On the Caudrey-Beals-Coifman System and the Gauge Group Action

The generalized Zakharov-Shabat systems with complex-valued Cartan elements and the systems studied by Caudrey, Beals and Coifman (CBC systems) and their gauge equivalent are studies. This includes: the properties of fundamental analytical solutions (FAS) for the gauge-equivalent to CBC systems and the minimal set of scattering data; the description of the class of nonlinear evolutionary equations solvable by the inverse scattering method and the recursion operator, related to such systems; the hierarchies of Hamiltonian structures.Comment: 12 pages, no figures, contribution to the NEEDS 2007 proceedings (Submitted to J. Nonlin. Math. Phys.

A Riemann-Hilbert Problem for an Energy Dependent Schr\"odinger Operator

\We consider an inverse scattering problem for Schr\"odinger operators with energy dependent potentials. The inverse problem is formulated as a Riemann-Hilbert problem on a Riemann surface. A vanishing lemma is proved for two distinct symmetry classes. As an application we prove global existence theorems for the two distinct systems of partial differential equations $u_t+(u^2/2+w)_x=0, w_t\pm u_{xxx}+(uw)_x=0$ for suitably restricted, complementary classes of initial data

Dressing chain for the acoustic spectral problem

The iterations are studied of the Darboux transformation for the generalized Schroedinger operator. The applications to the Dym and Camassa-Holm equations are considered.Comment: 16 pages, 6 eps figure

N-wave interactions related to simple Lie algebras. Z_2- reductions and soliton solutions

The reductions of the integrable N-wave type equations solvable by the inverse scattering method with the generalized Zakharov-Shabat systems L and related to some simple Lie algebra g are analyzed. The Zakharov- Shabat dressing method is extended to the case when g is an orthogonal algebra. Several types of one soliton solutions of the corresponding N- wave equations and their reductions are studied. We show that to each soliton solution one can relate a (semi-)simple subalgebra of g. We illustrate our results by 4-wave equations related to so(5) which find applications in Stockes-anti-Stockes wave generation.Comment: 18 pages, 1 figure, LaTeX 2e, IOP-style; More clear exposition. Introduction and Section 5 revised. Some typos are correcte

The Cauchy two-matrix model

We introduce a new class of two(multi)-matrix models of positive Hermitean matrices coupled in a chain; the coupling is related to the Cauchy kernel and differs from the exponential coupling more commonly used in similar models. The correlation functions are expressed entirely in terms of certain biorthogonal polynomials and solutions of appropriate Riemann-Hilbert problems, thus paving the way to a steepest descent analysis and universality results. The interpretation of the formal expansion of the partition function in terms of multicolored ribbon-graphs is provided and a connection to the O(1) model. A steepest descent analysis of the partition function reveals that the model is related to a trigonal curve (three-sheeted covering of the plane) much in the same way as the Hermitean matrix model is related to a hyperelliptic curve.Comment: 34 pages, 2 figures. V2: changes only to metadat

Partially integrable systems in multidimensions by a variant of the dressing method. 1

In this paper we construct nonlinear partial differential equations in more than 3 independent variables, possessing a manifold of analytic solutions with high, but not full, dimensionality. For this reason we call them ``partially integrable''. Such a construction is achieved using a suitable modification of the classical dressing scheme, consisting in assuming that the kernel of the basic integral operator of the dressing formalism be nontrivial. This new hypothesis leads to the construction of: 1) a linear system of compatible spectral problems for the solution $U(\lambda;x)$ of the integral equation in 3 independent variables each (while the usual dressing method generates spectral problems in 1 or 2 dimensions); 2) a system of nonlinear partial differential equations in $n$ dimensions ($n>3$), possessing a manifold of analytic solutions of dimension ($n-2$), which includes one largely arbitrary relation among the fields. These nonlinear equations can also contain an arbitrary forcing.Comment: 21 page

