| **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

Abstract

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)*