Upon having presented a bird's eye view of history of integrable systems, we give a brief review of certain earlier advances (arXiv:1401.2122 & arXiv:1812.02263) in the longstanding problem of search for partial differential systems in four independent variables, often referred to as (3+1)-dimensional or 4D systems, that are integrable in the sense of soliton theory. Namely, we review a recent construction for a large new class of (3+1)-dimensional integrable systems with Lax pairs involving contact vector fields. This class contains inter alia two infinite families of such systems, thus establishing that there is significantly more integrable (3+1)-dimensional systems than it was believed for a long time.
In fact, the construction under study yields (3+1)-dimensional integrable generalizations of many well-known dispersionless integrable (2+1)-dimensional systems like the dispersionless KP equation, as well as a first example of a (3+1)-dimensional integrable system with an algebraic, rather than rational, nonisospectral Lax pair.
To demonstrate the versatility of the construction in question, we employ it here to produce novel integrable (3+1)-dimensional generalizations for the following (2+1)-dimensional integrable systems: dispersionless BKP, dispersionless asymmetric Nizhnik--Veselov--Novikov, dispersionless Gardner, and dispersionless modified KP equations, and the generalized Benney system.
We develop a new integration technique allowing one to construct a rich manifold of particular solutions to multidimensional generalizations of classical $C$- and $S$-integrable Partial Differential Equations (PDEs). Generalizations of (1+1)-dimensional $C$-integrable and (2+1)-dimensional $S$-integrable $N$-wave equations are derived among examples. Examples of multidimensional second order PDEs are represented as well.
This paper develops a modification of the dressing method based on the inhomogeneous linear integral equation with integral operator having nonempty kernel. Method allows one to construct the systems of multidimensional Partial Differential Equations (PDEs) having the differential polynomial forms in any dimension n. Associated solution space is not full, although it is parametrized by a certain number of arbitrary functions of (n-1)-variables. We consider 4-dimensional generalization of the classical (2+1)-dimensional S-integrable N-wave equation as an example.
In this paper we study generalized classes of volume preserving multidimensional integrable systems via Nambu--Poisson mechanics. These integrable systems belong to the same class of dispersionless KP type equation. Hence they bear a close resemblance to the self dual Einstein equation. All these dispersionless KP and dToda type equations can be studied via twistor geometry, by using the method of Gindikin's pencil of two forms. Following this approach we study the twistor construction of our volume preserving systems.
A class of multidimensional integrable hierarchies connected with commutation of general (unreduced) (N+1)-dimensional vector fields containing derivative over spectral variable is considered. They are represented in the form of generating equation, as well as in the Lax-Sato form. A dressing scheme based on nonlinear vector Riemann problem is presented for this class. The hierarchies connected with Manakov-Santini equation and Dunajski system are considered as illustrative examples.
We transfer the scheme of constructing differential reductions, developed recently for the case of the Manakov-Santini hierarchy, to the general multidimensional case. We consider in more detail the four-dimensional case, connected with the second heavenly equation and its generalization proposed by Dunajski. We give a characterization of differential reductions in terms of the Lax-Sato equations as well as in the framework of the dressing method based on nonlinear Riemann-Hilbert problem.
A method of looking for boundary conditions consistent with the integrability property of multidimensional Kadomtsev-Petviashvili (KP) type equations is discussed. The method is based on involutions of the Lax pair taken at the border plane. New classes of boundary conditions for the KP and Hirota equations are proposed consistent with the Lax pair. The boundary problem on the stripe 0<y<1 for the KP equation is discussed, its exact solutions are found. Ward's problem on discrete versions of the generalized Toda chains is briefly discussed.
We represent an algorithm reducing a big class of systems of ($M+1$)-dimensional nonlinear partial differential equations (PDEs) to the systems of $M$-dimensional first order PDEs. Thus, we integrate the original system with respect to only one independent variable reducing its dimensionality by one. For this reason we call such systems partially integrable ones. In particular, if M=1, then $M$-dimensional PDEs become ODEs. Our method may be referred to as a generalization of the method of characteristics. Possible application in hydrodynamics is discussed.
