KdV Hierarchy Research Articles

Algebro-Geometric Solutions of the Generalized Virasoro Constraints

We will describe algebro-geometric solutions of the KdV hierarchy whose $\tau$-functions in addition satisfy a generalization of the Virasoro constraints (and, in particular, a generalization of the string equation). We show that these solutions are closely related to embeddings of the positive half of the Virasoro algebra into the Lie algebra of differential operators on the circle. Our results are tested against the case of Witten-Kontsevich $\tau$-function. As by-products, we exhibit certain links of our methods with double covers of the projective line equipped with a line bundle and with ${\rm Gl}(n)$-opers on the punctured disk.

The partition function of the extended r-reduced Kadomtsev–Petviashvili hierarchy

We derive a particular solution of the extended r-reduced KP hierarchy, which is specified by a generalized string equation . The work is a generalization to arbitrary of Buryakʼs recent results of a solution to the extended open KdV hierarchy which corresponds to r = 2.

Higher spin Lifshitz theory and integrable systems

In this note we construct asymptotically Lifshitz spacetimes in the Chern-Simons formulation of three dimensional higher spin gravity and relate the resulting theories to integrable systems which are elements of the KdV hierarchy.

Generalized Reflection Coefficients

I consider general reflection coefficients for arbitrary one-dimensional whole line differential or difference operators of order 2. These reflection coefficients are semicontinuous functions of the operator: their absolute value can only go down when limits are taken. This implies a corresponding semicontinuity result for the absolutely continuous spectrum, which applies to a very large class of maps. In particular, we can consider shift maps (thus recovering and generalizing a result of Last–Simon) and flows of the Toda and KdV hierarchies (this is new). Finally, I evaluate an attempt at finding a similar general setup that gives the much stronger conclusion of reflectionless limit operators in more specialized situations.

Adler–Gelfand–Dickey Approach to Classical 𝒲-Algebras Within the Theory of Poisson Vertex Algebras

We put the Adler-Gelfand-Dickey approach to classical W-algebras in the framework of Poisson vertex algebras. We show how to recover the bi-Poisson structure of the KP hierarchy, together with its generalizations and reduction to the N-th KdV hierarchy, using the formal distribution calculus and the lambda-bracket formalism. We apply the Lenard-Magri scheme to prove integrability of the corresponding hierarchies. We also give a simple proof of a theorem of Kupershmidt and Wilson in this framework. Based on this approach, we generalize all these results to the matrix case. In particular, we find (non-local) bi-Poisson structures of the matrix KP and the matrix N-th KdV hierarchies, and we prove integrability of the N-th matrix KdV hierarchy.

Auxiliary differential polynomials for the first Painlevé hierarchy

This article concerns with the first Painleve hierarchy, i.e. the $2n$-th order analogues of the first Painleve equation. Several higher order analogues of the first Painleve equation are proposed by several authors however, we investigate one derived from the KdV hierarchy by studying singularity manifold. We give the auxiliary differential polynomials of the first Painleve hierarchy.

On the Solutions of Some Boundary Value Problems for the General Kdv Equation

This paper is concerned with a class of partial differential equations, which are the linear combinations, with constant coefficients, of the classical flows of the KdV hierarchy. A boundary value problem with inhomogeneous boundary conditions of a certain special form is studied. We construct some class of solutions of the problem using the inverse spectral method.

Quasimodular instanton partition function and the elliptic solution of Korteweg–de Vries equations

The Gauge/Bethe correspondence relates Omega-deformed N=2 supersymmetric gauge theories to some quantum integrable models, in simple cases the integrable models can be treated as solvable quantum mechanics models. For SU(2) gauge theory with an adjoint matter, or with 4 fundamental matters, the potential of corresponding quantum model is the elliptic function. If the mass of matter takes special value then the potential is an elliptic solution of KdV hierarchy. We show that the deformed prepotential of gauge theory can be obtained from the average densities of conserved charges of the classical KdV solution, the UV gauge coupling dependence is assembled into the Eisenstein series. The gauge theory with adjoint mass is taken as the example.

Symplectic deformations of integrable field theories and AdS/CFT

Relativistic integrable field theories like the sine-Gordon equation have an infinite set of conserved charges. In a light-front formalism these conserved charges are closely related to the integrable modified KdV hierarchy at the classical level. The latter hierarchy admits a family of symplectic structures which we argue can be viewed as deformations of the relativistic sine-Gordon symplectic structure. These deformed theories are integrable but no longer relativistic and the basic excitations of the theory, the solitons, have an interesting non-relativistic dispersion relation that in a certain limit becomes the dispersion relation of dyonic giant magnons of string theory in the AdS/CFT correspondence. We argue that the deformed classical theories can be lifted to quantum theories when the sine-Gordon theory is embedded in a larger theory that describes the string world-sheet sigma model in AdS5×S5.

