Solitons in Nonlocal Media

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Introduction
1.1 Solitons in Nonlinear Physics
Nonlinearity has an important role in many disciplines such as physics, economics,
chemistry, biology and so on. In fact, most natural phenomena are intrinsically non-
linear, being linear only when small excitations are considered. Up to the twentieth
century, scientists focused on linear phenomena, firstly because of the large availability
of analytical solutions and secondly, but not less important, because of the possibility to
use superimposition principle, which provides the complete knowledge of a system after
studying its response to limited sets of excitations. This principle is largely adopted in
engineering and physics, for example in harmonic analysis. Einstein’s general relativity,
one of the most successful physical theories in 1900’s, is based on nonlinear equations.
With the advent of modern computers in the 50’s, the available computation power
allowed to study nonlinear problems numerically: among pioneering work I remind the
Fermi, Pasta and Ulam paper concerning energy distribution in a nonlinear vibrating
string (1; 2) and the Lorenz article about chaos in meteorology (3).
One of the most striking features of nonlinear systems is the formation of waves with
an invariant profile along their propagation due to the interplay between linear and
nonlinear effects, called solitons. Strictly speaking, solitons are solutions of integrable
models 1 , which can be solved by the inverse scattering technique (4). In non integrable
models, shape-preserving solutions are called solitary waves but, as usual in the spe-
1 In this context integrability means that the differential equations composing the model encompass
an infinite set of conserved quantities.
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