We construct a new class of N-dimensional Lie algebras and apply them to integrable systems. In this paper, we obtain a nonisospectral KdV integrable hierarchy by introducing a nonisospectral spectral problem. Then, a coupled nonisospectral KdV hierarchy is deduced by means of the corresponding higher-dimensional loop algebra. It follows that the K symmetries, τ symmetries and their Lie algebra of the coupled nonisospectral KdV hierarchy are investigated. The bi-Hamiltonian structures of the both resulting hierarchies are derived by using the trace identity. Finally, we derive a multi-component nonisospectral KdV hierarchy related to the N-dimensional loop algebra, which generalizes the coupled results to an arbitrary number of components.
Generation of higher-dimensional isospectral-nonisospectral integrable hierarchies associated with a new class of higher-dimensional column-vector loop algebras
With the nonuniform media taken into account, the nonisospectral and variable-coefficient Korteweg-de Vries equation, which describes various physical situations such as fluid dynamics and plasma, is under investigation in this paper. With appropriate selection of wave functions, the Darboux transformation is constructed, by which the multi-soliton solutions are derived and graphs are presented. The spectral parameters, coefficients and initial phase are discussed analytically and numerically to demonstrate their respective effect on the soliton dynamics, which plays a role in achieving the fe…
This work presents a classical Lie point symmetry analysis of a two-component, non-isospectral Lax pair of a hierarchy of partial differential equations in $2+1$ dimensions, which can be considered as a modified version of the Camassa-Holm hierarchy in $2+1$ dimensions. A classification of reductions for this spectral problem is performed. Non-isospectral reductions in $1+1$ dimensions are considered of remarkable interest.
In this paper we discuss the conditions under which the coupled KdV and coupled Harry Dym hierarchies possess inverse (negative) parts. We further investigate the structure of nonlocal parts of tensor invariants of these hierarchies, in particular, the nonlocal terms of vector fields, conserved one-forms, recursion operators, Poisson and symplectic operators. We show that the invertible cKdV hierarchies possess Poisson structures that are at most weakly nonlocal while coupled Harry Dym hierarchies have Poisson structures with nonlocalities of the third order.
The construction of Miura and Bäcklund transformations for $A_n$ mKdV and KdV hierarchies are presented in terms of gauge transformations acting upon the zero curvature representation. As in the well known $sl(2)$ case, we derive and relate the equations of motion for the two hierarchies. Moreover, the Miura-gauge transformation is not unique, instead, it is shown to be connected to a set of generators labeled by the exponents of $A_n$ The construction of generalized gauge-Bäcklund transformation for the $A_n$-KdV hierarchy is obtained as a composition of Miura and Bäcklund-gauge transformatio…
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 c…
Supersymmetry is formulated for integrable models based on the $sl(2|1)$ loop algebra endowed with a principal gradation. The symmetry transformations which have half-integer grades generate supersymmetry. The $sl(2|1)$ loop algebra leads to N=2 supersymmetric mKdV and sinh-Gordon equations. The corresponding N=1 mKdV and sinh-Gordon equations are obtained via reduction induced by twisted automorphism. Our method allows for a description of a non-local symmetry structure of supersymmetric integrable models.