Quantum Torus Symmetry of the KP, KdV and BKP Hierarchies

In this paper, we construct the quantum Torus symmetry of the KP hierarchy and further derive the quantum torus constraint on the tau function of the KP hierarchy. That means we give a nice representation of the quantum Torus Lie algebra in the KP system by acting on its tau function. Comparing to the $W_{\infty}$ symmetry, this quantum Torus symmetry has a nice algebraic structure with double indices. Further by reduction, we also construct the quantum Torus symmetries of the KdV and BKP hierarchies and further derive the quantum Torus constraints on their tau functions. These quantum Torus constraints might have applications in the quantum field theory, supersymmetric gauge theory and so on.

Multi-soliton solutions of a variable-coefficient KdV hierarchy

In this paper, Hirota’s bilinear method is extended to a new KdV hierarchy with variable coefficients. As a result, one-soliton solution, two-soliton solution and three-soliton solutions are obtained, from which the uniform formula of \(N\)-soliton solution is derived. Thanks to the arbitrariness of the included functions, these obtained solutions possess rich local structural features like the ridge soliton and the concave column soliton. It is shown that the Hirota’s bilinear method can be used for constructing multi-soliton solutions of some other hierarchies of nonlinear partial differential equations with variable coefficients.

On construction of symmetries and recursion operators from zero-curvature representations and the Darboux–Egoroff system

The Darboux–Egoroff system of PDEs with any number n ≥ 3 of independent variables plays an essential role in the problems of describing n -dimensional flat diagonal metrics of Egoroff type and Frobenius manifolds. We construct a recursion operator and its inverse for symmetries of the Darboux–Egoroff system and describe some symmetries generated by these operators. The constructed recursion operators are not pseudodifferential, but are Bäcklund autotransformations for the linearized system whose solutions correspond to symmetries of the Darboux–Egoroff system. For some other PDEs, recursion operators of similar types were considered previously by Papachristou, Guthrie, Marvan, Pobořil, and Sergyeyev. In the structure of the obtained third and fifth order symmetries of the Darboux–Egoroff system, one finds the third and fifth order flows of an ( n − 1 ) -component vector modified KdV hierarchy. The constructed recursion operators generate also an infinite number of nonlocal symmetries. In particular, we obtain a simple construction of nonlocal symmetries that were studied by Buryak and Shadrin in the context of the infinitesimal version of the Givental–van de Leur twisted loop group action on the space of semisimple Frobenius manifolds. We obtain these results by means of rather general methods, using only the zero-curvature representation of the considered PDEs.

A Darboux transformation of the super KdV hierarchy and a super lattice potential KdV equation

Abstract A hierarchy of super KdV equations is derived from sl ( 2 | 1 ) supermatrix-valued spectral problem. Each equation in the hierarchy is shown to be bi-super-Hamiltonian. Moreover, a Darboux transformation of the hierarchy is constructed. As the compatibility condition of a pair of the Darboux transformations, we obtain a super lattice potential KdV (lpKdV) equation with two discrete variables.

On generating (2 + 1)-dimensional hierarchies of evolution equations

Two isospectral problems are constructed with the help of a 6-dimensional Lie algebra. By using the Tu scheme, a (1+1)-dimensional expanding integrable couplings of the KdV hierarchy is obtained and the corresponding Hamiltonian structure is established. In addition, the 2-order matrix operators proposed by Tuguizhang are extended to the case where some 4-order matrices are given. Based on the extension, a new hierarchy of 2+1 dimensions is obtained by the Hamiltonian operator of the above (1+1)-dimensional case and the TAH scheme. The new hierarchy of 2+1 dimensions can be reduced to a coupled (2+1)-dimensional nonlinear equation and furthermore it can be reduced to the (2+1)-dimensional KdV equation which has important physics applications. The Hamiltonian structure for the (2+1)-dimensional hierarchy is derived with the aid of an extended trace identity. To the best of our knowledge, generating the (2+1)-dimensional equation hierarchies by virtue of the TAH scheme has not been studied in detail except to previous little work by Tu et al.

