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Necessary integrability conditions for evolutionary lattice equations

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Abstract

We study the structure of solutions of the Lax equation Dt(G) = [F,G] for formal series in powers of the shift operator. We show that if an equation with a given series F of degree m admits a solution G of degree k, then it also admits a solution H of degree m such that Hk = Gm. We use this property to derive necessary integrability conditions for scalar evolutionary lattices.

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Authors and Affiliations

  1. Landau Institute for Theoretical Physics, RAS, Chernogolovka, Russia

    V. E. Adler

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  1. V. E. Adler
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Correspondence to V. E. Adler.

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Prepared from an English manuscript submitted by the author; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 181, No. 2, pp. 276–295, November, 2014.

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Adler, V.E. Necessary integrability conditions for evolutionary lattice equations. Theor Math Phys 181, 1367–1382 (2014). https://doi.org/10.1007/s11232-014-0218-2

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