Symbolic Software for the Study of Integrability of Nonlinear Partial Differential and Differential-Difference Equations

本项目的目标是通过计算非线性偏微分方程（PDES）和微分差分方程（DDEs）的守恒定律、高阶对称性和递推算子，进一步发展和实现强大的符号算法，用于研究可积性。基于尺度不变性的概念，实现的算法将允许人们计算显式守恒密度、哈密顿量、广义对称性和协对称性。在算子方面，该算法可用于计算递归算子及其共轭算子、辛算子和余辛（哈密顿）算子。设想的产品是Mathematica语法中的一个全面的符号包，它将补充和构建以前的软件，以计算pde和dde的不变量和对称性。此外，该项目还进一步开发了符号软件来计算偏微分方程和偏微分方程的精确解。求解方法基于多线性均匀化技术。一旦实现，该软件将允许用户计算封闭形式的解，如孤子解、同斜轨道和异斜轨道，用于大型非线性PDES。将研究将该方法推广到DDEs。该软件将以Mathematica语法编写。这些包将完全是象征性的，并产生分析输出。新的数学技术的发展，除了现有方法和算法的改进和推广，是拟议工作的重要组成部分。并与现有算法进行比较。其目的是为研究孤子理论、动力系统和数学物理中可积性的各个方面的研究人员提供高质量的符号软件。具体来说，我们的求解器和可积性测试仪可用于分析网络、弹性介质、化学动力学、材料科学、生物科学、流体动力学、等离子体物理学和非线性光学中出现的非线性偏微分方程和偏微分方程。

The objective of this project is to further develop and implement powerful symbolic algorithmsfor the investigation of integrability, through the calculation of conservation laws,higher-order symmetries, and recursion operators of nonlinear partial differential equations (PDES) and differential-difference equations (DDEs).Based on the concept of scaling invariance, the implemented algorithms will allow one to compute explicitly conserved densities, Hamiltonians, generalized symmetries, and cosymmetries. On the operator side, the algorithms can be used to compute the recursion operator and its conjugate, and the symplectic and cosymplectic (Hamiltonian) operators. The envisioned product is a comprehensive symbolic package in Mathematica syntax that will complement and build on previous software to compute invariants and symmetries for PDEs and DDEs.In addition, the project further develops symbolic software to compute exact solutions of PDEs and DDEs. The solution methods are based on multi-linear homogenization techniques. Once implemented, the software will allow the user to compute closed form solutions, such as soliton solutions and homoclinic and heteroclinic orbits, for large classes of nonlinear PDES. The extension of the method to DDEs will be investigated.The software will be written in Mathematica syntax. The packages will be entirely symbolic and produce analytical output. The development of novel mathematical techniques, in addition to the refinement and generalization of existing methods and algorithms, is an essential part of the proposed work. A comparison with existing algorithms will also be carried out.The aim is to provide high-quality symbolic software to researchers working on various aspects of integrability in soliton theory, dynamical systems, and mathematical physics. Specifically, our solvers and integrability testers are useful for the analysis of nonlinear PDEs and DDEs that arise in networks, elastic media, chemical kinetics, material sciences, biosciences, fluid dynamics, plasma physics, and nonlinear optics.