Number Theory and the Symmetry Classification of Integrable Systems

Zero-curvature representations (ZCRs) are one of the main tools in the theory of integrable PDEs. In particular, Lax pairs for (1+1)-dimensional PDEs can be interpreted as ZCRs.In [arXiv:1303.3575], for any (1+1)-dimensional scalar evolution equation E, we defined a family of Lie algebras F(E) which are responsible for all ZCRs of E in the following sense. Representations of the algebras F(E) classify all ZCRs of the equation E up to local gauge transformations. In [arXiv:1804.04652] we showed that, using these algebras, one obtains necessary conditions for existence of a Bäcklund transformation between two given equations. The algebras F(E) are defined in [arXiv:1303.3575] in terms of generators and relations.In this preprint we show that, using the algebras F(E), one obtains some necessary conditions for integrability of the considered PDEs, where integrability is understood in the sense of soliton theory. Using these conditions, we prove non-integrability for some scalar evolution equations of order 5. Also, we prove a result announced in [arXiv:1303.3575] on the structure of the algebras F(E) for certain classes of equations of orders 3, 5, 7, which include KdV, mKdV, Kaup-Kupershmidt, Sawada-Kotera type equations.In this approach, ZCRs may depend on partial derivatives of arbitrary order, which may be higher than the order of the equation E. The algebras F(E) generalize Wahlquist-Estabrook prolongation algebras, which are responsible for a much smaller class of ZCRs.

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“…So the set V 0 is empty. Using formulas (34), (35), we see that properties (25), (26), (27) are equivalent to…”

Section: The Algebras F P (E A)mentioning

confidence: 99%

On Lie algebras responsible for integrability of (1+1)-dimensional scalar evolution PDEs

Igonin

Manno

2020

Journal of Geometry and Physics

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“…To simplify computations we shall use the so-called symbolic transform (see e.g. [8,9,26] and references therein), i.e., we replace a matrix differential operator R = j R j [u]D j x with the formal series R = j R j [u]z j and employ the formula…”

Section: Resultsmentioning

confidence: 99%

On symmetries and conservation laws of the Majda–Biello system

Vodova-Jahnova

2015

Nonlinear Analysis: Real World Applications

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In 2003, A.J. Majda and J.A. Biello derived and studied the so-called reduced equations for equatorial baroclinic-barotropic waves, to which we refer as to the Majda-Biello system. The equations in question describe the nonlinear interaction of long-wavelength equatorial Rossby waves and barotropic Rossby waves with a significant midlatitude projection in the presence of suitable horizontally and vertically sheared zonal mean flows.Below we present a Hamiltonian structure for Majda-Biello system and describe all generalized symmetries and conservation laws for the latter. It turns out that there are only three symmetries corresponding to x-translations, t-translations and to a scaling of t, x, u and v, and four conservation laws, one of which is associated to the conservation of energy, the second conserved quantity is just the Hamiltonian functional and the other two are Casimir functionals of the Hamiltonian operator admitted by our system. Our result provides inter alia a rigorous proof of the fact that the Majda-Biello system has just the conservation laws mentioned in the paper by Majda and Biello.2010 Mathematics Subject Classification. 37K05, 37K10.

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“…, u p−1 ) in the proof of Theorem 1 implies that G is analytic as well. Then the g-valued functionsÃ,B given by (25) are also analytic on a neighborhood of a ∈ E.…”

Section: Zcrs Gauge Transformations and The Algebras F P (E A)mentioning

confidence: 99%

Lie algebras responsible for zero-curvature representations of scalar evolution equations

Igonin

Manno

2019

Journal of Geometry and Physics

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Zero-curvature representations (ZCRs) are one of the main tools in the theory of integrable PDEs. In particular, Lax pairs for (1+1)-dimensional PDEs can be interpreted as ZCRs.For any (1+1)-dimensional scalar evolution equation E, we define a family of Lie algebras F(E) which are responsible for all ZCRs of E in the following sense. Representations of the algebras F(E) classify all ZCRs of the equation E up to local gauge transformations. To achieve this, we find a normal form for ZCRs with respect to the action of the group of local gauge transformations.As we show in other publications, using these algebras, one obtains some necessary conditions for integrability of the considered PDEs (where integrability is understood in the sense of soliton theory) and necessary conditions for existence of a Bäcklund transformation between two given equations. Examples of proving non-integrability and applications to obtaining non-existence results for Bäcklund transformations are presented in other publications as well.In our approach, ZCRs may depend on partial derivatives of arbitrary order, which may be higher than the order of the equation E. The algebras F(E) generalize Wahlquist-Estabrook prolongation algebras, which are responsible for a much smaller class of ZCRs.In this paper we describe general properties of F(E) and present generators and relations for these algebras. In other publications we study the structure of F(E) for equations of KdV, Krichever-Novikov, Kaup-Kupershmidt, Sawada-Kotera types. Among the obtained algebras, one finds infinite-dimensional Lie algebras of certain matrix-valued functions on rational and elliptic algebraic curves.2010 Mathematics Subject Classification. 37K30, 37K35.

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Number Theory and the Symmetry Classification of Integrable Systems

Cited by 9 publications

References 26 publications

On Lie algebras responsible for integrability of (1+1)-dimensional scalar evolution PDEs

On Lie algebras responsible for integrability of (1+1)-dimensional scalar evolution PDEs

On symmetries and conservation laws of the Majda–Biello system

Lie algebras responsible for zero-curvature representations of scalar evolution equations

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