Higher genus hyperelliptic reductions of the Benney equations

Abstract

Abstract. It was shown by Gibbons and Tsarev (1996 Phys. Lett. A 211 19, 1999 Phys. Lett. A 258 263) that N -parameter reductions of the Benney equations correspond to N -parameter families of conformal maps. Here, we consider a specific set of these, the hyperelliptic reductions. The mapping function for this is calculated explicitly by inverting a second-kind Abelian integral on the stratum Θ 1 of the Jacobi variety of a genus g (g ≥ 3) hyperelliptic curve. This is done using a method based on the result of Jorgenson (1992 Israel Journal of Mathematics 77 273).

On symmetries of the Gibbons–Tsarev equation

“…Then S is a symmetry ofĒ * if and only if S (z yy + z x z xy − z y z xx + 1) = 0, S (ϕ (i) x −X (i) ) = 0, S (ϕ (i) y −Ȳ (i) ) = 0 (37) modulo equations (3) and (23). Using formulas (29), variables x, y, z can be expressed in terms of ϕ (1) , ϕ (2) , ϕ (3) . Consequently, we can rewrite (36) and (37) in terms of ϕ (i) and z Ξ , |Ξ| > 0, alone.…”

“…If n ≥ 6, then the coefficients f (1) , f (2) , f (3) are zero. Obviously, Z = 0 and we obtain the invisible symmetries…”

Generalised elliptic functions

“…Large expansions of this type were first introduced in [11], in which used the generalised σ-function to construct explicit reductions of the Benney equations, (see also [9], [10] and [23]). Since then they have been an integral tool in the investigation of Abelian functions.…”

Relationship between the prime form and the sigma function for some cyclic (r,s) curves

“…In section 7, the Prime form is introduced; the Frobenius-Stickelberger results are then used to give a simple expression for this in terms of derivatives of σ. One of the motivations of this study was to permit the explicit construction of Schwartz-Christoffel maps giving formulae for certain reductions of the Benney hierarchy [BG1,BG2,G]; we will discuss this briefly in the last section.…”