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Infinitely many conservation laws for two integrable lattice hierarchies associated with a new discrete Schrödinger spectral problem. (English) Zbl 1020.37045
Summary: In this letter, by means of considering the matrix form of a new Schrödinger discrete spectral operator equation, and constructing opportune time evolution equations, and using discrete zero curvature representation, two discrete integrable lattice hierarchies proposed by M. Boiti, M. Buschi, F. Pempinelli and B. Pinari [J. Phys. A 36, 139-149 (2003)] are re-derived. From the matrix Lax representations, we demonstrate the existence of infinitely many conservation laws for the two lattice hierarchies and give the corresponding conserved densities and the associated fluxes by means of formulae. Thus their integrability is further confirmed. Specially we obtain the infinitely many conservation laws for a new discrete version of the KdV equation. A connection between the conservation laws of the discrete KdV equation and the ones of the KdV equation is discussed by two examples.

##### MSC:
 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 35P05 General topics in linear spectral theory for PDEs 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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##### References:
 [1] Boiti, M.; Bruschi, M.; Pempinelli, F.; Prinari, B., J. phys. A: math. gen., 36, 139, (2003) · Zbl 1067.39032 [2] Shabat, A., (), 331 [3] Boiti, M.; Pempinelli, F.; Prinari, B.; Spire, A., Inverse problems, 17, 515, (2001) [4] Bruschi, M.; Ragnisco, O., J. phys. A: math. gen., 14, 1075, (1981) [5] Bruschi, M.; Ragnisco, O., Lett. nuovo cimento, 29, 321, (1980) [6] Tsuchida, T.; Ujino, H.; Wadati, M., J. math. phys., 39, 4785, (1998) [7] Tsuchida, T.; Ujino, H.; Wadati, M., J. phys. A, 32, 2239, (1999) [8] Zhu, Z.N.; Wu, X.; Xue, W.; Zhu, Z.M., Phys. lett. A, 296, 280, (2002) [9] Zhu, Z.N.; Xue, W.; Wu, X.; Zhu, Z.M., J. phys. A: math. gen., 35, 5079, (2002) [10] Zhu, Z.N.; Zhu, Z.M.; Wu, X.; Xue, W., J. phys. soc. jpn., 71, 1864, (2002) [11] Blaszak, M.; Marciniak, K., J. math. phys., 35, 4661, (1994) [12] Levi, D.; Grundland, A.M., J. phys. A: math. gen., 35, L67, (2002) · Zbl 1004.37050
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