The new examples are found of the constraints which link the 1+2-dimensional and multifield integrable equations and lattices. The vector and matrix generalizations of the Nonlinear Schrödinger equation and the Ablowitz-Ladik lattice are considered among the other multifield models. It is demonstrated that using of the symmetries belonging to the hierarchies of these equations leads, in particular, to the KP equation and twodimensional analogs of the dressing chain, Toda lattice and dispersive long waves equations. In these examples the multifield equation and its symmetry have meaning of the …
We present a discrete inverse scattering transform for all ABS equations excluding Q4. The nonlinear partial difference equations presented in the ABS hierarchy represent a comprehensive class of scalar affine-linear lattice equations which possess the multidimensional consistency property. Due to this property it is natural to consider these equations living in an N-dimensional lattice, where the solutions depend on N distinct independent variables and associated parameters. The direct scattering procedure, which is one-dimensional, is carried out along a staircase within this multidimensiona…
A linearizable version of multidimensional system of $n$-wave type nonlinear PDEs is proposed. This system is derived using the spectral representation of its solution via the procedure similar to the dressing method for the ISTM-integrable nonlinear PDEs. The proposed system is shown to be completely integrable, particular solution is represented.
On the base of Lie algebraic and differential geometry methods, a wide class of multidimensional nonlinear integrable systems is obtained, and the integration scheme for such equations is proposed.
The notion of multidimensional quadrilateral lattice is introduced. It is shown that such a lattice is characterized by a system of integrable discrete nonlinear equations. Different useful formulations of the system are given. The geometric construction of the lattice is also discussed and, in particular, it is clarified the number of initial--boundary data which define the lattice uniquely.
A deformed differential calculus is developed based on an associative star-product. In two dimensions the Hamiltonian vector fields model the algebra of pseudo-differential operator, as used in the theory of integrable systems. Thus one obtains a geometric description of the operators. A dual theory is also possible, based on a deformation of differential forms. This calculus is applied to a number of multidimensional integrable systems, such as the KP hierarchy, thus obtaining a geometrical description of these systems. The limit in which the deformation disappears corresponds to taking the d…
We use deformations of Lie algebra homomorphisms to construct deformations of dispersionless integrable systems arising as symmetry reductions of anti--self--dual Yang--Mills equations with a gauge group Diff$(S^1)$.
We review the recent approach to the construction of (3+1)-dimensional integrable dispersionless partial differential systems based on their contact Lax pairs and the related $R$-matrix theory for the Lie algebra of functions with respect to the contact bracket. We discuss various kinds of Lax representations for such systems, in particular, linear nonisospectral contact Lax pairs and nonlinear contact Lax pairs as well as the relations among the two. Finally, we present a large number of examples with finite and infinite number of dependent variables, as well as the reductions of these exampl…
We introduce a general setting for multidimensional dispersionless integrable hierarchy in terms of differential $m$-form $Ω_m$ with the coefficients satisfying the Plücker relations, which is gauge-invariantly closed and its gauge-invariant coordinates (ratios of coefficients) are (locally) holomorphic with respect to one of the variables (the spectral variable). We demonstrate that this form defines a hierarchy of dispersionless integrable equations in terms of commuting vector fields locally holomorphic in the spectral variable. The equations of the hierarchy are given by the gauge-invarian…
We present a simple novel construction of recursion operators for integrable multidimensional dispersionless systems that admit a Lax pair whose operators are linear in the spectral parameter and do not involve the derivatives with respect to the latter. New examples of recursion operators obtained using our technique include {\em inter alia} those for the general heavenly equation, which describes a class of anti-self-dual solutions of the vacuum Einstein equations, and a six-dimensional equation resulting from a system of Ferapontov and Khusnutdinova.