Using methods of math.DG/0304245 and [I.S.Krasil'shchik and P.H.M.Kersten, Symmetries and recursion operators for classical and supersymmetric differential equations, Kluwer, 2000], we accomplish an extensive study of the N=1 supersymmetric Korteweg-de Vries equation. The results include: a description of local and nonlocal Hamiltonian and symplectic structures, five hierarchies of symmetries, the corresponding hierarchies of conservation laws, recursion operators for symmetries and generating functions of conservation laws. We stress that the main point of the paper is not just the results on…
We discuss commuting flows and conservation laws for Lax hierarchies on noncommutative spaces in the framework of the Sato theory. On commutative spaces, the Sato theory has revealed essential aspects of the integrability for wide class of soliton equations which are derived from the Lax hierarchies in terms of pseudo-differential operators. Noncommutative extension of the Sato theory has been already studied by the author and Kouichi Toda, and the existence of various noncommutative Lax hierarchies are guaranteed. In the present paper, we present conservation laws for the noncommutative Lax h…
For each of the simple Lie algebras $\mathfrak{g}=A_l$, $D_l$ or $E_6$, we show that the all-genera one-point FJRW invariants of $\mathfrak{g}$-type, after multiplication by suitable products of Pochhammer symbols, are the coefficients of an algebraic generating function and hence are integral. Moreover, we find that the all-genera invariants themselves coincide with the coefficients of the unique calibration of the Frobenius manifold of $\mathfrak{g}$-type evaluated at a special point. For the $A_4$ (5-spin) case we also find two other normalizations of the sequence that are again integral an…
Using the fact that Miura transformation can be expressed in the form of gauge transformation connecting the KdV and mKdV equations, we discuss the derivation of the Bäcklund transformation and its Miura-gauge transformation connecting both hierarchies.
We give 4 formulations of the Modified KP hierarchy and show that they are equivalent. We also discuss the reductions of the MKP hierarchy to the modified $n$-KdV hierarchies. As a byproduct, we find an astonishingly simple explicit description of all polynomial tau-functions of the KP, the MKP and the $n$-KdV hierarchies, and (implicitly) also for the modified $n$-KdV hierarchy.
It is well-known that the finite-gap solutions of the KdV equation can be generated by its recursion operator.We generalize the result to a special form of Lax pair, from which a method to constrain the integrable system to a lower-dimensional or fewer variable integrable system is proposed. A direct result is that the $n$-soliton solutions of the KdV hierarchy can be completely depicted by a series of ordinary differential equations (ODEs), which may be gotten by a simple but unfamiliar Lax pair. Furthermore the AKNS hierarchy is constrained to a series of univariate integrable hierarchies. T…
We propose a scheme for nonlinearizing linear equations to generate integrable nonlinear systems of both the AKNS and the KN classes, based on the simple idea of dimensional analysis and detecting the building blocks of the Lax pair. Along with the well known equations we discover a novel integrable hierarchy of higher order nonholonomic deformations for the AKNS family, e.g. for the KdV, the mKdV, the NLS and the SG equation, showing thus a two-fold universality of the recently found deformation for the KdV equation.
According to a conjecture of E. Witten proved by M. Kontsevich, a certain generating function for intersection indices on the Deligne -- Mumford moduli spaces of Riemann surfaces coincides with a certain tau-function of the KdV hierarchy. The generating function is naturally generalized under the name the {\em total descendent potential} in the theory of Gromov -- Witten invariants of symplectic manifolds. The papers arXiv: math.AG/0108100 and arXive: math.DG/0108160 contain two equivalent constructions, motivated by some results in Gromov -- Witten theory, which associate a total descendent p…
We develop a theory of Lagrangian reduction on loop groups for completely integrable systems after having exchanged the role of the space and time variables in the multi-time interpretation of integrable hierarchies. We then insert the Sobolev norm $H^1$ in the Lagrangian and derive a deformation of the corresponding hierarchies. The integrability of the deformed equations is altered and a notion of weak integrability is introduced. We implement this scheme in the AKNS and SO(3) hierarchies and obtain known and new equations. Among them we found two important equations, the Camassa-Holm equati…
In this paper, we construct the additional flows of the noncommutative Kadomtsev-Petviashvili(KP) hierarchy and the additional symmetry flows constitute an infinite dimensional Lie algebra $W_{1+\infty}$. In addition, the generating function of the additional symmetries can also be proved to have a nice form in terms of wave functions and this generating symmetry is used to construct the noncommutative KP hierarchy with self-consistent sources and the constrained noncommutative KP hierarchy. The above results will be further generalized to the noncommutative Gelfand-Dickey hierarchies which co…
It is shown that the one-dimensional generalized N=4 supersymmetric Toda lattice (TL) hierarchy (nlin.Si/0311030) contains the N=4 super-KdV hierarchy with the first flow time in the role of space coordinate. Two different N=2 superfield forms of the generalized N=4 supersymmetric TL equation, which are useful when solving the N=4 super-KdV and (1,1)-GNLS hierarchies, are discussed.
