Multidimensional integrable systems, differential-difference equations and the symmetry approach

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Integrable partial differential equations are of great interest in modern mathematics, where they have important connections with algebra and differential geometry, and in physics, where they describe many important physical models. Many concepts of modern mathematical physics such as solitons, instantons and quantum groups have their origin in theory of integrable systems.One of the most important and challenging problems in the theory of integrable systems is how to recognize when a given equation is integrable. This project is devoted to multidimensional and differential-difference polynomial integrable systems. The combination of the Perturbative Symmetry Approach and number theoretical methods is proposed for studying integrability of multidimensional and differential-difference equations as well as for obtaining complete classification results of integrable systems of this type.The aims of the proposed research are the following:I. Obtain a global (at arbitrary order) classification of integrable scalar and coupled polynomial homogeneous equations in 2+1 dimensions. II. Extend the perturbative symmetry approach to higher dimensional partial differential equations.III. Extend the symbolic method to differential-difference equations.IV. Obtain a global classification of integrable scalar and coupled differential-difference equations.AMS 2000 subject classification: 37K1 0, 37K35