Waves Derived from Galactic Orbits. Solitons and Breathers
We show how it is possible to define collections of non-interacting particles moving in the same potential so that solitons or breathers are formed. The displacements of particles in this case obey a partial differential equation (PDE). Thus it is possible to derive PDEs from a given Hamiltonian. We demonstrate the above methodology using, as an example, a cubic potential and we derive a Korteweg-de Vries equation.
We apply this methodology in galactic models and show that near resonances, where “Third Integrals” of motion can be defined, the motion of stars can be described in terms of solutions of a Sine-Gordon PDE. This equation admits kink or anti-kink solitons solutions.
In particular in the case of the Inner Lindblad Resonance (ILR), applying the Third Integral on a string of stars having as initial conditions the successive consequents of one orbit on a Poincaré surface of section a Frenkel-Kontorova Hamiltonian is constructed. The corresponding equations of motion are an infinite set of discrete Sine-Gordon equations. This set of equations admits solutions that represent localized oscillations on a grid that are known as Discrete Breathers. An analytic breather solution is derived and compared with the corresponding numerical solution in the case of a perturbed isochrone model.
The advantage of this methodology is that it takes into account the distribution of phases of stars moving under the same value of the third integral. Because of their nature, soliton solutions resist to dispersion and they can be a natural building block to construct more stable non-linear density waves in galaxies.
KeywordsPeriodic Orbit Invariant Curve Unstable Periodic Orbit Stable Periodic Orbit Discrete Breather
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