Classical<i>R</i>-matrix theory of dispersionless systems: II. (2   1) dimension theory

B aszak, Maciej; Szablikowski, Błażej M.

doi:10.1088/0305-4470/35/48/309

Cited by 39 publications

(65 citation statements)

References 19 publications

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“…[28]. Although many interesting systems of the form (1) arise as dispersionless limits of multidimensional soliton equations [39] or within the R-matrix approach [3] and, therefore, should be regarded as integrable, no intrinsic definition of the integrability for multidimensional quasilinear systems has been proposed until recently. In particular, the standard symmetry approach [32,33], which proves to be extremely effective in the case of higher order evolution equations and systems, does not seem to work in this context.…”

Section: Introductionmentioning

confidence: 99%

On the Integrability of (2+1)-Dimensional Quasilinear Systems

Ferapontov

Хуснутдинова

2004

Commun. Math. Phys.

177

295

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A (2+1)-dimensional quasilinear system is said to be 'integrable' if it can be decoupled in infinitely many ways into a pair of compatible n-component onedimensional systems in Riemann invariants. Exact solutions described by these reductions, known as nonlinear interactions of planar simple waves, can be viewed as natural dispersionless analogs of n-gap solutions. It is demonstrated that the requirement of the existence of 'sufficiently many' n-component reductions provides the effective classification criterion. As an example of this approach we classify integrable (2+1)-dimensional systems of conservation laws possessing a convex quadratic entropy.MSC: 35L40, 35L65, 37K10.

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Section: Introductionmentioning

confidence: 99%

On the Integrability of (2+1)-Dimensional Quasilinear Systems

Ferapontov

Хуснутдинова

2004

Commun. Math. Phys.

177

295

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“…which is equivalent to the zero curvature condition 2 = 0, h d makes it possible to retrieve [23,45] the corresponding connection ϒ by constructing a mapping…”

Section: Remark 22mentioning

confidence: 99%

The Novel Lie-Algebraic Approach to Studying Integrable Heavenly Type Multi-Dimensional Dynamical Systems

Blackmore¹,

Eo²,

Ak³

2017

J Generalized Lie Theory Appl

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The review is devoted to a novel Lie-algebraic approach to studying integrable heavenly type multi-dimensional dynamical systems and its relationships to old and recent investigations of the classical Buhl problem of describing compatible linear vector field equations, its general Pfeiffer and modern Lax-Sato type special solutions. Eespecially we analyze the related Lie-algebra structures and integrability properties of a very interesting class of nonlinear dynamical systems called the dispersionless heavenly type equations, which were initiated by Pleban′ski and later analyzed in a series of articles. The AKS-algebraic and related -structure schemes are used to study the orbits of the corresponding co-adjoint actions, which are intimately related to the classical Lie-Poisson structures on them. It is demonstrated that their compatibility condition coincides with the corresponding heavenly type equations under consideration. It is shown that all these equations originate in this way and can be represented as a Lax compatibility condition for specially constructed loop vector fields on the torus. The infinite hierarchy of conservations laws related to the heavenly equations is described, and its analytical structure connected with the Casimir invariants, is mentioned. In addition, typical examples of such equations, demonstrating in detail their integrability via the scheme devised herein, are presented. The relationship of a very interesting Lagrange-d'Alembert type mechanical interpretation of the devised integrability scheme with the Lax-Sato equations is also discussed.

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“…Recently some of these integrable hydrodynamic chains were rediscovered (see [2]) and studied in [2], [4], [5], [22], [25], [24]). …”

Section: the First Commuting Flow Ismentioning

confidence: 99%