What is more, can we obtain similar results to another integrable or nonintegrable system
with homoclinic or heteroclinic breather wave?
Namely, the Darboux-Halphen system is an algebraically nonintegrable system
[33,48] whose general solution can be expressed in terms of elliptic integrals .
Nevertheless, we need to consider the manifestations of nonintegrable systems
. That is the cause of apparition of chaos because an integrable system motion is either periodic or quasiperiodic.
Nonlinear waves in integrable and nonintegrable systems
Contributors cover the life and work of Prigogine, asymmetry in nonequilibrium statistical mechanics in time, quantum and classical dynamics of nonintegrable systems
, statistical mechanics of a gravitational plasma, inverse problems for reaction-diffusion systems with applications to geographical population genetics, Carnot efficiency, genome-wide sequence analysis in the modeling of the replication of mammals as it applies to DNA in chromatin, and biological rhythms as temporal dissipative structures.
Moreover, since the rules governing the behavior of each system are unique to its contingent mutually transforming interactions, there is typically no privileged perspective from which to explain nonintegrable systems
The abundant solutions solved by these two methods suggest that the rich structures of nonlinear systems do not only exist in the integrable systems but also in the nonintegrable systems
. Furthermore, there are some types of localized solutions decaying in all directions, for instance, the dromions and ring solitons have not been found by these two methods; those will be left for us to do more research.