt University, Edinburgh, July 95

Variational Approach to Soliton Dynamics in The Discrete Nonlinear Schrödinger Equation

B. Malomed ¹, Michael Weinstein

¹Department of Applied Mathematics,
School of Mathematical Sciences,
Tel Aviv University, Ramat Aviv,
Tel Aviv 69978, Israel


Using a variational technique, based on an effective Lagrangian, we analyze static and dynamical properties of solitons in the 1D discrete nonlinear Schr\"{o}dinger equation with an arbitrary power nonlinearity. We demonstrate that, at low values of the nonlinearity power, there is no bistability and no threshold for formation of the soliton. Above a critical value of the power, which lies between the cubic and quintic nonlinearities, but at the powers which are smaller than that corresponding to the quintic nonlinearity, there is still no threshold. However, in a certain interval of the energies, the soliton becomes multistable; there are two stable and one unstable solutions at the same energy. The quintic nonlinearity plays another critical role: in this case, and at stronger nonlinearities, there exists a static energy threshold for formation of the soliton. For all the nonlinearities stronger than quintic, one always has two solitons above the threshold, one stable and one unstable, the stable solution corresponding to a narrower soliton. We also find dynamical energy threshold for formation of a soliton from an arbitrarily narrow initial configuration.

Conference paper.


Last modified Mon Apr 8 16:24:58 GB-Eire 1996 (DBD)