Related papers: Geometries and Symmetries of Soliton equations and…
This paper shows that the left-invariant geodesic flow on the symplectic group relative to the Frobenius metric is an integrable system that is not contained in the Mishchenko-Fomenko class of rigid body metrics. This system may be…
The models of the non-linear optics in which solitons were appeared are considered. These models are of paramount importance in studies of non-linear wave phenomena. The classical examples of phenomena of this kind are the self-focusing,…
We apply our recent formalism establishing new connections between the geometry of moving space curves and soliton equations, to the nonlinear Schr\"{o}dinger equation (NLS). We show that any given solution of the NLS gets associated with…
We propose a new analyzing method, which is called the tautological flow method, to analyze the integrability of partial difference equations (P$\Delta$Es) based on that of partial differential equations (PDEs). By using this method, we…
Two new approaches to solving first-order quasilinear elliptic systems of PDEs in many dimensions are proposed. The first method is based on an analysis of multimode solutions expressible in terms of Riemann invariants, based on links…
A manifold is multisymplectic, or more specifically n-plectic, if it is equipped with a closed nondegenerate differential form of degree n+1. In our previous work with Baez and Hoffnung, we described how the `higher analogs' of the…
We study the space of periodic solutions of the elliptic $\sinh$-Gordon equation by means of spectral data consisting of a Riemann surface $Y$ and a divisor $D$. We show that the space $M_g^{\mathbf{p}}$ of real periodic finite type…
We study locally harmonic maps between a Riemann surface or Lorentz surface $M$ and a Riemann surface or Lorentz surface $N$. {All four cases are studied in a unified way}. All four cases are written using a unified formalism. Therefore…
For any toric automorphism with only real eigenvalues a Riemannian metric with an integrable geodesic flow on the suspension of this automorphism is constructed. A qualitative analysis of such a flow on a three-solvmanifold constructed by…
We present a detailed study of a parametric Lie algebra encompassing the symmetry algebras of various models, both continuous and discrete. This algebraic structure characterizes the isotropic oscillator (with positive, purely imaginary,…
We derive the soliton matrices corresponding to an arbitrary number of higher-order normal zeros for the matrix Riemann-Hilbert problem of arbitrary matrix dimension, thus giving the complete solution to the problem of higher-order…
This paper is devoted to the problem of classification, up to smooth isomorphisms or up to orbital equivalence, of smooth integrable vector fields on 2-dimensional surfaces, under some nondegeneracy conditions. The main continuous…
We study local normal forms for completely integrable systems on Poisson manifolds in the presence of additional symmetries. The symmetries that we consider are encoded in actions of compact Lie groups. The existence of Weinstein's…
Elliptic soliton solutions, i.e., a hierarchy of functions based on an elliptic seed solution, are constructed using an elliptic Cauchy kernel, for integrable lattice equations of Kadomtsev-Petviashvili (KP) type. This comprises the lattice…
This paper shows that the Ablowitz-Ladik hierarchy of equations (a well-known integrable discretization of the Non-linear Schrodinger system) can be explicitly viewed as a hierarchy of commuting flows which: (a) are Hamiltonian with respect…
In this paper, we investigate the relationship between algebraic soliton metrics and soliton metrics for geometric evolution equations on Lie groups. After discussing the general relationship between algebraic soliton metrics and soliton…
The negative symmetry flows are incorporated into the Riemann-Hilbert problem for the homogeneous $A_m$-hierarchy and its $\hat{gl} (m+1, C)$ extension. A loop group automorphism of order two is used to define a sub-hierarchy of $\hat{gl}…
A connection between differential geometry and soliton equations is discussed
We introduce filtrations in chiral homology complexes of smooth elliptic curves, exploiting the mixed Hodge structure on cohomology groups of configuration spaces. We use these to relate the chiral homology of a smooth elliptic curve with…
On a smooth, closed Riemannian manifold, we study the question of proportionality of components, also called synchronization, of vector-valued solutions to nonlinear elliptic Schr\"odinger systems with constant coefficients. In particular,…