# Exact Solutions to the Sine-Gordon Equation - Nonlinear Sciences > Exactly Solvable and Integrable Systems

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Abstract: A systematic method is presented to provide various equivalent solutionformulas for exact solutions to the sine-Gordon equation. Such solutions areanalytic in the spatial variable $x$ and the temporal variable $t,$ and theyare exponentially asymptotic to integer multiples of $2\pi$ as $x\to\pm\infty.$The solution formulas are expressed explicitly in terms of a real triplet ofconstant matrices. The method presented is generalizable to other integrableevolution equations where the inverse scattering transform is applied via theuse of a Marchenko integral equation. By expressing the kernel of thatMarchenko equation as a matrix exponential in terms of the matrix triplet andby exploiting the separability of that kernel, an exact solution formula to theMarchenko equation is derived, yielding various equivalent exact solutionformulas for the sine-Gordon equation.

Autor: Tuncay Aktosun, Francesco Demontis, Cornelis van der Mee

Fuente: https://arxiv.org/