Complex plane representations and stationary states in cubic and quintic resonant systems

Weakly nonlinear energy transfer between normal modes of strongly resonant PDEs is captured by the corresponding effective resonant systems. In a previous article, we have constructed a large class of such resonant systems (with specific representatives related to the physics of Bose-Einstein condensates and Anti-de Sitter spacetime) that admit special analytic solutions and an extra conserved quantity. Here, we develop and explore a complex plane representation for these systems modelled on the related cubic Szego and LLL equations. To demonstrate the power of this representation, we use it to give simple closed form expressions for families of stationary states bifurcating from all individual modes. The conservation laws, the complex plane representation and the stationary states admit furthermore a natural generalization from cubic to quintic nonlinearity. We demonstrate how two concrete quintic PDEs of mathematical physics fit into this framework, and thus directly benefit from the analytic structures we present: the quintic nonlinear Schroedinger equation in a one-dimensional harmonic trap, studied in relation to Bose-Einstein condensates, and the quintic conformally invariant wave equation on a two-sphere, which is of interest for AdS/CFT-correspondence.Comment: v2: version accepted for publicatio

The fractional orthogonal derivative

This paper builds on the notion of the so-called orthogonal derivative, where an n-th order derivative is approximated by an integral involving an orthogonal polynomial of degree n. This notion was reviewed in great detail in a paper in J. Approx. Theory (2012) by the author and Koornwinder. Here an approximation of the Weyl or Riemann-Liouville fractional derivative is considered by replacing the n-th derivative by its approximation in the formula for the fractional derivative. In the case of, for instance, Jacobi polynomials an explicit formula for the kernel of this approximate fractional derivative can be given. Next we consider the fractional derivative as a filter and compute the transfer function in the continuous case for the Jacobi polynomials and in the discrete case for the Hahn polynomials. The transfer function in the Jacobi case is a confluent hypergeometric function. A different approach is discussed which starts with this explicit transfer function and then obtains the approximate fractional derivative by taking the inverse Fourier transform. The theory is finally illustrated with an application of a fractional differentiating filter. In particular, graphs are presented of the absolute value of the modulus of the transfer function. These make clear that for a good insight in the behavior of a fractional differentiating filter one has to look for the modulus of its transfer function in a log-log plot, rather than for plots in the time domain.Comment: 32 pages, 7 figures. The section between formula (4.15) and (4.20) is correcte

A fast and spectrally convergent algorithm for rational-order fractional integral and differential equations

A fast algorithm (linear in the degrees of freedom) for the solution of linear variable-coefficient rational-order fractional integral and differential equations is described. The approach is related to the ultraspherical method for ordinary differential equations [27], and involves constructing two different bases, one for the domain of the operator and one for the range of the operator. The bases are constructed from direct sums of suitably weighted ultraspherical or Jacobi polynomial expansions, for which explicit representations of fractional integrals and derivatives are known, and are carefully chosen so that the resulting operators are banded or almost-banded. Geometric convergence is demonstrated for numerous model problems when the variable coefficients and right-hand side are sufficiently smooth