From embedded solitons to 4d dynamical systems

Abstract

The term ``embedded soliton'' was coined in 1999 to describe a new type of soliton (discovered in 1997) whose internal frequencies lie within the spectrum of the radiation modes of certain nonlinear systems. In 2005 it was discovered that ``embedded lattice solitons'' (ELS) can also exist in discrete systems. The present communication shows that a discrete higher-order NLS equation with exact ELS leads naturally to a four-dimensional dynamical system that can be cast in the form $\phi _{n+4}=F\left( \phi _{n},\ldots,\phi _{n+3}\right) $, where F is a nonlinear function. In all the particular cases studied in this communication, at least two of the four Lyapunov coefficients associated with the system are positive, thus indicating a chaotic behavior.