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 fin…
We consider a certain super extension, called the super tau-cover, of a bihamiltonian integrable hierarchy which contains the Hamiltonian structures including both the local and non-local ones as odd flows. In particular, we construct the super tau-cover of the principal hierarchy associated with an arbitrary Frobenius manifold, and the super tau-cover of the Korteweg-de Vries (KdV) hierarchy. We also show that the Virasoro symmetries of these bihamiltonan integrable hierarchies can be extended to symmetries of the associated super tau-covers.
We continue the study of the quantization of supersymmetric integrable KdV hierarchies. We consider the N=2 KdV model based on the $sl^{(1)}(2|1)$ affine algebra but with a new algebraic construction for the L-operator, different from the standard Drinfeld-Sokolov reduction. We construct the quantum monodromy matrix satisfying a special version of the reflection equation and show that in the classical limit, this object gives the monodromy matrix of N=2 supersymmetric KdV system. We also show that at both the classical and the quantum levels, the trace of the monodromy matrix (transfer matrix)…
We use the connection between the Frobrenius manifold and the Douglas string equation to further investigate Minimal Liouville gravity. We search a solution of the Douglas string equation and simultaneously a proper transformation from the KdV to the Liouville frame which ensure the fulfilment of the conformal and fusion selection rules. We find that the desired solution of the string equation has explicit and simple form in the flat coordinates on the Frobenious manifold in the general case of (p,q) Minimal Liouville gravity.
The quantization procedure for both N=1 and N=2 supersymmetric Korteweg-de Vries (SUSY KdV) hierarchies is constructed. Namely, the quantum counterparts of the monodromy matrices, built by means of the integrated vertex operators, are shown to satisfy a specialization of reflection equation, leading to the quantum integrable theory. The relation of such models to the study of integrable perturbed superconformal and topological models is discussed.
We prove that any classical affine W-algebra W(g,f), where g is a classical Lie algebra and f is an arbitrary nilpotent element of g, carries an integrable Hamiltonian hierarchy of Lax type equations. This is based on the theories of generalized Adler type operators and of generalized quasideterminants, which we develop in the paper. Moreover, we show that under certain conditions, the product of two generalized Adler type operators is a Lax type operator. We use this fact to construct a large number of integrable Hamiltonian systems, recovering, as a special case, all KdV type hierarchies con…
In this paper we prove that the generating series of the Hodge integrals over the moduli space of stable curves is a solution of a certain deformation of the KdV hierarchy. This hierarchy is constructed in the framework of the Dubrovin-Zhang theory of the hierarchies of the topological type. It occurs that our deformation of the KdV hierarchy is closely related to the hierarchy of the Intermediate Long Wave equation.
