Symbolic representation and classification of integrable systems

Михайлов, А. В.; Новиков, В. С.; Wang, Jing Ping

doi:10.1017/cbo9780511721564.006

Cited by 29 publications

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“…In this section we briefly recall the basic definitions and notations of the perturbative symmetry approach (for details see [20,21]). We also present the integrability test which we will subsequently apply to isolate integrable generalizations of the Camassa-Holm equation.…”

Section: Integrability Test: Perturbative Symmetriesmentioning

confidence: 99%

Two-component generalizations of the Camassa–Holm equation

2017

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Recommended by Tamara Grava AbstractA classification of integrable two-component systems of non-evolutionary partial differential equations that are analogous to the Camassa-Holm equation is carried out via the perturbative symmetry approach. Independently, a classification of compatible pairs of Hamiltonian operators of specific forms is carried out, in order to obtain bi-Hamiltonian structures for the same systems of equations. Using reciprocal transformations, some exact solutions and Lax pairs are also constructed for the systems considered.

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Section: Integrability Test: Perturbative Symmetriesmentioning

confidence: 99%

Two-component generalizations of the Camassa–Holm equation

2017

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“…or consult one of the papers [18,21,39]. All usual operations from differential algebra translate naturally.…”

Section: The Gel'fand-dikiȋ Transformationmentioning

confidence: 99%

Global Classification of Two-Component Approximately Integrable Evolution Equations

Kamp

2009

Found Comput Math

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“…We obtain compatible Hamiltonian and symplectic structure for a new twocomponent fifth-order integrable system recently found by Mikhailov, Novikov and Wang, and show that this system possesses a hereditary recursion operator and infinitely many commuting symmetries and conservation laws, as well as infinitely many compatible Hamiltonian and symplectic structures, and is therefore completely integrable. The system in question admits a reduction to the Kaup-Kupershmidt equation.PACS 02.30.Ik In the course of an ongoing classification of integrable polynomial evolution systems in two independent and two dependent variables Mikhailov, Novikov and Wang [8] (see also [9]) have found a systemHere and belowNote that (1) is one of just two new integrable systems found in [8], and therefore it is natural to explore its properties in order to find out whether it enjoys any features not present in the other higher-order integrable systems.Upon setting v ≡ 0 the system (1) reduces [8] to the well-known Kaup-Kupershmidt equation, see e.g. [4,15] and references therein for more details on the latter.…”

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“…In the course of an ongoing classification of integrable polynomial evolution systems in two independent and two dependent variables Mikhailov, Novikov and Wang [8] (see also [9]) have found a system u t = − …”

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On complete integrability of the Mikhailov–Novikov–Wang system

Vojčák

2011

Journal of Mathematical Physics

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Abstract. We obtain compatible Hamiltonian and symplectic structure for a new twocomponent fifth-order integrable system recently found by Mikhailov, Novikov and Wang, and show that this system possesses a hereditary recursion operator and infinitely many commuting symmetries and conservation laws, as well as infinitely many compatible Hamiltonian and symplectic structures, and is therefore completely integrable. The system in question admits a reduction to the Kaup-Kupershmidt equation.PACS 02.30.Ik In the course of an ongoing classification of integrable polynomial evolution systems in two independent and two dependent variables Mikhailov, Novikov and Wang [8] (see also [9]) have found a systemHere and belowNote that (1) is one of just two new integrable systems found in [8], and therefore it is natural to explore its properties in order to find out whether it enjoys any features not present in the other higher-order integrable systems.Upon setting v ≡ 0 the system (1) reduces [8] to the well-known Kaup-Kupershmidt equation, see e.g. [4,15] and references therein for more details on the latter.Using the so-called symbolic method Mikhailov et al. [8] proved that the system (1) possesses infinitely many generalized symmetries (in the sense of [10]) of orders m ≡ 1, 5 mod 6. However, this result alone neither provides an explicit construction for the symmetries in question nor does it necessarily entail the existence of a recursion operator, see e.g. [1,14] and references therein. On the other hand, no recursion operator, symplectic or (bi-)Hamiltonian structure for (1) was found so far. In view of the important role played by these quantities in establishing integrability, see e.g. [2,10] and references therein, it is natural to ask whether (1) admits any such quantities at all, and it is the goal of the present paper to show that this is indeed the case and thus (1) indeed is a completely integrable system.To this end we have first found a few low-order symmetries and cosymmetries of (1) and subsequently constructed nonlocal parts of the operators in question (and then the operators per se) using these quantities, cf. e.g. [5,7,11]. As a result, we arrive at the following assertion.

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Symbolic representation and classification of integrable systems

Cited by 29 publications

References 55 publications

Two-component generalizations of the Camassa–Holm equation

Two-component generalizations of the Camassa–Holm equation

Global Classification of Two-Component Approximately Integrable Evolution Equations

On complete integrability of the Mikhailov–Novikov–Wang system

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