Classification of Integrable (2+1)-Dimensional Quasilinear Hierarchies

Ferapontov, E. V.; Хуснутдинова, К. Р.; Pavlov, Maxim V.

doi:10.1007/s11232-005-0117-7

Cited by 24 publications

(38 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We introduce N field variables a k such that ∂ i a k = H (k)i H (1)i (i.e., a k ≡ a k1 ; see (19) and (20)). Then algebraic system (16) for Egorov systems of hydrodynamic type (15) ∂ t k a n = ∂ t 1 a kn (a) (22) can be written in the form…”

Section: The Generalized Hodograph Methodsmentioning

confidence: 99%

See 1 more Smart Citation

Integrability of the Egorov systems of hydrodynamic type

Pavlov

2007

Theor Math Phys

Self Cite

View full text Add to dashboard Cite

show abstract

Section: The Generalized Hodograph Methodsmentioning

confidence: 99%

“…If the rotation coefficients β ik are symmetric (see (17)), then Egorov systems of hydrodynamic type (15) can be written together in symmetric potential form (19).…”

Section: Conjugate Curvilinear Coordinate Systemsmentioning

confidence: 99%

Integrability of the Egorov systems of hydrodynamic type

Pavlov

2007

Theor Math Phys

Self Cite

View full text Add to dashboard Cite

show abstract

“…To illustrate this procedure we consider a simple example g = u 3 − 1 2 (u 1 ) 2 which automatically satisfies (15). The corresponding h, as specified by (14), is given by h = µ + αu 1 + βu 2 + γu 3 + δu 4 − u 1 u 2 , which can be reduced to a canonical form h = u 4 − u 1 u 2 by redefining u 4 appropriately (this transformation freedom allows one to absorb arbitrary integration constants arising at each step of the construction). Thus, the first three equations of the chain are…”

Section: Introductionmentioning

confidence: 99%

“…The existence of such Lax pairs was established in [40,14] for any equation (16) provided g satisfies the compatibility conditions (15).…”

Section: Introductionmentioning

confidence: 99%

Differential-geometric approach to the integrability of hydrodynamic chains: the Haantjes tensor

2007

Self Cite

View full text Add to dashboard Cite

The integrability of an m-component system of hydrodynamic type, u t = V (u)u x , by the generalized hodograph method requires the diagonalizability of the m × m matrix V (u). This condition is known to be equivalent to the vanishing of the corresponding Haantjes tensor. We generalize this approach to hydrodynamic chains -infinite-component systems of hydrodynamic type for which the ∞×∞ matrix V (u) is 'sufficiently sparse'. For such systems the Haantjes tensor is well-defined, and the calculation of its components involves finite summations only. We illustrate our approach by classifying broad classes of conservative and Hamiltonian hydrodynamic chains with the zero Haantjes tensor. We prove that the vanishing of the Haantjes tensor is a necessary condition for a hydrodynamic chain to possess an infinity of semi-Hamiltonian hydrodynamic reductions, thus providing an easy-to-verify necessary condition for the integrability.MSC: 35L40, 35L65, 37K10.

show abstract

“…in both above equations one can obtain (2) and the infinite series of conservation laws for ∂A m = 0, m > k + 1 (8) and corresponding 2+1 quasilinear equations (see, for instance, (4)) was found in [10], [11], [9], [12], [19], [23], [32]). 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%