Exact Cylindrical Soliton Solutions of the Sine-Gordon Equation, The Sinh-Gordon Equation and The Periodic Toda Equation(Hirota's Method in Soliton Theory)

Key Takeaway: Cylindrical solitons can be accurately predicted using the series expansions of Bessel functions for the sine-Gordon equation, sinh-Gordon equation, and periodic Toda equation.

Abstract

We consider the sine-Gordon equation, the sinh-Gordon equation and the periodic Toda equation which are written respectively as Δ u +sin u =0, Δ u +sinh u =0 and Δ u n -exp (- u n + u n -1 )+exp (- u n +1 + u n )=0, ( u n = u n + N ' ), where Δ ≡∂ 2 /∂ x 2 +∂ 2 /∂ y 2 . The exact analytic solutions of the above equations corresponding to the cylindrical solitons have been obtained. The solutions are expressed by the series expansions of the Bessel functions.