Riemann-Hilbert Problems with canonical normalization and families of commuting operators

Abstract

We start with a Riemann-Hilbert Problems (RHP) with canonical normalization whose sewing functions depends on several additional variables. Using Zakharov-Shabat theorem we are able to construct a family of ordinary differential operators for which the solution of the RHP is a common fundamental analytic solution. This family of operators obviously commute. Thus we are able to construct new classes of integrable nonlinear evolution equations.

Dressing method and quadratic bundles related to symmetric spaces. Vanishing boundary conditions

“…The theory of complete quadratic bundles like this one is more complicated than in the case we have considered in that report. The latter represents certain interest in relation to N-wave type equations with cubic non-linearity recently derived by Gerdjikov [11].…”

Riemann-Hilbert problem, integrability and reductions

“…Indeed, following Zakharov and Shabat [61,62] it became possible to develop the dressing method for explicit calculation of the soliton solutions. Later results by Dickey, Gelfand [14] and Mark Adler [2] established the fact that the solution of the RHP in each of the sectors Ω k for λ ≫ 1 can be represented as an asymptotic series in the form [18]:…”

“…where the subscript + means that only terms with positive powers of λ in the corresponding expansion in powers of λ are retained. In [18] it was demonstrated that the first few coefficients of Q(x, t, λ) were sufficient to parameterize U and V of a polynomial in λ Lax pair. Indeed, for a Lax operator linear in λ we have…”

“…The formulae (2. 19) and (2.20) allow one to calculate effectively compatible Lax pairs when both L and M are quadratic in λ or polynomials in λ of any order greater than 2, see [18]. One can also impose on Q + (x, t, λ) reductions of Z h -type; the resulting Lax pair will naturally possess Z h as reduction group.…”

On new types of integrable 4-wave interactions

“…In the present paper, which is a natural extension of [8], we propose an alternative approach to the same class of equations using as a starting point the Riemann-Hilbert problem (RHP) [37,38,33,34,29,36,23]; the importance of the canonical normalization of RHP was noticed in [11,6]. Our aim is to show that this allows one to construct rings of commuting operators and in addition gives a tool to study their spectral properties.…”