A linear system, which generates a Moyal-deformed two-dimensional soliton equation as integrability condition, can be extended to a three-dimensional linear system, treating the deformation parameter as an additional coordinate. The supplementary integrability conditions result in a first order differential equation with respect to the deformation parameter, the flow of which commutes with the flow of the deformed soliton equation. In this way, a deformed soliton hierarchy can be extended to a bigger hierarchy by including the corresponding deformation equations. We prove the extended hierarch…
To each partition function of cohomological field theory one can associate an Hamiltonian integrable hierarchy of topological type. The Givental group acts on such partition functions and consequently on the associated integrable hierarchies. We consider the Hirota and Lax formulations of the deformation of the hierarchy of N copies of KdV obtained by an infinitesimal action of the Givental group. By first deforming the Hirota quadratic equations and then applying a fundamental lemma to express it in terms of pseudo-differential operators, we show that such deformed hierarchy admits an explici…
The generalization of the N=2 supersymmetric chiral matrix (k|n,m)--GNLS hierarchy (Lett. Math. Phys. 45 (1998) 63, solv-int/9711009) to the case when matrix entries are bosonic and fermionic unconstrained N=2 superfields is proposed. This is done by exhibiting the corresponding matrix Lax--pair representation in terms of N=2 unconstrained superfields. It is demonstrated that when matrix entries are chiral and antichiral N=2 superfields, it reproduces the N=2 chiral matrix (k|n,m)-GNLS hierarchy, while in the scalar case, k=1, it is equivalent to the N=2 supersymmetric multicomponent hierarchy…
This is the third in a series of papers attempting to describe a uniform geometric framework in which many integrable systems can be placed. A soliton hierarchy can be constructed from a splitting of an infinite dimensional group $L$ as positive and negative subgroups L_+, L_- and a commuting sequence in the Lie algebra of L_+. Given f in L_-, there is a formal inverse scattering solution u_f of the hierarchy. When there is a 2 co-cycle that vanishes on both subalgebras of L_+ and L_-, Wilson constructed for each f in L_- a tau function tau_f for the hierarchy. In this third paper, we prove th…
A technique based on extended Lax Pairs is first considered to derive variable-coefficient generalizations of various Lax-integrable NLPDE hierarchies recently introduced in the literature. As illustrative examples, we consider generalizations of KdV equations and three variants of generalized MKdV equations. It is demonstrated that the techniques yield Lax- or S-integrable NLPDEs with both time- AND space-dependent coefficients which are thus more general than almost all cases considered earlier via other methods such as the Painleve Test, Bell Polynomials, and various similarity methods. How…
We derive explicit formulas for lambda-brackets of the affine classical W-algebras attached to the minimal and short nilpotent elements of any simple Lie algebra g. This is used to compute explicitly the first non-trivial PDE of the corresponding intgerable generalized Drinfeld-Sokolov hierarchies. It turns out that a reduction of the equation corresponding to a short nilpotent is Svinolupov's equation attached to a simple Jordan algebra, while a reduction of the equation corresponding to a minimal nilpotent is an integrable Hamiltonian equation on 2h-3 functions, where h is the dual Coxeter n…
In this paper we study various aspects of the double ramification (DR) hierarchy, introduced by the first author, and its quantization. We extend the notion of tau-symmetry to quantum integrable hierarchies and prove that the quantum DR hierarchy enjoys this property. We determine explicitly the genus $1$ quantum correction and, as an application, compute completely the quantization of the $3$- and $4$-KdV hierarchies (the DR hierarchies for Witten's $3$- and $4$-spin theories). We then focus on the recursion relation satisfied by the DR Hamiltonian densities and, abstracting from its geometri…
The importance of the theory of pseudo-differential operators in the study of non linear integrable systems is point out. Principally, the algebra $Ξ$ of nonlinear (local and nonlocal) differential operators, acting on the ring of analytic functions $u_{s}(x, t)$, is studied. It is shown in particular that this space splits into several classes of subalgebras $Σ_{jr}, j=0,\pm 1, r=\pm 1$ completely specified by the quantum numbers: $s$ and $(p,q)$ describing respectively the conformal weight (or spin) and the lowest and highest degrees. The algebra ${\huge Σ}_{++}$ (and its dual $Σ_{--}$) of l…
In this paper we define a quantization of the Double Ramification Hierarchies of [Bur15b] and [BR14], using intersection numbers of the double ramification cycle, the full Chern class of the Hodge bundle and psi-classes with a given cohomological field theory. We provide effective recursion formulae which determine the full quantum hierarchy starting from just one Hamiltonian, the one associated with the first descendant of the unit of the cohomological field theory only. We study various examples which provide, in very explicit form, new $(1+1)$-dimensional integrable quantum field theories w…
In this paper we begin the study of the relationship between the local Gromov-Witten theory of Calabi-Yau rank two bundles over the projective line and the theory of integrable hierarchies. We first of all construct explicitly, in a large number of cases, the Hamiltonian dispersionless hierarchies that govern the full descendent genus zero theory. Our main tool is the application of Dubrovin's formalism, based on associativity equations, to the known results on the genus zero theory from local mirror symmetry and localization. The hierarchies we find are apparently new, with the exception of t…
