This paper surveys the classification of integrable evolution equations whose field variables take values in an associative algebra, which includes matrix, Clifford, and group algebra valued systems. A variety of new examples of integrable systems possessing higher order symmetries are presented. Symmetry reductions lead to an associative algebra-valued version of the Painleve transcendent equations. The basic theory of Hamiltonian structures for associative algebra-valued systems is developed and the biHamiltonian structures for several examples are found.

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Integrable Evolution Equations on Associative Algebras

Generalization of the modified KdV equation to a multi-component system, that is expressed by \(\frac{\partial u_i}{\partial t} + 6 \bigl( \sum_{j,k=0}^{M-1} C_{jk} u_j u_k \bigr) \frac{\partial u_i}{\partial x} + \frac{\partial^3 u_{i}}{\partial x^3} =0\), i =0, 1, …, M -1, is studied. We apply a new extended version of the inverse scattering method to this system. It is shown that this system has an infinite number of conservation laws and multi-soliton solutions. Further, the initial value problem of the model is solved.

The Coupled Modified Korteweg-de Vries Equations

Algebro-geometric solutions of the coupled modified Korteweg–de Vries hierarchy

A vector potential KdV equation and vector Ito equation: soliton solutions, bilinear Bäcklund transformations and Lax pairs

We propose integrable discretizations of derivative nonlinear Schrodinger (DNLS) equations such as the Kaup–Newell equation, the Chen–Lee–Liu equation and the Gerdjikov–Ivanov equation by constructing Lax pairs. The discrete DNLS systems admit the reduction of complex conjugation between two dependent variables and possess bi-Hamiltonian structure. Through transformations of variables and reductions, we obtain novel integrable discretizations of the nonlinear Schrodinger (NLS), modified KdV (mKdV), mixed NLS, matrix NLS, matrix KdV, matrix mKdV, coupled NLS, coupled Hirota, coupled Sasa–Satsuma and Burgers equations. We also discuss integrable discretizations of the sine-Gordon equation, the massive Thirring model and their generalizations.

http://iopscience.iop.org/article/10.1088/0305-4470/35/36/310/pdf

Integrable discretizations of derivative nonlinear Schrödinger equations

The authors extend previous results on the linear spectral problem introduced by Fordy and Kulish (1983). The odd-order isospectral flows admit both a KdV and MKdV type reduction. The non-linear terms are related to the curvature tensor of the corresponding Hermitian symmetric space. Their KdV equations are themselves reductions of known matrix KdV equations. They discuss the conserved densities and Hamiltonian structure associated with these equations.

C Athorne

Papers

Generalised KdV and MKdV equations associated with symmetric spaces