Davey-Stewartson Hierarchy Research Articles

On the dispersionless Davey–Stewartson hierarchy: Zakharov–Shabat equations, twistor structure, and Lax–Sato formalism

AbstractIn this paper, we continue the study of the Davey–Stewartson system, which is one of the most important (2 + 1)‐dimensional integrable models. As we showed in the previous paper, the dDS (dispersionless Davey–Stewartson) system arises from Hamiltonian vector fields Lax pair. A new hierarchy of compatible partial differential equations (PDEs) defining infinitely many symmetries, which is associated with the dDS system, is defined in this paper. We show that this hierarchy arises from the commutation condition of a particular series of one‐parameter Hamiltonian vector fields.

The semi-classical limit of Davey–Stewartson hierarchy

The dispersionless Davey–Stewartson hierarchy arises from the commutation condition of a particular series of one-parameter Hamiltonian vector fields. In this paper, we show the connection between the Davey–Stewartson hierarchy and the dispersionless Davey–Stewartson hierarchy.

Negative Times of the Davey-Stewartson Integrable Hierarchy

We use example of the Davey-Stewartson hierarchy to show that in addition to the standard equations given by Lax operator and evolutions of times with positive numbers, one can consider time evolutions with negative numbers and the same Lax operator. We derive corresponding Lax pairs and integrable equations.

On (2+1)-dimensional expanding integrable model of the Davey-Stewartson hierarchy

This paper mainly investigates the reductions of an integrable coupling of the Levi hierarchy and an expanding model of the (2+1)-dimensional Davey-Stewartson hierarchy. It is shown that the integrable coupling system of the Levi hierarchy possesses a quasi-Hamiltonian structure under certain constraints. Based on the Lie algebras construct, The type abstraction hierarchy scheme is used to gener?ate the (2+1)-dimensional expanding integrable model of the Davey-Stewartson hierarchy.

A Corresponding Lie Algebra of a Reductive homogeneous Group and Its Applications* *Supported by the Innovation Team of Jiangsu Province hosted by China University of Mining and Technology (2014), the National Natural Science Foundation of China under Grant No. 11371361, the Fundamental Research Funds for the Central Universities (2013XK03), and the Natural Science Foundation of Shandong Province under Grant No.

With the help of a Lie algebra of a reductive homogeneous space G/K, where G is a Lie group and K is a resulting isotropy group, we introduce a Lax pair for which an expanding (2+1)-dimensional integrable hierarchy is obtained by applying the binormial-residue representation (BRR) method, whose Hamiltonian structure is derived from the trace identity for deducing (2+1)-dimensional integrable hierarchies, which was proposed by Tu, et al. We further consider some reductions of the expanding integrable hierarchy obtained in the paper. The first reduction is just right the (2+1)-dimensional AKNS hierarchy, the second-type reduction reveals an integrable coupling of the (2+1)-dimensional AKNS equation (also called the Davey-Stewartson hierarchy), a kind of (2+1)-dimensional Schrödinger equation, which was once reobtained by Tu, Feng and Zhang. It is interesting that a new (2+1)-dimensional integrable nonlinear coupled equation is generated from the reduction of the part of the (2+1)-dimensional integrable coupling, which is further reduced to the standard (2+1)-dimensional diffusion equation along with a parameter. In addition, the well-known (1+1)-dimensional AKNS hierarchy, the (1+1)-dimensional nonlinear Schrödinger equation are all special cases of the (2+1)-dimensional expanding integrable hierarchy. Finally, we discuss a few discrete difference equations of the diffusion equation whose stabilities are analyzed by making use of the von Neumann condition and the Fourier method. Some numerical solutions of a special stationary initial value problem of the (2+1)-dimensional diffusion equation are obtained and the resulting convergence and estimation formula are investigated.

An (2+1)-dimensional Expanding Model of the Davey-Stewartson Hierarchy As Well As Its Hamiltonian Structure

An (2+1)-dimensional Expanding Model of the Davey-Stewartson Hierarchy As Well As Its Hamiltonian Structure

Surfaces in the four-space and the Davey–Stewartson equations

We show that any equation from the Davey–Stewartson hierarchy induces an infinite family of geometrically different deformations of tori in R 4 preserving the Willmore functional. We expose a derivation of the Weierstrass representation for surfaces in the four-space, which is not unique in difference from the case of surfaces in the three-space. This non-uniqueness implies that the spectral curve of a torus in R 4 is not uniquely defined as a complex curve formed by the Floquet multipliers.

