Solitons and breathers from the third integral of motion in galaxies

Abstract

We show how it is possible to derive partial differential equations from the third integral of motion of galactic orbits. An example is given in the case of orbits near the inner Lindblad resonance. In this case, a Sine–Gordon equation is derived. Applying the third integral on a string of stars having as initial conditions the successive consequents of one orbit on a Poincare surface of section, a discrete set of Sine–Gordon equations is derived corresponding to a Frenkel–Kontorova Hamiltonian system. Analytical soliton solutions are derived. In particular, we show that breathers can be formed in galaxies by collections of orbits. A good agreement between analytical and numerical solutions is found. The new formulation opens the possibility that, instead of single orbits, collections of stars forming solitons and breathers, which are more dispersion resistant, can be superimposed to construct non-linear density waves in galaxies. Such collections are more natural building blocks of density waves, as they take into account a more realistic distribution and correlation of phases.