Two-phase solutions of focusing NLS equation are classically constructed out of an appropriate Riemann surface of genus two, and expressed in terms of the corresponding theta-function. We show here that in a certain limiting regime such solutions reduce to some elementary ones called "Solitons on unstable condensate". This degeneration turns out to be conveniently studied by means of basic tools from the theory of Riemann-Hilbert problems. In particular no acquaintance with Riemann surfaces and theta-function is required for such analysis.

We address the problem of long-time asymptotics for the solutions of the Korteweg-de Vries equation under low regularity assumptions. We consider decreasing initial data admitting only a finite number of moments. For the so-called "soliton region", an improved asymptotic estimate is provided, in comparison with the one in [8]. Our analysis is based on the dbar steepest descent method proposed by P

Abstract. For some positive integers g and n we consider a subgroup G g,n of the 2g-dimensional modular group keeping invariant a certain locus W g,n in the Siegel upper half plane of degree g. We address the problem of describing a fundamental domain for the modular action of the subgroup on W g,n . Our motivation comes from geometry: g and n represent the genus and the number of ovals of a generic real Riemann surface of separated type; the locus W g,n contains the corresponding period matrix computed with respect to some specific basis in the homology. In this paper we formulate a general procedure to solve the problem when g is even and n equals one. For g equal to two or four the explicit calculations are worked out in full detail.

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