Symmetry and Complete Integrability of Nonlinear Difference Equations

非线性差分方程的对称性和完全可积性

• 批准号: 14340053
• 负责人: SATSUMA Junkichi
• 金额: $ 7.3万
• 依托单位: Aoyama Gakuin University (2005)The University of Tokyo (2002-2004)
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2005
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14340053/
• 关键词: Soliton; Complete Integrability; Difference Equations; Painleve Equations; Symmetry; Ultradiscrete System; 超離散; セルオートマトン; 可積分系; 離散力学系

The purpose of this research is to clarify the mathematical structure of singularity confinement condition for nonlinear difference equations, to extend the condition to ultradiscrete systems, and to study algebraic and geometrical structure of discrete integrable systems. Some of our results are as follows.1. We have studied the relation between continuous and discrete Painleve equations and given a determinant-type solution for them. We also presented a method to describe q-Painleve equations in a unified way. Furthermore, we studied their transformation structure in detail.2. We have presented a few of new integrable nonlinear equations and studied solutions of a coupled KP equation and the higher order nonlinear Schrodinger equations.3.We have proposed an ultradiscrete Sine-Gordon equation, which is considered to be a cellular automaton with soliton-type solutions. We have also given an ultradiscrete modified KdV equation and have shown that the equation directly relates with a box and ball system. Moreover, we proposed a method to ultradiscretize variables without positivity.4.We have studied periodic ultradiscrete systems and determined fundamental period of their solutions exactly and have given an algolithm to constitute the conserved quantities explicitly. We also clarify the relation between conserved quantities of ultradiscrete and continuous systems.

这项研究的目的是阐明非线性差异方程的奇异性约束条件的数学结构，将条件扩展到超级转移系统，并研究离散整合系统的代数和几何结构。我们的一些结果如下1。我们已经研究了连续和离散的painleve方程之间的关系，并为它们提供了决定性型解决方案。我们还提出了一种以统一的方式描述Q-Painleve方程的方法。此外，我们详细研究了它们的转换结构2。我们已经提出了一些新的可集成的非线性方程和耦合KP方程和更高级的非线性schrodinger方程的解决方案。3。我们提出了一个超级差正弦的正弦 - 戈登方程解决方案。我们还给出了超凝结的修改KDV方程，并表明该方程与盒子和球系统直接相关。此外，我们提出了一种超级差异的方法。4。我们已经研究了周期性的超级转移系统，并确定了其溶液的基本周期，并给出了算法，以明确构成保守数量。我们还阐明了保守量的超级物质和连续系统之间的关系。

项目成果

Ultradiscrete Miura transformation

超离散 Miura 变换

• 发表时间: 2007
• 期刊: Phys. Lett. A Vol. 362
• 作者: K.Kobayashi;Y.Hatta;K.Yasu;S.Minamihata;N.Hodoshima;T.Arai;M.Shindo;S. Kubo
• 通讯作者: S. Kubo

Extending the SIR epidemic model

• DOI: 10.1016/j.physa.2003.12.035
• 发表时间: 2004-05-15
• 期刊: PHYSICA A-STATISTICAL MECHANICS AND ITS APPLICATIONS
• 影响因子: 3.3
• 作者: Satsuma, J;Willox, R;Carstea, AS
• 通讯作者: Carstea, AS

Exact solutions for discrete and ultradiscrete modified KdV equations and their relation to box-ball systems

离散和超离散修正 KdV 方程的精确解及其与盒球系统的关系

• 发表时间: 2006
• 期刊: Journal of Physics A : Mathematical and General 39
• 作者: Yokokawa;R.;Takeuchi;S.;Kon;T.;Nishiura M.;Sutoh;K.;Fujita H.;Mikio Murata
• 通讯作者: Mikio Murata

Sasa-Satsuma higher-order nonlinear Schrodinger equation and its bilinearization and multisoliton solutions

• DOI: 10.1103/physreve.68.016614
• 发表时间: 2003-07-01
• 期刊: PHYSICAL REVIEW E
• 影响因子: 2.4
• 作者: Gilson, C;Hietarinta, J;Ohta, Y
• 通讯作者: Ohta, Y

An ultradiscretization of the Sine-Gordon equation

正弦-戈登方程的超离散化

• 发表时间: 2004
• 期刊: Phys.Lett.A 331
• 作者: 加納悟;竹内明香;遠藤 乾;SATSUMA Junkichi(共著)
• 通讯作者: SATSUMA Junkichi(共著)