Calculation of Volterra kernels for solutions of nonlinear differential equations

We consider vector-valued autonomous differential equations of the form x' = f(x) + phi with analytic f and investigate the nonanticipative solution operator phi bar right arrow A(phi) in terms of its Volterra series. We show that Volterra kernels of order > 1 occurring in the series expansion of the solution operator A are continuous functions, and establish recurrence relations between the kernels allowing their explicit calculation. A practical tensor calculus is provided for the finite-dimensional case. In addition to analytically calculating the kernels, we present an algorithm to numerically obtain them from the output x(t) through sampling the input space by linear combinations of delta functions. We call this "differential sampling". It is a nonlinear analogue of the classical method of impulse response. We prove a continuity theorem stating that, in the finite-dimensional case, approximate delta functions give rise to approximate Volterra kernels and that continuity holds in the sense of weak convergence. Finally, we discuss a practical implementation of differential sampling and relate it to the Wiener method

Numerical methods for calculating poles of the scattering matrix with applications in grating theory

Waveguide and resonant properties of diffractive structures are often explained through the complex poles of their scattering matrices. Numerical methods for calculating poles of the scattering matrix with applications in grating theory are discussed. A new iterative method for computing the matrix poles is proposed. The method takes account of the scattering matrix form in the pole vicinity and relies upon solving matrix equations with use of matrix decompositions. Using the same mathematical approach, we also describe a Cauchy-integral-based method that allows all the poles in a specified domain to be calculated. Calculation of the modes of a metal-dielectric diffraction grating shows that the iterative method proposed has the high rate of convergence and is numerically stable for large-dimension scattering matrices. An important advantage of the proposed method is that it usually converges to the nearest pole.Comment: 9 pages, 2 figures, 4 table

Effective range function below threshold

We demonstrate that the kernel of the Lippmann-Schwinger equation, associated with interactions consisting of a sum of the Coulomb plus a short range nuclear potential, below threshold becomes degenerate. Taking advantage of this fact, we present a simple method of calculating the effective range function for negative energies. This may be useful in practice since the effective range expansion extrapolated to threshold allows to extract low-energy scattering parameters: the Coulomb-modified scattering length and the effective range.Comment: 14 pages, 1 figur