Cubic String Boundary Value Problems and Cauchy Biorthogonal Polynomials

Details Cubic String Boundary Value Problems and Cauchy Biorthogonal Polynomials Cubic String Boundary Value Problems and Cauchy Biorthogonal Polynomials

2009-04-14

arxiv.org/abs/0904.2188v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

14 pages. Partially replaces arXiv:0711.4082

Scientific paper

Cauchy Biorthogonal Polynomials appear in the study of special solutions to the dispersive nonlinear partial differential equation called the Degasperis-Procesi (DP) equation, as well as in certain two-matrix random matrix models. Another context in which such biorthogonal polynomials play a role is the cubic string; a third order ODE boundary value problem -f'''=zg f which is a generalization of the inhomogeneous string problem studied by M.G. Krein. A general class of such boundary value problems going beyond the original cubic string problem associated with the DP equation is discussed under the assumption that the source of inhomogeneity g is a discrete measure. It is shown that by a suitable choice of a generalized Fourier transform associated to these boundary value problems one can establish a Parseval type identity which aligns Cauchy biorthogonal polynomials with certain natural orthogonal systems on L^2_g.

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