A Sinh-Lattice Equation–Nearly Integrable Solitons–

A sinh-lattice (SHL) equation \((uₙ₊₁-uₙ)-(uₙ-uₙ₋₁)-üₙ=0\) or (d 2 /d t 2 )ln [(1+ v n 2 ) 1/2 + v n ]= v n +1 + v n -1 -2 v n with v n =sinh ( u n - u n -1 ), which is a version of a kink-generating sine-lattice (SL) equation \(sin(uₙ₊₁-uₙ)-sin(uₙ-uₙ₋₁)-üₙ=0\), is studied. It is shown both analytically and numerically that the SHL equation yields, though not mathematically exact, but well-defined, one-and two-soliton solutions for soliton wave number k up to k ≃1.0. In such regime the equation obeyed by v n share the soliton-generating properties with the Toda lattice equation (d 2 /d t 2 )ln (1+ v n )= v n +1 + v n -1 -2 v n . A brief discussion is given on connection of the SHL equation with the Toda lattice equation, the SL equation and the modified KdV equation.