In this paper we use the Riemann–Hilbert problem, with jumps supported on appropriate curves in complex plane, for matrix biorthogonal polynomials and apply it to find Sylvester systems of differential equations orthogonal its second kind functions as well. For aim, type Pearson weights are shown be instrumental. Several applications given, order increasing complexity. First, a general discussi...

The capacity of coupled nonlinear Schrr odinger equations to support multipulse solutions (multibump solitary-waves) is investigated. A detailed analysis is undertaken for a system of quadratically coupled equations that describe the phenomena of second harmonic generation and parametric wave interaction in non-centrosymmetric optical materials. Utilising the framework of homoclinic bifurcation...

A restarted nonsymmetric Lanczos algorithm is given for computing eigenvalues and both right and left eigenvectors. The restarting limits the storage so that finding eigenvectors is practical. Restarting also makes it possible to deal with roundoff error in new ways. We give a scheme for avoiding near-breakdown and discuss maintaining biorthogonality. A system of linear equations can be solved ...

Soliton equations are infinite-dimensional integrable systems described by nonlinear evolution equations. As one of the soliton equations, long wave equation takes on profound significance of theory and reality. By using the method of nonlinearization, the relation between long wave equation and second-order eigenvalue problem is generated. Based on the nonlinearized Lax pairs, Euler-Lagrange f...

Approximation only by derivative (or more generally momentum) expansions, combined with reparametrization invariance, turns the continuous renormalization group for quantum eld theory into a set of partial di erential equations which at xed points become non-linear eigenvalue equations for the anomalous scaling dimension . We review how these equations provide a powerful and robust means of dis...