Projective differential geometry of higher reductions of the two-dimensional Dirac equation

We investigate reductions of the two-dimensional Dirac equation imposed by the requirement of the existence of a differential operator D n of order n mapping its eigenfunctions to adjoint eigenfunctions. For first order operators these reductions (and multicomponent analogs thereof) lead to the Lame equations descriptive of orthogonal coordinate systems. Our main observation is that nth order reductions coincide with the projective-geometric ‘Gauss–Codazzi’ equations governing special classes of line congruences in the projective space P 2 n−1 , which is the projectivised kernel of D n . In the second order case this leads to the theory of W-congruences in P 3 which belong to a linear complex, while the third order case corresponds to isotropic congruences in P 5. Higher reductions are compatible with odd order flows of the Davey–Stewartson hierarchy. All these flows preserve the kernel D n , thus defining nontrivial geometric evolutions of line congruences. Multicomponent generalizations are also discussed. The correspondence between geometric picture and the theory of integrable systems is established; the definition of the class of reductions and all geometric objects in terms of the multicomponent KP hierarchy is presented. Generating forms for reductions of arbitrary order are constructed.

Induced Surfaces and Their Integrable Dynamics Ii. Generalized Weierstrass Representations in 4‐d Spaces and Deformations via Ds Hierarchy

Extensions of the generalized Weierstrass representation to generic surfaces in 4‐D Euclidean and pseudo‐Euclidean spaces are given. Geometric characteristics of surfaces are calculated. It is shown that integrable deformations of such induced surfaces are generated by the Davey–Stewartson hierarchy. Geometrically, these deformations are characterized by the invariance of an infinite set of functionals over surface. The Willmore functional (the total squared mean curvature) is the simplest of them. Various particular classes of surfaces and their integrable deformations are considered.

Weierstrass Representations for Surfaces in 4D Spaces and Their Integrable Deformations via DS Hierarchy

Generalized Weierstrass representations for generic surfaces conformally immersed into four-dimensional Euclidean and pseudo-Euclidean spaces of different signatures are presented. Integrable deformations of surfaces in these spaces generated by the Davey–Stewartson hierarchy of integrable equations are proposed. The Willmore functional of a surface is invariant under such deformations.

The Davey-Stewartson System and the Bäcklund Transformations

We consider the (coupled) Davey-Stewartson (DS) system and its B\"{a}cklund transformations (BT). Relations among the DS system, the double Kadomtsev-Petviashvili (KP) system and the Ablowitz-Ladik hierarchy (ALH) are established. The DS hierarchy and the double KP system are equivalent. The ALH is the BT of the DS system in a certain reduction. {From} the BT of coupled DS system we can obtain new coupled derivative nonlinear Schr\"{o}dinger equations.

Topological field theories and integrable models

We show that the classical non-Abelian pure Chern–Simons action is related to nonrelativistic models in (2+1) dimensions, via reductions of the gauge connection in Hermitian symmetric spaces. In such models the matter fields are coupled to gauge Chern–Simons fields, associated with the isotropy subgroup of the considered symmetric space. Thus a relation between the Chern–Simons theory and the Davey–Stewartson hierarchy is established in a natural way. The Bäcklund transformations are interpreted in terms of Chern–Simons constraints. Moreover, certain nonintegrable Heisenberg models can be embedded into the pure Chern–Simons theory. The main classical and quantum properties of these systems are discussed.

A useful auxiliary linear system for a Davey-Stewartson hierarchy

A system of linear auxiliary equations which leads to a Davey-Stewartson hierarchy is rewritten. The dressing method is used to derive higher order Davey-Stewartson equations. The system of linear auxiliary equations is shown to be more convenient than the conventional one when one considers the initial-boundary value problem.

Nets in [formula omitted], their integrable evolutions and the DS hierarchy

Differential equations which describe classical nets in the three-dimensional Euclidean space R 3 are amenable to the inverse spectral transform method. An infinite hierarchy of commuting integrable evolutions (deformations) of conjugate nets is presented. It is shown that this hierarchy coincides with the hierarchy of equations for the wavefunction of the Davey-Stewartson (DS) system. The DS hierarchy itself represents the hierarchy of the iso-conjugate deformations of conjugate nets formulated in terms of rotation coefficients.