New Degeneration of Fay's Identity and its Application to Integrable Systems

Abstract

In this paper we prove a new degenerated version of Fay's trisecant identity. The new identity is applied to construct new algebro-geometric solutions of the multi-component nonlinear Schrödinger equation. This approach also provides an independent derivation of known algebro-geometric solutions to the Davey-Stewartson equations.

New construction of algebro-geometric solutions to the Camassa–Holm equation and their numerical evaluation

“…In what follows we choose a canonical homology basis in H 1 (R g ) satisfying (2.14) and take a, b ∈ R g such that τ a = b. Denote by a contour connecting the points a and b which does not intersect the canonical homology basis. Then the action of τ on the generators (A, B, ) of the relative homology group H 1 (R g , {a, b}) is given by (see [27] for more details)…”

“…for some N ∈ Z g . In the case where τ a = a and τ b = b, the action of τ on the generators (A, B, ) of the relative homology group H 1 (R g , {a, b}) reads (see [27] for more details)…”

“…Let a, b ∈ R g and denote by {A, B, S b } the generators of the homology group H 1 (R g \ {a, b}) of the punctured Riemann surface R g \{a, b}, where S b is a positively oriented small contour around b such that S b • = 1. It was proved in [27] that in the case where τ a = b, the A-cycles in the homology group H 1 (R g \ {a, b}) are stable under τ . Therefore, by the uniqueness of the normalized differential of the third kind Ω b−a which has residue 1 at b and residue −1 at a, we get…”

Transverse stability of periodic traveling waves in Kadomtsev-Petviashvili equations: A numerical study

“…3. In particular there seems to be a 'breathing' structure similar to the so-called breathers in the nonlinear Schrödinger equation (NLS), see, e.g., [12] (for the relation between NLS and KP solutions see [9,21]).…”

Computation of the topological type of a real Riemann surface

“…However, the choice of a basis, where certain cycles are invariant under the automorphisms, is often convenient in applications. In the context of solutions to integrable PDEs on general compact Riemann surfaces as for the Kadomtsev-Petviashvili (KP) (see [13]) and the Davey-Stewartson (DS) equations (see [30,26]), smoothness conditions are formulated conveniently in a homology basis adapted to the anti-holomorphic involution τ defined on the surface. For instance, on a real surface there exists a canonical homology basis (A, B) (that we call for simplicity symmetric homology basis in the following) satisfying the conditions (see [35,38])…”