Spectrum transformation and conservation laws of lattice potential KdV equation

doi:10.1007/s11464-016-0542-y

Front. Math. China

2017, Vol. 12

Issue (2) : 403-416 https://doi.org/10.1007/s11464-016-0542-y

RESEARCH ARTICLE

Spectrum transformation and conservation laws of lattice potential KdV equation

Senyue LOU¹,Ying SHI²,Da-jun ZHANG³(

)

¹. Faculty of Science, Ningbo University, Ningbo 315211, China
². School of Science, Zhejiang University of Science and Technology, Hangzhou 310023, China
³. Department of Mathematics, Shanghai University, Shanghai 200444, China

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Abstract

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.

Keywords Conserved quantities analytic property conformal map inverse scattering transform lattice potential KdV equation

Corresponding Author(s): Da-jun ZHANG

Issue Date: 27 December 2016

Cite this article:

Senyue LOU,Ying SHI,Da-jun ZHANG. Spectrum transformation and conservation laws of lattice potential KdV equation[J]. Front. Math. China, 2017, 12(2): 403-416.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-016-0542-y
https://academic.hep.com.cn/fmc/EN/Y2017/V12/I2/403

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