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Register a new userCritical points, Lauricella functions and Whitham-type equations
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Added by
Boris Konopelchenko
Publication date
2014
fields
Physics
and research's language is
English
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A large class of semi-Hamiltonian systems of hydrodynamic type is interpreted as the equations governing families of critical points of functions obeying the classical linear Darboux equations for conjugate nets.The distinguished role of the Euler-Poisson-Darboux equations and associated Lauricella-type functions is emphasised. In particular, it is shown that the classical g-phase Whitham equations for the KdV and NLS equations are obtained via a g-fold iterated Darboux-type transformation generated by appropriate Lauricella functions.
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Based on the well-established theory of discrete conjugate nets in discrete differential geometry, we propose and examine discrete analogues of important objects and notions in the theory of semi-Hamiltonian systems of hydrodynamic type. In particular, we present discrete counterparts of (generalised) hodograph equations, hyperelliptic integrals and associated cycles, characteristic speeds of Whitham type and (implicitly) the corresponding Whitham equations. By construction, the intimate relationship with integrable system theory is maintained in the discrete setting.
We prove that integrability of a dispersionless Hirota type equation implies the symplectic Monge-Ampere property in any dimension $geq 4$. In 4D this yields a complete classification of integrable dispersionless PDEs of Hirota type through a list of heavenly type equations arising in self-dual gravity. As a by-product of our approach we derive an involutive system of relations characterising symplectic Monge-Ampere equations in any dimension. Moreover, we demonstrate that in 4D the requirement of integrability is equivalent to self-duality of the conformal structure defined by the characteristic variety of the equation on every solution, which is in turn equivalent to the existence of a dispersionless Lax pair. We also give a criterion of linerisability of a Hirota type equation via flatness of the corresponding conformal structure, and study symmetry properties of integrable equations.
The Laplacian growth (the Hele-Shaw problem) of multi-connected domains in the case of zero surface tension is proven to be equivalent to an integrable systems of Whitham equations known in soliton theory. The Whitham equations describe slowly modulated periodic solutions of integrable hierarchies of nonlinear differential equations. Through this connection the Laplacian growth is understood as a flow in the moduli space of Riemann surfaces.
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We prove sharp pointwise decay estimates for critical Dirac equations on $mathbb{R}^n$ with $ngeq 2$. They appear for instance in the study of critical Dirac equations on compact spin manifolds, describing blow-up profiles, and as effective equations in honeycomb structures. For the latter case, we find excited states with an explicit asymptotic behavior. Moreover, we provide some classification results both for ground states and for excited states.
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