We consistently develop a recently proposed scheme of matrix extension of dispersionless integrable systems for the general case of multidimensional hierarchies, concentrating on the case of dimension $d\geqslant 4$. We present extended Lax pairs, Lax-Sato equations, matrix equations on the background of vector fields and the dressing scheme. Reductions, construction of solutions and connections to geometry are discussed. We consider separately a case of Abelian extension, for which the Riemann-Hilbert equations of the dressing scheme are explicitly solvable and give an analogue of Penrose for…
The $2n$ dimensional manifold with two mutually commutative operators of differentiation is introduced. Nontrivial multidimensional integrable systems connected with arbitrary graded (semisimple) algebras are constructed. The general solution of them is presented in explicit form.
We represent the Fourier form of the dressing method, which is effective for construction of multidimensional integral-differential equations together with their solutions. Example of integrable (but non-physical) expansion of Intermediate Long Wave equation in (2+1)-dimensions is considered.
The negative symmetry flows are incorporated into the Riemann-Hilbert problem for the homogeneous $A_m$-hierarchy and its $\hat{gl} (m+1, C)$ extension. A loop group automorphism of order two is used to define a sub-hierarchy of $\hat{gl} (m+1, C)$ hierarchy containing only the odd symmetry flows. The positive and negative flows of the $\pm 1$ grade coincide with equations of the multidimensional Toda model and of topological-anti-topological fusion.
We introduce a hierarchy of integrable PDEs in 2+1 dimensions arising from the commutation of 2-dimensional vector fields. We also solve the associated Cauchy problems, using the recently developed Inverse Scattering Transform for 1-parameter families of multidimensional vector fields.
We show that well-chosen Lagrangians for a class of two-dimensional integrable lattice equations obey a closure relation when embedded in a higher dimensional lattice. On the basis of this property we formulate a Lagrangian description for such systems in terms of Lagrangian multiforms. We discuss the connection of this formalism with the notion of multidimensional consistency, and the role of the lattice from the point of view of the relevant variational principle.
Integrable hierarchies associated with the singular sector of the KP hierarchy, or equivalently, with $\dbar$-operators of non-zero index are studied. They arise as the restriction of the standard KP hierarchy to submanifols of finite codimension in the space of independent variables. For higher $\dbar$-index these hierarchies represent themselves families of multidimensional equations with multidimensional constraints. The $\dbar$-dressing method is used to construct these hierarchies. Hidden KdV, Boussinesq and hidden Gelfand-Dikii hierarchies are considered too.
We consider Hamiltonian systems restricted to the hypersurfaces of contact type and obtain a partial version of the Arnold-Liouville theorem: the system not need to be integrable on the whole phase space, while the invariant hypersurface is foliated on an invariant Lagrangian tori. In the second part of the paper we consider contact systems with constraints. As an example, the Reeb flows on Brieskorn manifolds are considered.
We present a new approach to construction of recursion operators for multidimensional integrable systems which have a Lax-type representation in terms of a pair of commuting vector fields. It is illustrated by the examples of the Manakov--Santini system which is a hyperbolic system in N dependent and N + 4 independent variables, where N is an arbitrary natural number, the six-dimensional generalization of the first heavenly equation, the modified heavenly equation, and the dispersionless Hirota equation.
We develop the method for constructing Lax representations of PDEs via the twisted extensions of their algebras of contact symmetries by generalizing the construction to the Lie--Rinehart algebras. We present examples of application of the proposed technique.
The $2n$ dimensional manifold with two mutually commutative operators of differentiation is introduced. Nontrivial multidimensional integrable systems connected with arbitrary graded (semisimple) algebras are constructed. The general solution of them is presented in explicit form.
We consider two-dimensional lattice equations defined on an elementary square of the Cartesian lattice and depending on the variables at the corners of the quadrilateral. For such equations the property often associated with integrability is that of "multidimensional consistency" (MDC): it should be possible to extend the equation from two to higher dimensions so that the embedded two-dimensional lattice equations are compatible. Usually compatibility is checked using "Consistency-Around-a-Cube" (CAC). In this context it is often assumed that the equations on the six sides of the cube are the …
In the present paper we introduce a multi-dimensional version of the R-matrix approach to the construction of integrable hierarchies. Applying this method to the case of the Lie algebra of functions with respect to the contact bracket, we construct integrable hierarchies of (3+1)-dimensional dispersionless systems of the type recently introduced by one of us in arXiv:1401.2122.
