Real Theta-Function Solutions of the Kadomtsev–petviashvili Equation

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

“…Following the work by Dubrovin and Natanzon [6] on smoothness of algebro-geometric solutions of the Kadomtsev Petviashvili (KP1) equation in the case where R g admits real ovals we get Proposition 3.3. Let R g be a hyperelliptic curve of genus g which admits an anti-holomorphic involution τ .…”

“…In this part, we review known results [20], [6] about the theta divisor of real Riemann surfaces. Let us choose the canonical homology basis satisfying (A.2) and consider the Jacobian J = J(R g ) of the real Riemann surface R g .…”

“…The first is to increase the number of spatial dimensions to two. This leads to the Davey-Stewartson equations (DS), i ψ t + ψ xx − α 2 ψ yy + 2 (Φ + ρ |ψ| 2 ) ψ = 0, 6) where α = i, 1 and ρ = ±1; ψ(x, y, t) and Φ(x, y, t) are functions of the real variables x, y and t, the latter being real valued and the former being complex valued. In what follows, DS1 ρ denotes the Davey-Stewartson equation when α = i, and DS2 ρ the Davey-Stewartson equation when α = 1.…”

Finite-gap solutions of the Davey-Stewartson equations

“…In particular, we shall give sufficient conditions for the regularity of these solutions. It is important to emphasize that all these restrictions for the reductions (a), (b), (d), (e) are similar to those used for the reductions of KP to KPI and KP2 (see [7], [8]). But for DS2-the construction has no analogue in the KP equation theory.…”

30 years of finite-gap integration theory

“…Of course, this 'non-spectral' approach produces in general complex-valued and singular solutions. Isolating the smooth real-valued solutions needed separate non-trivial work, which was mainly done by Dubrovin & Natanzon (1989). The same scheme was applied by Krichever to integrate the whole KP (or Zakharov-Shabat) scalar hierarchy of equations obtained as compatibility conditions of the linear evolution equations, with higher order differential operators (of mutually prime order) in the x variable on the right-hand side.…”