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In this paper, the existence of the homoclinic orbit is proved under certain parametric conditions, by studying qualitative properties of ODE on GAIM of the sine-Gordon equation.
By the Lie symmetry group, the reduction for divergence-free vector-fields (DFVs) is studied, and the following results are found. A n-dimensional DFV can be locally reduced to a (n - 1)-dimensional DFV if it admits a one-parameter symmetry group that is spatial and divergenceless. More generally, a n-dimensional DFV admitting a r-parameter, spatial, divergenceless Abelian (commutable) symmetry...
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