Summary
We extend the generalized inverse method recently introduced by Zakharovet al. in order to solve differential-difference and difference-difference equations and give the explicit form of the nondegenerateN-soliton solution. As an example we consider a discrete version of the sine-Gordon equation.
Riassunto
Il « metodo inverso generalizzato » recentemente introdotto da Zakharovet al. è esteso ed applicato per risolvere equazioni non lineari alle differenze finite. Per tali equazioni viene data la forma esplicita della soluzione aN solitoni non degenere. Come esempio si considera un analogo discreto dell’equazione di sine-Gordon.
Реэюме
Мы расщиряем обобшенный универсальный метод, недавно предложенный Захаровым и др., для рещения дифференциально-раэ ностных и раэностно-раэностных уравнений. Для зтих уравнений приводится явная форма рещения сN солитонами. Как пример, рассматривается дискретный аналог уравнения синус-Гордона.
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References
V. E. Zakharov andA. V. Mikhailov:Sov. Phys. JETP,47, 1017 (1979);V. E. Zakharov andS. V. Manakov:Soviet Science Review A, edited byV. Khalatnikov (New York, N. Y., 1979), p. 133.
V. E. Zakharov:The inverse scattering method, to be published.
N. I. Muskelishvili:Singular Integral Equations (Groningen, 1953).
Equation (4.3) reduces to the one derived byS. J. Orfanidis:Phys. Rev. D,18, 3828 (1978), by the position θ(n, t) = [u(n + 1,t) +u(n, t)]/2, θ(n, t) =u t (n, t)/2.
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The research reported in this paper has been supported in part by the C.N.R. grant No. 78.00919.02.
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Levi, D., Ragnisco, O. & Bruschi, M. Extension of the Zakharov-Shabat generalized inverse method to solve differential-difference and difference-difference equations. Nuov Cim A 58, 56–66 (1980). https://doi.org/10.1007/BF02730220
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DOI: https://doi.org/10.1007/BF02730220