Miura Map Research Articles

Stationary coupled KdV systems and their Stäckel representations

AbstractIn this article, we investigate stationary coupled Korteweg–de Vries (cKdV) systems and prove that every ‐field stationary cKdV system can be written, after a careful reparameterization of jet variables, as a classical separable Stäckel system in different ways. For each of these parameterizations, we present an explicit map between the jet variables and the separation variables of the system. Finally, we show that each pair of Stäckel representations of the same stationary cKdV system, when considered in the phase space extended by Casimir variables, is connected by an appropriate finite‐dimensional Miura map, which leads to an ‐Hamiltonian formulation for the stationary cKdV system.

Miura maps for Stäckel systems

We introduce the concept of Miura maps between parameter-dependent algebraic curves of hyper-elliptic type. These Miura maps induce Miura maps between Stäckel systems defined (on the extended phase space) by the considered algebraic curves. This construction yields a new way of generating multi-Hamiltonian representations for Stäckel systems.

Virasoro symmetries of the constrained dispersionless mKP hierarchy

This paper aims at the auto-Bäcklund transformations for the dispersionless mKP hierarchy, dispersionless Miura map between the constrained dispersionless mKP hierarchy and the constrained dispersionless Harry Dym hierarchy and Virasoro symmetries of the constrained dispersionless mKP hierarchy. The additional symmetries of the constrained dispersionless mKP hierarchy are constructed in a manner similar to that of the dispersionless mKP hierarchy. By comparison, a new Laurent series Yl is introduced. The additional flows form a subalgebra of the Virasoro algebra.

Stationary Flows Revisited

In this paper we revisit the subject of stationary flows of Lax hierarchies of a coupled KdV class. We explain the main ideas in the standard KdV case and then consider the dispersive water waves (DWW) case, with respectively 2 and 3 Hamiltonian representations. Each Hamiltonian representation gives us a different form of stationary flow. Comparing these, we construct Poisson maps, which, being non-canonical, give rise to bi-Hamiltonian representations of the stationary flows. An alternative approach is to use the Miura maps, which we do in the case of the DWW hierarchy, which has two ''modifications''. This structure gives us 3 sequences of Poisson related stationary flows. We use the Poisson maps to build a tri-Hamiltonian representation of each of the three stationary hierarchies. One of the Hamiltonian representations allows a multi-component squared eigenfunction expansion, which gives N degrees of freedom Hamiltonians, with first integrals. A Lax representation for each of the stationary flows is derived from the coupled KdV matrices. In the case of 3 degrees of freedom, we give a generalisation of our Lax matrices and Hamiltonian functions, which allows a connection with the rational Calogero-Moser (CM) system. This gives a coupling of the CM system with other potentials, along with a Lax representation. We present the particular case of coupling one of the integrable Hénon-Heiles systems to CM.

Integrable systems and the boundary dynamics of higher spin gravity on AdS3

We introduce a new set of boundary conditions for three-dimensional higher spin gravity with gauge group SL(3, ℝ) × SL(3, ℝ), where its dynamics at the boundary is described by the members of the modified Boussinesq integrable hierarchy. In the asymptotic region the gauge fields are written in the diagonal gauge, where the excitations go along the generators of the Cartan subalgebra of sl(3, ℝ) ⊕ sl(3, ℝ). We show that the entire integrable structure of the modified Boussinesq hierarchy, i.e., the phase space, the Poisson brackets and the infinite number of commuting conserved charges, are obtained from the asymptotic structure of the higher spin theory. Furthermore, its known relation with the Boussinesq hierarchy is inherited from our analysis once the asymptotic conditions are re-expressed in the highest weight gauge. Hence, the Miura map is recovered from a purely geometric construction in the bulk. Black holes that fit within our boundary conditions, the Hamiltonian reduction at the boundary, and the generalization to higher spin gravity with gauge group SL(N, ℝ) × SL(N, ℝ) are also discussed.

Integrability and asymptotic behaviour of a differential-difference matrix equation

In this paper we consider the matrix lattice equation Un,t(Un+1−Un−1)=g(n)I, in both its autonomous (g(n)=2) and nonautonomous (g(n)=2n−1) forms. We show that each of these two matrix lattice equations are integrable. In addition, we explore the construction of Miura maps which relate these two lattice equations, via intermediate equations, to matrix analogs of autonomous and nonautonomous Volterra equations, but in two matrix dependent variables. For these last systems, we consider cases where the dependent variables belong to certain special classes of matrices, and obtain integrable coupled systems of autonomous and nonautonomous lattice equations and corresponding Miura maps. Moreover, in the nonautonomous case we present a new integrable nonautonomous matrix Volterra equation, along with its Lax pair. Asymptotic reductions to the matrix potential Korteweg–de Vries and matrix Korteweg–de Vries equations are also given.

