Frontiers of Mathematics in China >
Spectrum transformation and conservation laws of lattice potential KdV equation
Received date: 27 Dec 2015
Accepted date: 22 Apr 2016
Published date: 20 Feb 2017
Copyright
Many multi-dimensional consistent discrete systems have soliton solutions with nonzero backgrounds, which brings difficulty in the investigation of integrable characteristics. In this paper, we derive infinitely many conserved quantities for the lattice potential Korteweg-de Vries equation whose solutions have nonzero backgrounds. The derivation is based on the fact that the scattering data a(z) is independent of discrete space and time and the analytic property of Jost solutions of the discrete Schrödinger spectral problem. The obtained conserved densities are asymptotic to zero when |n| (or |m|) tends to infinity. To obtain these results, we reconstruct a discrete Riccati equation by using a conformal map which transforms the upper complex plane to the inside of unit circle. Series solution to the Riccati equation is constructed based on the analytic and asymptotic properties of Jost solutions.
Senyue LOU , Ying SHI , Da-jun ZHANG . Spectrum transformation and conservation laws of lattice potential KdV equation[J]. Frontiers of Mathematics in China, 2017 , 12(2) : 403 -416 . DOI: 10.1007/s11464-016-0542-y
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