Spectrum transformation and conservation laws of lattice potential KdV equation

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• 作者：Senyue Lou ; Ying Shi ; Da-jun Zhang
• 关键词：Conserved quantities ; analytic property ; conformal map ; inverse scattering transform ; lattice potential KdV equation
• 刊名：Frontiers of Mathematics in China
• 出版年：2017
• 出版时间：April 2017
• 年：2017
• 卷：12
• 期：2
• 页码：403-416
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics, general;
• 出版者：Higher Education Press
• ISSN：1673-3576
• 卷排序：12

文摘

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.