Screening operators and parabolic inductions for affine [formula omitted]-algebras

(Affine) W-algebras are a family of vertex algebras defined by the generalized Drinfeld-Sokolov reductions associated with a finite-dimensional reductive Lie algebra g over C, a nilpotent element f in [g,g], a good grading Γ and a symmetric invariant bilinear form κ on g. We introduce free field realizations of W-algebras by using Wakimoto representations of affine Lie algebras, where W-algebras are described as the intersections of kernels of screening operators. We call these Wakimoto free fields realizations of W-algebras. As applications, under certain conditions that are valid in all cases of type A, we construct parabolic inductions for W-algebras, which we expect to induce the parabolic inductions of finite W-algebras defined by Premet and Losev. In type A, we show that our parabolic inductions are a chiralization of the coproducts for finite W-algebras defined by Brundan-Kleshchev. In type BCD, we are able to obtain some generalizations of the coproducts in some special cases. This paper also contains an appendix by Shigenori Nakatsuka on the compatibility of screening operators with Miura maps.

Integrable Potentials by Darboux Transformations in Rings and Quantum and Classical Problems

We study a problem in associative rings of left and right factorization of a polynomial differential operator regarded as an evolution operator. In a direct sum of rings, the polynomial arising in the problem of dividing an operator by an operator for two commuting operators leads to a time-dependent left/right Darboux transformation based on an intertwining relation and either Miura maps or generalized Burgers equations. The intertwining relations lead to a differential equation including differentiations in a weak sense. In view of applications to operator problems in quantum and classical mechanics, we derive the direct quasideterminant or dressing chain formulas. We consider the transformation of creation and annihilation operators for specified matrix rings and study an example of the Dicke model.

Structure of classical (finite and affine) $\mathcal W$-algebras

First, we derive an explicit formula for the Poisson bracket of the classical finite \mathcal W -algebra \mathcal W^{\mathrm {fin}}(\mathfrak g,f) , the algebra of polynomial functions on the Slodowy slice associated to a simple Lie algebra \mathfrak g and its nilpotent element f . On the other hand, we produce an explicit set of generators and we derive an explicit formula for the Poisson vertex algebra structure of the classical affine \mathcal W -algebra \mathcal W(\mathfrak g,f) . As an immediate consequence, we obtain a Poisson algebra isomorphism between \mathcal W^{\mathrm {fin}}(\mathfrak g,f) and the Zhu algebra of \mathcal W(\mathfrak g,f) . We also study the generalized Miura map for classical \mathcal W -algebras.

Miura maps and inverse scattering for the Novikov–Veselov equation

We use the inverse scattering method to construct classical solutions for the Novikov-Veselov (NV) equation, solving a problem posed by Lassas, Mueller, Siltanen, and Stahel. We exploit Bogadanov's Miura-type map which transforms solutions of the modified Novikov-Veselov (mNV) equation into solutions of the NV equation. We show that the Cauchy data of conductivity type considered by Lassas, Mueller, Siltanen, and Stahel correspond precisely to the range of the Miura map, so that it suffices to study the mNV equation. We solve the mNV equation using the scattering transform associated to the defocussing Davey-Stewartson II equation.

A New Kind of Bäcklund Transformation for Partial Differential Equations

The existence of auto-Backlund transformations is generally understood as being a characteristic feature of integrable equations. The nature of such transformations changes depending on whether the equation under consideration is a partial differential equation (PDE) or an ordinary differential equation (ODE). We show in this paper how certain properties of completely integrable systems such as Hamiltonian structures and Miura maps can be used in order to derive ODE-type auto-Backlund transformations for a certain class of PDEs. We apply this method to two integrable equations which are nonisospectral extensions of the integrated modified inverse Korteweg–de Vries equation and of the integrated modified inverse Broer–Kaup system, and show that ODE-type auto-Backlund transformations exist for both of them. These auto-Backlund transformations involve shifts on the functions appearing as coefficients in the equations and they mimic the auto-Backlund transformations for the second and fourth Painleve equations. We believe these ODE-type auto-Backund transformations for PDEs to be new. Our derivation is in fact valid for a wide class of equations that includes both integrable and nonintegrable ODEs and PDEs.

Systematic method of generating new integrable systems via inverse Miura maps

We provide a new natural interpretation of the Lax representation for an integrable system; that is, the spectral problem is the linearized form of a Miura transformation between the original system and a modified version of it. On the basis of this interpretation, we formulate a systematic method of identifying modified integrable systems that can be mapped to a given integrable system by Miura transformations. Thus, this method can be used to generate new integrable systems from known systems through inverse Miura maps; it can be applied to both continuous and discrete systems in 1 + 1 dimensions as well as in 2 + 1 dimensions. The effectiveness of the method is illustrated using examples such as the nonlinear Schrödinger (NLS) system, the Zakharov–Ito system (two-component KdV), the three-wave interaction system, the Yajima–Oikawa system, the Ablowitz–Ladik lattice (integrable space-discrete NLS), and two (2 + 1)-dimensional NLS systems.