The evolutionary status of the semiregular variable QYSge

Repeated spectroscopic observations made with the 6m telescope of yielded new data on the radial-velocity variability of the anomalous yellow supergiant QYSge. The strongest and most peculiar feature in its spectrum is the complex profile of NaI D lines, which contains a narrow and a very wide emission components. The wide emission component can be seen to extend from -170 to +120 km/s, and at its central part it is cut by an absorption feature, which, in turn, is split into two subcomponents by a narrow (16km/s at r=2.5) emission peak. An analysis of all the Vr values leads us to adopt for the star a systemic velocity of Vr=-21.1 km/s, which corresponds to the position of the narrow emission component of NaI. The locations of emission-line features of NaI D lines are invariable, which point to their formation in regions that are external to the supergiant's photosphere. Differential line shifts of about 10km/s are revealed. The absorption lines in the spectrum of QYSge have a substantial width of FWHM~45 km/s. The method of model atmospheres is used to determine the following parameters: Teff=6250K, lg g=2.0, and microturbulence Vt=4.5km/s. The metallicity of the star is found to be somewhat higher than the solar one with an average overabundance of iron-peak elements of [Met/H]=+0.20. The star is found to be slightly overabundant in carbon and nitrogen, [C/Fe]=+0.25, [N/Fe]=+0.27. The alpha-process elements Mg, Si, and Ca are slightly overabundant [alpha/H]=+0.12. The strong sodium excess, [Na/Fe]=+0.75, is likely to be due to the dredge-up of the matter processed in the NeNa cycle. Heavy elements of the s-process are underabundant relative to the Sun. On the whole, the observed properties of QYSge do not give grounds for including this star into the group of RCrB or RVTau-type type objects.Comment: 29 pages, 8 figures, 4 tables; accepted by Astrophys. Bulleti

Complete integrability of shock clustering and Burgers turbulence

We consider scalar conservation laws with convex flux and random initial data. The Hopf-Lax formula induces a deterministic evolution of the law of the initial data. In a recent article, we derived a kinetic theory and Lax equations to describe the evolution of the law under the assumption that the initial data is a spectrally negative Markov process. Here we show that: (i) the Lax equations are Hamiltonian and describe a principle of least action on the Markov group that is in analogy with geodesic flow on $SO(N)$; (ii) the Lax equations are completely integrable and linearized via a loop-group factorization of operators; (iii) the associated zero-curvature equations can be solved via inverse scattering. Our results are rigorous for $N$-dimensional approximations of the Lax equations, and yield formulas for the limit $N \to \infty$. The main observation is that the Lax equations are a $N \to \infty$ limit of a Markovian variant of the $N$-wave model. This allows us to introduce a variety of methods from the theory of integrable systems

Leading Order Temporal Asymptotics of the Modified Non-Linear Schrodinger Equation: Solitonless Sector

Using the matrix Riemann-Hilbert factorisation approach for non-linear evolution equations (NLEEs) integrable in the sense of the inverse scattering method, we obtain, in the solitonless sector, the leading-order asymptotics as $t$ tends to plus and minus infinity of the solution to the Cauchy initial-value problem for the modified non-linear Schrodinger equation: also obtained are analogous results for two gauge-equivalent NLEEs; in particular, the derivative non-linear Schrodinger equation.Comment: 29 pages, 5 figures, LaTeX, revised version of the original submission, to be published in Inverse Problem

An Integrable Shallow Water Equation with Linear and Nonlinear Dispersion

We study a class of 1+1 quadratically nonlinear water wave equations that combines the linear dispersion of the Korteweg-deVries (KdV) equation with the nonlinear/nonlocal dispersion of the Camassa-Holm (CH) equation, yet still preserves integrability via the inverse scattering transform (IST) method. This IST-integrable class of equations contains both the KdV equation and the CH equation as limiting cases. It arises as the compatibility condition for a second order isospectral eigenvalue problem and a first order equation for the evolution of its eigenfunctions. This integrable equation is shown to be a shallow water wave equation derived by asymptotic expansion at one order higher approximation than KdV. We compare its traveling wave solutions to KdV solitons.Comment: 4 pages, no figure