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

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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.