A notion of curvature is introduced in multivariable operator theory. The curvature invariant of a Hilbert module over C[z(1),., z(d)] is a nonnegative real number which has significant extremal properties, which tends to be an integer, and which is hard to compute directly. It is shown that for graded Hilbert modules, the curvature agrees with the Euler characteristic of a certain finitely gen...

In recent years, there has been a series of results of Fokas and collaborators on boundary value problems for soliton equations (see [3] for a comprehensive review). The method of Fokas in [3] goes beyond existence and uniqueness. In fact, it reduces these problems to Riemann-Hilbert factorisation problems in the complex plane, thus generalising the existing theory which reduces initial value p...

Under investigation in this work is the coupled Sasa-Satsuma equation, which can describe propagations of two optical pulse envelopes birefringent fibers. The Riemann-Hilbert problem for equation formulated on basis corresponding $5 \times 5$ matrix spectral problem, allows us to present a suitable representation solution equation. Then Deift-Zhou steepest descent method used analyze long time ...

Abstract The paper aims at developing the Riemann–Hilbert (RH) approach for modified Camassa–Holm (mCH) equation on line with non-zero boundary conditions, in case when solution is assumed to two different constants sides of line. We present detailed properties spectral functions associated initial data Cauchy problem mCH and obtain a representation this terms an RH problem.

In this paper, we study an inhomogeneous Hilbert boundary value problem with a finite index and condition on circle for generalized Cauchy-Riemann equation singular coefficient. To solve problem, conducted complete of the solvability theory analytic functions infinite due to number points special type vorticity. Based these results, have derived formula general solution studied existence soluti...

We study the Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion and with monotonically increasing initial data using the Riemann-Hilbert (RH) approach. The solution of the Cauchy problem, in the zero dispersion limit, is obtained using the steepest descent method for oscillatory Riemann-Hilbert problems. The asymptotic solution is completely described by a scalar func...

We present a detailed analysis of the solution of the focusing nonlinear Schrödinger equation with initial condition ψ(x, 0) = N sech(x) in the limit N → ∞. We begin by presenting new and more accurate numerical reconstructions of the N -soliton by inverse scattering (numerical linear algebra) for N = 5, 10, 20, and 40. We then recast the inverse-scattering problem as a Riemann-Hilbert problem ...

The problem that is studied concerns a yawed wedge under a free surface, moving at a uniform speed. The model involves a trailing cavity whose boundary is a dividing streamline through the vertex of the wedge. The cavity closure mechanism is described according to the Tulin– Terent’ev single-spiral-vortex model. A closed-form solution to the governing nonlinear boundaryvalue problem is found by...