In this paper, the geometry size effects of the ringshape quasi-soliton solutions in the perturbed Sine-Gordon equations (SGE) in the circular symmetric annular Josephson junction are investigated with numerical method. The perturbed SGE, when the inner radius r0 of the circular symmetric annular junctions is larger then or equal to 7.5λJ, there exists a kind of ring-shape quasi-soliton solutions, which correspond to the quasi-fluxon of excitation in the circular symmetric annular Josephson junction, and the first zero-field step also. The range of the first zero-field step depend strongly on the size of the inner radius r0 and widths (αr0-r0) of the circular symmetric annular junctions, but the geometry size effects of the nature of the ring-shape quasi-solitons are more complicated.
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