Related papers: Integrable Systems and Harmonic Maps into Lie Grou…
In recent years, significant progress has been made in the study of integrable systems from a gauge theoretic perspective. This development originated with the introduction of $4$d Chern-Simons theory with defects, which provided a…
For some integrable systems, such as the open Toda molecule, the spectral curve of the Lax representation becomes the graph $C = \{(\lambda,z) \mid z = A(\lambda)\}$ of a function $A(\lambda)$. Those integrable systems provide an…
Difference-difference systems are suggested corresponding to the Cartan matrices of any simple or affine Lie algebra. In the cases of the algebras $A_N$, $B_N$, $C_N$, $G_2$, $D_3$, $A_1^{(1)}$, $A_2^{(2)}$, $D^{(2)}_N$ these systems are…
An integrable theory is developed for the perturbation equations engendered from small disturbances of solutions. It includes various integrable properties of the perturbation equations: hereditary recursion operators, master symmetries,…
Toda equations associated with twisted loop groups are considered. Such equations are specified by Z-gradations of the corresponding twisted loop Lie algebras. The classification of Toda equations related to twisted loop Lie algebras with…
We present a notion of symmetry for 1+1-dimensional integrable systems which is consistent with their group theoretic description and reproduces in special cases the known Baecklund transformation for the generalized Korteweg-deVries…
In this paper, the theory of harmonic maps is extended. The soliton or traveling wave solutions of Euler's equations of the extended harmonic maps are studied. In certain cases, the chaotic behaviors of these partial equations can be found…
On the base of Lie algebraic and differential geometry methods, a wide class of multidimensional nonlinear systems is obtained, and the integration scheme for such equations is proposed.
In this paper are studied the harmonic maps between two generalized Lagrange spaces. At the same time, it is proved that the solutions of $C^2$ class of certain ODEs or PDEs are harmonic maps between certain convenient generalized Lagrange…
A new class of integrable maps, obtained as lattice versions of polynomial dynamical systems is introduced. These systems are obtained by means of a discretization procedure that preserves several analytic and algebraic properties of a…
The theory of Poisson-Lie groups and Lie bialgebras plays a major role in the study of one dimensional integrable systems; many families of integrable systems can be recovered from a Lax pair which is constructed from a Lie bialgebra…
Harmonic maps from $\BR^2$ or one-connected domain ${\O}\subset \BR^2$ into $GL(m, \BC)$ and $U(m)$ are treated. The GBDT version of the B\"acklund-Darboux transformation is applied to the case of the harmonic maps. A new general formula on…
We develop the theory of equivariant harmonic self-maps of compact cohomogeneity one manifolds and construct new harmonic self-maps of the compact Lie groups SO(4L+2), L >= 1, with degree -3, of SO(8), SO(14) and SO(26) with degree -5 each,…
In this paper we prove a compactness theorem for a sequence of harmonic maps which are defined on a converging sequence of Riemannian manifolds.
In this paper, by selecting appropriate spectral matrices within the loop algebra of symplectic Lie algebra sp(6), we construct two distinct classes of integrable soliton hierarchies. Then, by employing the Tu scheme and trace identity, we…
We give a review of the systematic construction of hierarchies of soliton flows and integrable elliptic equations associated to a complex semi-simple Lie algebra and finite order automorphisms. For example, the non-linear Schr\"odinger…
In this paper, we discuss the associated family of harmonic maps $\mathcal{F}: M \rightarrow G/K$ from a Riemann surface $M$ into inner symmetric spaces of compact or non-compact type which are either algebraic or totally symmetric. These…
In this paper, the classification in [Lin,Wei,Ye] of solutions to Toda systems of type $A$ with singular sources is generalized to Toda systems of types $C$ and $B$. Like in the $A$ case, the solution space is shown to be parametrized by…
One fruitful motivating principle of much research on the family of integrable systems known as ``Toda lattices'' has been the heuristic assumption that the periodic Toda lattice in an affine Lie algebra is directly analogous to the…
Differential-difference integrable exponential type systems are studied corresponding to the Cartan matrices of semi-simple or affine Lie algebras. For the systems corresponding to the algebras $A_2$, $B_2$, $C_2$, $G_2$ the complete sets…