A note on the dispersionless BKP hierarchy

We discuss the Miura map as well as the Poisson algebras associated with the dispersionless Dym hierarchy. Particularly, we study explicitly the bi-Hamiltonian structure of a truncated Dym system with two variables, in which a new hierarchy flow generated by logarithmic Hamiltonians appears. We then show that this new hierarchy emerges naturally from the topological recursion relation in the Landau–Ginzburg formulation.

Bäcklund transformations for fourth Painlevé hierarchies

Bäcklund transformations (BTs) for ordinary differential equations (ODEs), and in particular for hierarchies of ODEs, are a topic of great current interest. Here, we give an improved method of constructing BTs for hierarchies of ODEs. This approach is then applied to fourth Painlevé ( P IV ) hierarchies recently found by Gordoa et al. [Publ. Res. Inst. Math. Sci. (Kyoto) 37 (2001) 327–347]. We show how the known pattern of BTs for P IV can be extended to our P IV hierarchies. Remarkably, the BTs required to do this are precisely the Miura maps of the dispersive water wave hierarchy. We also obtain the important result that the fourth Painlevé equation has only one nontrivial fundamental BT, and not two such as is frequently stated.

Global Well-Posedness of mKdV in L 2 (𝕋, ℝ)

ABSTRACT We show that the Miura map L 2(𝕋) → H −1(𝕋), r↦r x + r 2 is a global fold and then apply our results on global well-posedness of KdV in H −1(𝕋) to show that mKdV is globally well-posed in L 2(𝕋).

The Miura map on the line

We study relations between properties of the Miura map r ↦ q=B(r)=r′+r2 and Schrödinger operators Lq=−d2/dx2+q, where r and q are real-valued functions or distributions (possibly not decaying at infinity) from various classes. In particular, we study B as a map from Lloc2(ℝ) to the local Sobolev space Hloc−1(ℝ)⁠, and the restriction of B to the Sobolev spaces Hrβ(ℝ) with rβ≥0. For example, we prove that the image of B on Lloc2(ℝ) consists exactly of those q∈Hloc−1(ℝ) such that the operator Lq is positive. We also investigate mapping properties of the Miura map in these spaces. As an application, we prove an existence result for solutions of the Korteweg-de Vries equation in H−1(ℝ) for initial data in the range B(L2(ℝ)) of the Miura map.

On therth dispersionless Toda hierarchy: factorization problem, additional symmetries and some solutions

For a family of Poisson algebras, parametrized by by an integer number r, and an associated Lie algebraic splitting, we consider the factorization of given canonical transformations. In this context we rederive the recently found r-th dispersionless modified KP hierachies and r-th dispersionless Dym hierarchies, giving a new Miura map among them. We also found a new integrable hierarchy which we call the r-th dispersionless Toda hierarchy. Moreover, additional symmetries for these hierarchies are studied in detail and new symmetries depending on arbitrary functions are explicitly constructed for the r-th dispersionless KP, r-th dispersionless Dym and r-th dispersionless Toda equations. Some solutions are derived by examining the imposition of a time invariance to the potential r-th dispersionless Dym equation, for which a complete integral is presented and therefore an appropriate envelope leads to a general solution. Symmetries and Miura maps are applied to get new solutions and solutions of the r-th dispesionless modified KP equation.

Global fold structure of the Miura map on L2(T)

The main purpose of this paper is to study the Miura transform r → r′ + r2-functions. More precisely, we prove that the Miura transform, viewed as map from L2(T) to H−1(T)⁠, has a global fold structure with a “Whitney type” singularity at L02(T)⁠, the space of periodic L2-functions with mean zero. Using the well-known fact that the Miura transform maps solutions of the modified Korteweg-de Vries equation (mKdV) to solutions of the Korteweg-de Vries equation (KdV), the above result can be used as a tool to obtain low-regularity well-posedness results for mKdV on the circle from corresponding low-regularity well-posedness results of KdV (and vice versa).

Painlevé Hierarchies and the Painlevé Test

In a series of recent papers, we derived several new hierarchies of higher-order analogues of the six Painleve equations. Here we consider one particular example of such a hierarchy, namely, a recently derived fourth Painleve hierarchy. We use this hierarchy to illustrate how knowing the Hamiltonian structures and Miura maps can allow finding first integrals of the ordinary differential equations derived. We also consider the implications of the second member of this hierarchy for the Painleve test. In particular, we find that the Ablowitz–Ramani–Segur algorithm cannot be applied to this equation. This represents a significant failing in what is now a standard test of singularity structure. We present a solution of this problem.

Computation of densities and fluxes of nonlinear differential‐difference equations

Direct methods to find conserved densities and fluxes of differentialdifference equations are presented and illustrated for the Kacvan Moerbeke (KvM) and modified Volterra lattices. A Miura map whi...