Hyperelliptic reduction of the Benney moment equations

Abstract

Abstract. We consider N -parameter reductions of the Benney moment equations. These were shown in Gibbons and Tsarev (1996 Phys. Lett. A 211 19, 1999 Phys. Lett. A 258 263 ) to correspond to N −parameter families of conformal maps and to satisfy a particular system of PDE. A specific known example of this, the (N = 2) elliptic reduction (L Yu and J Gibbons 2000 Inverse Problems 16 605 ) is described. We then consider an analogous reduction for a genus 2 hyperelliptic curve (N = 3). The mapping function is given by the inversion of a 2nd kind Abelian integral on the Θ−divisor. This is found explicitly following a method given by Enolskii, Pronine and Richter (2003 J. Nonlinear Science 13 157).

Higher genus hyperelliptic reductions of the Benney equations

“…Following [14] and [1], we will now consider reductions of the moment equations; that is the case where only a finite number, N, of the A n are independent. Here, the moment equations can be reduced to a diagonal system of hydrodynamic type with N Riemann invariants,λ i say, dependent on N characteristic speeds,p i .…”

“…The case of the n = 2 elliptic reduction was evaluated in [14] by Yu and Gibbons. The n = 3 genus 2 hyperelliptic reduction was studied in [1] by the authors. We now consider the case for g ≥ 3.…”

Integrable Equations in Nonlinear Geometrical Optics

“…, are admissible (the constant terms which have appeared in the cases (28) and (32) discussed above are, in fact, irrelevant). Hence, these polynomial deformations are given by…”

“…The dVN equation (32) and the whole dVN hierarchy have an infinite set of integrals of motion [15]. The simplest of them is given by…”

Wannier Functions for Quasiperiodic Finite-Gap Potentials

“…Higher genera theory was developed by K.Weierstrass and F.Klein and its exposition is fixed in classical monograph of H.Baker [2]; recent results in the area are given in [12,13,14,15] whilst various applications see e.g. in [18], [36], [34], [35], [19], [5], [6].…”