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This is a short review of the construction of quasi-periodic (algebraic-geometrical) solutions to hierarchies of nonlinear integrable equations. As is well known, the solutions are expressed through Riemann's theta-functions associated with algebraic curves. It is explained how solutions from this class can be treated within the framework of the approach to the integrable hierarchies developed by the Kyoto school. Three representative examples are considered in detail: the Kadomtsev-Petviashvili hierarchy, the 2D Toda lattice hierarchy and the B-version of the Kadomtsev-Petviashvili hierarchy. [ABSTRACT FROM AUTHOR]

We propose the most general approach to construction of the U – V pairs of hierarchies of soliton equations in two dimensions based on the theory of classical non-skew-symmetric r -matrices with spectral parameters and infinite-dimensional Lie algebras. We show that reduction in integrable hierarchies is connected with “special points” of classical r -matrices in which they become singular or degenerated. We prove that “Mikhailov’s reduction” or reduction with the help of automorphism is a partial case of our construction. We consider two types of integrable hierarchies and the corresponding soliton equations: the so-called “positive” and “negative flow” equations. We show that the “negative flow” equations can be written in the most general case without a specification of the concrete form of classical r -matrix. They coincide with a generalization of chiral field equation and its different reductions, with a generalization of abelian and non-abelian Toda field equations and new class of integrable equations which we call “double-shift” equations. For the case of “positive” hierarchies and “positive” flows we explicitly write general U – V pairs of “nominative” equations of hierarchy. We consider examples of new equations of such the types and present new soliton equations coinciding with elliptic deformation of dNS equation and its “negative flow” equation and ultimate generalization of the abelian and non-abelian modified Toda field equations. [ABSTRACT FROM AUTHOR]

We describe a variational framework for non-commuting flows, extending the theories of Lagrangian multiforms and pluri-Lagrangian systems, which have gained prominence in recent years as a variational description of integrable systems in the sense of multidimensional consistency. In the context of non-commuting flows, the manifold of independent variables, often called multi-time, is a Lie group whose bracket structure corresponds to the commutation relations between the vector fields generating the flows. Natural examples are provided by superintegrable systems for the case of Lagrangian 1-form structures, and integrable hierarchies on loop groups in the case of Lagrangian 2-forms. As particular examples we discuss the Kepler problem, the rational Calogero-Moser system, and a generalisation of the Ablowitz-Kaup-Newell-Segur system with non-commuting flows. We view this endeavour as a first step towards a purely variational approach to Lie group actions on manifolds. [ABSTRACT FROM AUTHOR]

Abstract: We study structural aspects of the Ablowitz–Ladik (AL) hierarchy in the light of its realization as a two-component reduction of the two-dimensional Toda hierarchy, and establish new results on its connection to the Gromov–Witten theory of local . We first of all elaborate on the relation to the Toeplitz lattice and obtain a neat description of the Lax formulation of the AL system. We then study the dispersionless limit and rephrase it in terms of a conformal semisimple Frobenius manifold with non-constant unit, whose properties we thoroughly analyze. We build on this connection along two main strands. First of all, we exhibit a manifestly local bi-Hamiltonian structure of the Ablowitz–Ladik system in the zero-dispersion limit. Second, we make precise the relation between this canonical Frobenius structure and the one that underlies the Gromov–Witten theory of the resolved conifold in the equivariantly Calabi–Yau case; a key role is played by Dubrovin’s notion of “almost duality” of Frobenius manifolds. As a consequence, we obtain a derivation of genus zero mirror symmetry for local in terms of a dual logarithmic Landau–Ginzburg model. [Copyright &y& Elsevier]

Abstract: In our recent paper, we proved the polynomiality of a Poisson bracket for a class of infinite-dimensional Hamiltonian systems of partial differential equations (PDEs) associated to semi-simple Frobenius structures. In the conformal (homogeneous) case, these systems are exactly the hierarchies of Dubrovin and Zhang, and the bracket is the first Poisson structure of their hierarchy. Our approach was based on a very involved computation of a deformation formula for the bracket with respect to the Givental–Lee Lie algebra action. In this paper, we discuss the structure of that deformation formula. In particular, we give an alternative derivation using a deformation formula for the weak quasi-Miura transformation that relates our hierarchy of PDEs with its dispersionless limit. [Copyright &y& Elsevier]

Abstract: A general approach is proposed for discretizing nonlinear dynamical systems and field theories on suitable functional spaces, defined over a regular lattice of points, in such a way that both their symmetry and integrability properties are preserved. A class of discrete KdV equations is introduced. Also, new hierarchies of discrete evolution equations of Gelfand–Dickey type are defined. [Copyright &y& Elsevier]

Abstract: The Bullough–Dodd model is an important two-dimensional integrable field theory which finds applications in physics and geometry. We consider a conformally invariant extension of it, and study its integrability properties using a zero curvature condition based on the twisted Kac–Moody algebra . The one- and two-soliton solutions as well as the breathers are constructed explicitly. We also consider integrable extensions of the Bullough–Dodd model by the introduction of spinor (matter) fields. The resulting theories are conformally invariant and present local internal symmetries. All the one-soliton solutions, for two examples of those models, are constructed using a hybrid of the dressing and Hirota methods. One model is of particular interest because it presents a confinement mechanism for a given conserved charge inside the solitons. [Copyright &y& Elsevier]

Abstract: We give a representation–theoretic interpretation of recent discovered coupled soliton equations using vertex operators construction of affinization of not simple but quadratic Lie algebras. In this setup we are able to obtain new integrable hierarchies coupled to each Drinfeld–Sokolov of A, B, C, D hierarchies and to construct their soliton solutions. [Copyright &y& Elsevier]

Inside the algebra L T N (R) of N × N -matrices with coefficients from a commutative algebra R over k = R or ℂ , that possess only a finite number of nonzero diagonals above the central diagonal, we consider two deformations of commutative Lie subalgebras generated by the n th power S n , n ⩾ 1 , of the matrix S of the shift operator and a maximal commutative subalgebra h of gl n (k) , where the evolution equations of the deformed generators are determined by a set of Lax equations, each corresponding to a different decomposition of L T N (R). This yields the h [ S n ] -hierarchy and its strict version. We show that both sets of Lax equations are equivalent to a set of zero curvature equations. Next we introduce two Cauchy problems linked with these sets of zero curvature equations and present sufficient conditions under which they can be solved. Moreover, we show that these conditions hold in the formal power series context. Next we introduce two L T N (R) -models, one for each hierarchy, a set of equations in each module and special vectors satisfying these equations from which the Lax equations of each hierarchy can be derived. We conclude by presenting a functional analytic context in which these special vectors can be constructed. Thus one obtains solutions of both hierarchies. [ABSTRACT FROM AUTHOR]