A Sine-Lattice (Sine-Form Discrete Sine-Gordon) Equation —One-and Two-Kink Solutions and Physical Models—

A sine-lattice (SL) (sine-form discrete sine-Gordon) equation defined as \(\sin (u_{n+1}-u_{n})-\sin (u_{n}-u_{n-1})-\ddot{u}_{n}{=}g \sin u_{n}\) is studied. By introducing a dependent variable transformation and the Hirota D–operators, it can be recast into a bilinear operator form similar to that of the sine-Gordon (SG) equation. This shows that the SL equation is much closer in its soliton properties to the SG equation than the discrete SG equation with the difference factor u n +1 + u n -1 -2 u n . It is shown that the SL equation yields, though not mathematically exact, but well-defined, one-kink and specific two-kink solutions. A numerical calculation which lends support to this is also presented. Several physical models are discussed where the SL equation appears as a more natural model field equation than the discrete SG equation.

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