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Integrable discretizations for a generalized sine-Gordon equation and the reductions to the sine-Gordon equation and the short pulse equation
In this paper, we propose fully discrete analogues of a generalized
sine-Gordon (gsG) equation .
The bilinear equations of the discrete KP hierarchy and the proper definition
of discrete hodograph transformations are the keys to the construction. Then we
derive semi-discrete analogues of the gsG equation from the fully discrete gsG
equation by taking the temporal parameter . Especially, one
full-discrete gsG equation is reduced to a semi-discrete gsG equation in the
case of (Feng {\it et al. Numer. Algorithms} 2023). Furthermore,
-soliton solutions to the semi- and fully discrete analogues of the gsG
equation in the determinant form are constructed. Dynamics of one- and
two-soliton solutions for the discrete gsG equations are discussed with plots.
We also investigate the reductions to the sine-Gordon (sG) equation and the
short pulse (SP) equation. By introducing an important parameter , we
demonstrate that the gsG equation reduces to the sG equation and the SP
equation, and the discrete gsG equation reduces to the discrete sG equation and
the discrete SP equation, respectively, in the appropriate scaling limit. The
limiting forms of the -soliton solutions to the gsG equation also correspond
to those of the sG equation and the SP equation
Integrable discretizations for a generalized sine-Gordon equation and the reductions to the sine-Gordon equation and the short pulse equation
In this paper, we propose fully discrete analogues of a generalized sine-Gordon (gsG) equation utx=(1+Ξ½β2x)sinu. The bilinear equations of the discrete KP hierarchy and the proper definition of discrete hodograph transformations are the keys to the construction. Then we derive semi-discrete analogues of the gsG equation from the fully discrete gsG equation by taking the temporal parameter bβ0. Especially, one full-discrete gsG equation is reduced to a semi-discrete gsG equation in the case of Ξ½=β1 (Feng {\it et al. Numer. Algorithms} 2023). Furthermore, N-soliton solutions to the semi- and fully discrete analogues of the gsG equation in the determinant form are constructed. Dynamics of one- and two-soliton solutions for the discrete gsG equations are discussed with plots. We also investigate the reductions to the sine-Gordon (sG) equation and the short pulse (SP) equation. By introducing an important parameter c, we demonstrate that the gsG equation reduces to the sG equation and the SP equation, and the discrete gsG equation reduces to the discrete sG equation and the discrete SP equation, respectively, in the appropriate scaling limit. The limiting forms of the N-soliton solutions to the gsG equation also correspond to those of the sG equation and the SP equation
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