A new set of discrete integrable equations, called face-centered quad equations, was recently obtained using new types of interaction-round-a-face solutions of the classical Yang-Baxter equation. These equations satisfy a new formulation of multidimensional consistency, known as consistency-around-a-face-centered-cube (CAFCC), which requires consistency of an overdetermined system of fourteen five-point equations on the face-centered cubic unit cell. In this paper a new formulation of CAFCC is introduced where so-called type-C equations are centered at faces of the face-centered cubic unit cel…
We prove that the cosine law for spherical triangles and spherical tetrahedra defines integrable systems, both in the sense of multidimensional consistency and in the sense of dynamical systems.
We introduce a novel systematic construction for integrable (3+1)-dimensional dispersionless systems using nonisospectral Lax pairs that involve contact vector fields. In particular, we present new large classes of (3+1)-dimensional integrable dispersionless systems associated to the Lax pairs which are polynomial and rational in the spectral parameter.
To capture a multidimensional consistency feature of integrable systems in terms of the geometry, we give a condition called \emph{geodesic compatibility} that implies the existence of integrals in involution of the geodesic flow. The geodesic compatibility condition is constructed from a concrete example namely the integrable Calogero's Goldfish system through the Poisson structure and the variational principle. The geometrical view of the geodesic compatibility gives a compatible parallel transport between two different Hamiltonian vector fields.
This paper develops the technique of constructing Lax representations for PDEs via non-central extensions generated by non-triivial exotic 2-cocycles of their contact symmetry algebras. We show that the method is applicable to the Lax representations with non-removable spectral parameters. Also we demonstrate that natural extensions of the symmetry algebras produce the integrable hierarchies associated to their PDEs.
In this letter I analyze a covering jet manifold scheme, its relation to the invariant theory of the associated vector fields, and applications to the Lax-Sato-type integrability of nonlinear dispersionless differential systems. The related contact geometry linearization covering scheme is also discussed. The devised techniques are demonstrated for such nonlinear Lax-Sato integrable equations as Gibbons-Tsarev, ABC, Manakov-Santini and the differential Toda singular manifold equations.
Commutation of multidimensional vector fields leads to integrable nonlinear dispersionless PDEs arising in various problems of mathematical physics and intensively studied in the recent literature. This report is aiming to solve the scattering and inverse scattering problem for integrable dispersionless PDEs, recently introduced just at a formal level, concentrating on the prototypical example of the Pavlov equation, and to justify an existence theorem for global bounded solutions of the associated Cauchy problem with small data.
In this work, we establish a connection between the extended Prelle-Singer procedure with other widely used analytical methods to identify integrable systems in the case of $n^{th}$-order nonlinear ordinary differential equations (ODEs). By synthesizing these methods we bring out the interlink between Lie point symmetries, contact symmetries, $λ$-symmetries, adjoint-symmetries, null forms, Darboux polynomials, integrating factors, Jacobi last multiplier and generalized $λ$-symmetries corresponding to the $n^{th}$-order ODEs. We also prove these interlinks with suitable examples. By exploiting …
We consider evolution equations for curves in the 3-dimensional sphere $S^3$ that are invariant under the group $SU(2,1)$ of pseudoconformal transformations, which preserves the standard contact structure on the sphere. In particular, we investigate how invariant evolutions of Legendrian and transverse curves induce well-known integrable systems and hierarchies at the level of their geometric invariants.
The usual propetry of s<-->t duality for scattering amplitudes, e.g. for Veneziano amplitude, is deeply connected with the 2-dimensional geometry. In particular, a simple geometric construction of such amplitudes was proposed in a joint work by this author and S.Saito (solv-int/9812016). Here we propose analogs of one of those amplitudes associated with multidimensional euclidean spaces, paying most attention to the 3-dimensional case. Our results can be regarded as a variant of "Regge calculus" intimately connected with ideas of the theory of integrable models.
