The perturbed sine-Gordon breather equation integrated by Riemann's method

Abstract

A general first-order perturbation solution in the neighborhood of the unperturbed breather solution is given for the nonlinear perturbed sine-Gordon equation. The inhomogeneous linear-hyperbolic differential equation is solved by Riemann's method. No methods of inverse scattering theory are used for the determination of the Riemann function. Instead, the Bäcklund transformation and a novel inversion relation are applied. The Riemann function may be expressed in terms of two-variable Lommel functions. It is shown that the thus-formulated Riemann function has the correct symmetry, unlike the discrete part. As an example, the asymptotic solution for a low-amplitude breather under a constant perturbation is given, showing that plane waves are radiated to both sides of the breather.

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Republished from Teoreticheskaya i Matematicheskaya Fizika, Vol. 107, No. 3, pp. 439–449, June, 1996.

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Mann, E. The perturbed sine-Gordon breather equation integrated by Riemann's method. Theor Math Phys 107, 775–783 (1996). https://doi.org/10.1007/BF02070385

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Keywords

  • Differential Equation
  • Plane Wave
  • Asymptotic Solution
  • Riemann Function
  • Perturbation Solution