Second Order Reductions of the WDVV Equations Related to Classical Lie Algebras

Hoevenaars, L. K.; Martini, Ruud

doi:10.1007/s11005-004-6029-x

Cited by 2 publications

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References 15 publications

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“…A generalization of this construction to multiply-connected domains [34] leads to a remarkable hierarchy of Hirota-like equations which constitute the so-called universal Whitham hierarchy [33]. Further examples of Hirota-type relations arise in the theory of the associativity (WDVV) equations [8,10,30], e.g., e uxx+uxy+uxt−uyt − e uxy+uyy+uyt−uxt + e uxt+uyt+utt−uxy = 0, e uxx+2uxt+uxy − e utt+2uxt+uyt = 1, e utt−uxx = sinh u xy / sinh u yt , coth u xy = coth u xt coth u yt , etc. These equations arise as certain differential constraints which should be considered along with a full set of the WDVV equations.…”

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confidence: 99%

Integrable Equations of the Dispersionless Hirota type and Hypersurfaces in the Lagrangian Grassmannian

Ferapontov

Hadjikos

Хуснутдинова

2009

International Mathematics Research Notices

119

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We investigate integrable second order equations of the form F (u xx , u xy , u yy , u xt , u yt , u tt ) = 0.Familiar examples include the Boyer-Finley equation u xx + u yy = e utt , the potential form of the dispersionless Kadomtsev-Petviashvili (dKP) equation u xt − 1 2 u 2 xx = u yy , the dispersionless Hirota equation (α − β)e uxy + (β − γ)e uyt + (γ − α)e utx = 0, etc. The integrability is understood as the existence of infinitely many hydrodynamic reductions. We demonstrate that the natural equivalence group of the problem is isomorphic to Sp(6), revealing a remarkable correspondence between differential equations of the above type and hypersurfaces of the Lagrangian Grassmannian. We prove that the moduli space of integrable equations of the dispersionless Hirota type is 21-dimensional, and the action of the equivalence group Sp(6) on the moduli space has an open orbit. MSC: 35Q58, 37K25, 53A40, 53B25, 53Z05.

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