Reductions and hodograph solutions of the dispersionless KP hierarchy

Mañas, Manuel; Alonso, Luis Martı́nez; Медина, Елена

doi:10.1088/0305-4470/35/2/316

Cited by 54 publications

(105 citation statements)

References 16 publications

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“…i = j, which was first derived in [17,18] in the theory of hydrodynamic reductions of Benney's moment equations, see also [21,29,30] for further discussion. For any solution µ i , v of the system (16) one can reconstruct λ i and w by virtue of (13).…”

Section: Examplesmentioning

confidence: 99%

“…Solutions of this type, known as nonlinear interactions of n planar simple waves, were extensively investigated in gas dynamics and magnetohydrodynamics in a series of publications [5,6,7,36,37,9,20]. Later, they appeared in the context of the dispersionless KP hierarchy [15,16,17,18,21,31,29,30] and the theory of integrable hydrodynamictype chains [34,35]. We will call a multidimensional system integrable if it possesses 'sufficiently many' n-component reductions of the form (2) for arbitrary n (the precise definition follows).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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: Examplesmentioning

confidence: 99%

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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show abstract

“…The system (2) has been thoroughly investigated in the subsequent publications [38,27,28,33,52,53,49,17] where, in particular, its Hamiltonian and integrability aspects were uncovered. Hydrodynamic reductions of the chain (2) were studied in [18,19,20,26,36].…”

Section: Introductionmentioning

confidence: 99%

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

2007

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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.

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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%

“…for any hydrodynamic chain determined by the the generating function of conservation laws (see (22) and (26)) p t = ∂ x ψ(u; p), one can seek in the form…”

Section: Explicit Hamiltonian Hydrodynamic Reductionsmentioning

confidence: 99%