Conformal maps and reductions of the Benney equations

Hyperelliptic reduction of the Benney moment equations

“…Its asymptotics are given as follows Similar expressions may be found in principle for higher moments A n . If λ(p) is required to satisfy Benney's equation (10) and to be of this form, then theλ n satisfy ∂λ n ∂t +p n ∂λ n ∂x = 0 (n = 1, 2, 3).…”

“…If we now hold p constant, this gives ∂λ ∂t + p ∂λ ∂x − ∂A 0 ∂x ∂λ ∂p = 0 (10) which is a Vlasov equation of the same form as (4). Thus (4) and (10) have the same characteristics.…”

“…However we will now consider families of distribution functions f , which are parameterised by finitely many N Riemann invariantsλ i (x, t). We are interested in the case [9], [10] where the function λ(p, x, t) is such that only N of the moments are independent. Then there are N characteristic speeds, assumed real and distinct, and N corresponding Riemann invariants (p i ,λ i ), so Benney's equations reduce to a diagonal system of hydrodynamic type with finitely many dependent variablesλ i ,…”

“…The construction of a more general family of solutions for equations of this type was outlined in [9] and [10]. Instead of considering the principal value integral (7), we now define a new function λ + (x, p, t) :…”

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“…[1]- [10]) . In this sense, it is worth-mentioning their connections [11,12] with some classical problems of the theory of conformal maps. Furthermore, we have recently proposed a∂-method [13]- [14] for studying dispersionless integrable hierarchies which reveals an intimate relation between these hierarchies and the theory of quasiconformal mappings [15]- [18].…”