Multidimensional Integrable Systems Research Articles

Multidimensional conservation laws and integrable systems II

AbstractIn this paper we continue investigation of a new property of two‐dimensional integrable systems—existence of infinitely many local three‐dimensional conservation laws for pairs of integrable two‐dimensional commuting flows. Multicomponent two‐dimensional hydrodynamic reductions of the Mikhalëv equation are considered. Infinitely many three‐dimensional local conservation laws for the Korteweg–de Vries pair of commuting flows are constructed. Thus, we show that pairs of commuting dispersive two‐dimensional systems also possess infinitely many local three‐dimensional conservation laws. They can be used for averaging of multiparametric families of solutions to the Mikhalëv equation.

Dispersionless Multi-Dimensional Integrable Systems and Related Conformal Structure Generating Equations of Mathematical Physics

Using diffeomorphism group vector fields on $\mathbb{C}$-multiplied tori and the related Lie-algebraic structures, we study multi-dimensional dispersionless integrable systems that describe conformal structure generating equations of mathematical physics. An interesting modification of the devised Lie-algebraic approach subject to spatial-dimensional invariance and meromorphicity of the related differential-geometric structures is described and applied in proving complete integrability of some conformal structure generating equations. As examples, we analyze the Einstein-Weyl metric equation, the modified Einstein-Weyl metric equation, the Dunajski heavenly equation system, the first and second conformal structure generating equations and the inverse first Shabat reduction heavenly equation. We also analyze the modified Pleba\'nski heavenly equations, the Husain heavenly equation and the general Monge equation along with their multi-dimensional generalizations. In addition, we construct superconformal analogs of the Whitham heavenly equation.

Multi-dimensional compactons and compact vortices

As though to compensate for the rarity of multidimensional integrable systems, non-integrable spatial extensions of many of the well known dispersive equations on the line exhibit a remarkable variety of solitary patterns unavailable in 1D. In the present work we elaborate upon the ubiquity of this effect and find it to be even more pronounced in compacton admitting systems which admit new patterns otherwise unavailable, in both the real (the sublinear ZK) and the complex realm (the sublinear NLS and complex KG). In the planar complex case, in addition to a countable number of radial solutions, for every spin number m we find both analytical and numerical evidence of a countable number of multi-modal compact vortices which have a strictly finite radius and, depending on m, either extend to the origin or vanish identically in a disk around it.

Recursion operators for dispersionless integrable systems in any dimension

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.

Multidimensional integrable systems and deformations of Lie algebra homomorphisms

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(S1).

U V Manifold and Integrable Systems in Spaces of Arbitrary Dimension

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.

Volume Preserving Multidimensional Integrable Systems and Nambu–Poisson Geometry

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 geometry for multidimensional integrable systems

A deformed differential calculus is developed based on an associative ★-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 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 correspond to taking the dispersionless limit in these hierarchies.

Integrability for multidimensional lattice models

The generating principle for the algebraic construction of the hierarchy of the d-simplex equations generalizing the Yang-Baxter equation in any dimension d is given. Following this principle, we construct the generalization of the Lax equations for multidimensional integrable systems. We show that this contribution leads, as in d=2, to the existence of an infinite series of conserved charges for such models. In the d=3 case a reinterpretation of these results is given in terms of functionals of loops, leading to the notion of extended gauge symmetry and gauge fields associated to quantum groups.

The action variable in dissipattvely perturbed hamiltonian systems

The action variable of a conservative one-dimensional Hamiltonian system decays exponentially when a linear disapative perturbation is added, irrespective of the Hamiltonian. In multidimensional integrable systems the product of the actions decays exponentially. Various features of these results are discussed, including the connection to adiabatic invariance and semiclassical implications.

Multidimensional nonlinear integrable systems and methods for constructing their solutions

A new method for constructing multidimensional nonlinear integrable systems and their solutions by means of a nonlocal Riemann problem is presented. This is the natural generalization of the method of the local Riemann problem to the case of several space variables and includes the well-known Zakharov-Shabat method of dressing by Volterra operators.