Yang-Baxter (YB) map systems (or set-theoretic analoga of entwining YB structures) are presented. They admit zero curvature representations with spectral parameter depended Lax triples L1, L2, L3 derived from symplectic leaves of 2 x 2 binomial matrices equipped with the Sklyanin bracket. A unique factorization condition of the Lax triple implies a 3-dimensional compatibility property of these maps. In case L1 = L2 = L3 this property yields the se--theoretic quantum Yang-Baxter equation, i.e. the YB map property. By considering periodic 'staircase' initial value problems on quadrilateral latti…
The search for new integrable (3+1)-dimensional partial differential systems is among the most important challenges in the modern integrability theory. It turns out that such a system can be associated to any pair of rational functions of one variable in general position, as established below using contact Lax pairs introduced in arXiv:1401.2122.
A new, completely integrable, two dimensional evolution equation is derived for an ion acoustic wave propagating in a magnetized, collisionless plasma. The equation is a multidimensional generalization of a modulated wavepacket with weak transverse propagation, which has resemblance to nonlinear Schrodinger (NLS) equation and has a connection to Kadomtsev-Petviashvili equation through a constraint relation. Higher soliton solutions of the equation are derived through Hirota bili- nearization procedure, and an exact lump solution is calculated exhibiting 2D structure. Some mathe- matical proper…
We use the consistency approach to classify discrete integrable 3D equations of the octahedron type. They are naturally treated on the root lattice $Q(A_3)$ and are consistent on the multidimensional lattice $Q(A_N)$. Our list includes the most prominent representatives of this class, the discrete KP equation and its Schwarzian (multi-ratio) version, as well as three further equations. The combinatorics and geometry of the octahedron type equations are explained. In particular, the consistency on the 4-dimensional Delaunay cells has its origin in the classical Desargues theorem of projective g…
We derive and analyze a three dimensional model of a figure skater. We model the skater as a three-dimensional body moving in space subject to a non-holonomic constraint enforcing movement along the skate's direction and holonomic constraints of continuous contact with ice and pitch constancy of the skate. For a static (non-articulated) skater, we show that the system is integrable if and only if the projection of the center of mass on skate's direction coincides with the contact point with ice and some mild (and realistic) assumptions on the directions of inertia's axes. The integrability is …
A generalization of the notion of a (pseudo-) Riemannian space is proposed in a framework of noncommutative geometry. In particular, there are parametrized families of generalized Riemannian spaces which are deformations of classical geometries. We also introduce harmonic maps on generalized Riemannian spaces into Hopf algebras and make contact with integrable models in two dimensions.
The pentagram map was introduced by R. Schwartz in 1992 and is now one of the most renowned discrete integrable systems. In the present paper we prove that this map, as well as all its known integrable multidimensional generalizations, can be seen as refactorization-type mappings in the Poisson-Lie group of pseudo-difference operators. This brings the pentagram map into the rich framework of Poisson-Lie groups, both describing new structures and simplifying and revealing the origin of its known properties. In particular, for multidimensional pentagram maps the Poisson-Lie group setting provide…
We develop a new method for the construction of one-dimensional integrable Lindblad systems, which describe quantum many body models in contact with a Markovian environment. We find several new models with interesting features, such as annihilation-diffusion processes, a mixture of coherent and classical particle propagation, and a rectified steady state current. We also find new ways to represent known classical integrable stochastic equations by integrable Lindblad operators. Our method can be extended to various other situations and it establishes a structured approach to the study of solva…
We describe the point and contact equivalence groupoids of an important class of two-dimensional quasilinear hyperbolic equations. In particular, we prove that this class is normalized in the usual sense with respect to point transformations, and its contact equivalence groupoid is generated by the first-order prolongation of its point equivalence groupoid, the contact vertex group of the wave equation and a family of contact admissible transformations between trivially Darboux-integrable equations.