Singular soliton operators and indefinite metrics

We consider singular real second order 1D Schrodinger operators such that all local solutions to the eigenvalue problems are x-meromorphic for all λ. All algebrogeometrical potentials (i.e. “singular finite-gap” and “singular solitons”) satisfy to this condition. A Spectral Theory is constructed for the periodic and rapidly decreasing potentials in the classes of functionswith singularities: The corresponding operators are symmetric with respect to some natural indefinite inner product as it was discovered by the present authors. It has a finite number of negative squares in the both (periodic and rapidly decreasing) cases. The time dynamics provided by the KdV hierarchy preserves this number. The right analog of Fourier Transform on Riemann Surfaces with good multiplicative properties (the R-Fourier Transform) is a partial case of this theory. The potential has a pole in this case at x = 0 with asymptotics u ∼ g(g + 1)/x 2. Here g is the genus of spectral curve.

Bihamiltonian Cohomologies and Integrable Hierarchies I: A Special Case

We present some general results on properties of the bihamiltonian cohomologies associated to bihamiltonian structures of hydrodynamic type, and compute the third cohomology for the bihamiltonian structure of the dispersionless KdV hierarchy. The result of the computation enables us to prove the existence of bihamiltonian deformations of the dispersionless KdV hierarchy starting from any of its infinitesimal deformations.

Asymptotic symmetries and thermodynamics of higher spin black holes inAdS3

We study black holes carrying higher spin charge in AdS3 within the framework of SL(N, R) x SL(N, R) Chern-Simons theory. Focussing attention on the N=4 case, we explicitly analyze the asymptotic symmetry algebra of black hole solutions with a chemical potential for spin-four charge. We demonstrate that the background describes an RG flow between an IR fixed point with W_4 symmetry and a UV fixed point with W-symmetry associated to a non-principal embedding of sl(2) in sl(4). Matching Chern-Simons equations with Ward identities of the deformed CFT, we show that the UV stress tensor is twisted by a certain U(1) current, and the flow is triggered by an operator with dimension 4/3 at the UV fixed point. We find independent confirmation of this picture via a consistent formulation of thermodynamics with respect to this UV fixed point. We further analyze the thermodynamics of multiple branches of black hole solutions for N=4,5 and find that the BTZ-branch, dominant at low temperatures, ceases to exist at higher temperatures following a merger with a thermodynamically unstable branch. We also point out an interesting connection between the RG flows and generalized KdV hierarchies.

$ \mathcal{W} $ symmetry and integrability of higher spin black holes

We obtain the asymptotic symmetry algebra of sl(3,R) x sl(3,R) Chern-Simons theory with Dirichlet boundary conditions for fixed chemical potential. These boundary conditions are obeyed by higher spin black holes. For each embedding of sl(2,R) into sl(3,R), we show that the asymptotic symmetry group is independent of the chemical potential. On the one hand, starting from AdS3 in the principal embedding, we show that the W3 x W3 symmetry is preserved upon turning on perturbatively spin 3 chemical potentials. On the other hand, starting from AdS3 in the diagonal embedding, we show that the W3^(2) x W3^(2) symmetry is preserved upon turning on finite spin 3/2 chemical potentials. We also make connections between the canonical Lagrangian formalism and integrability methods based on the third KdV (Boussinesq) hierarchy.

MOTIONS OF CARTAN CURVES IN n-DIMENSIONAL MINKOWSKI SPACE

Integrable systems satisfied by the curvatures of Cartan curves in Minkowski space under inextensible motions are identified. It is shown that motions of Cartan curves in three-, four- and n-dimensional (n > 4) Minkowski space are related to the KdV hierarchy, a coupled KdV system and a multi-component generalization of the coupled KdV system.

Lower estimates of the growth order of solutions to the second Painlevé hierarchy

This article concerns with the second Painlevé hierarchy, i.e., the 2nth order analogues of the second Painlevé equation. Though several higher order analogues of the second Painlevé equation are proposed by several authors, we investigate one derived from the KdV hierarchy by similarity reduction, and give the lower estimate of the growth order of transcendental meromorphic solutions to the second Painlevé hierarchy with an integer parameter. Moreover, the first lemma in Appendix supplements an incompleteness of Li and He [“On analytic properties of higher analogs of the second Painlevé equation,” J. Math. Phys. 43, 1106–1115 (2002)]10.1063/1.1420396 and Li [“Some properties of the solutions of higher analogue of the Painlevé equation,” Acta Math. Appl. Sin. 22, 59–64 (2006)]10.1007/s10255-005-0285-0.