Multiple precision pseudospectral computations of the radiation coefficient for weakly nonlocal solitary waves: Fifth-order Korteweg--DeVries equation

By performing multiple precision pseudospectral calculations using two different basis sets, we compute the radiation coefficient α(ε) for very small ε to resolve discrepancies between earlier numerical work of the author's and a small-ε perturbation theory of Grimshaw and Joshi. Multiple precision is needed because α is asymptotically proportional to exp(-π/2ε), and is therefore below the single precision roundoff threshold when ε<1/25. Because α decreases exponentially with 1/ε, we use a numerical method whose error decreases exponentially fast with the number of grid points N: a pseudospectral method or finite differences of as high as twenty-fourth order. Richardson iteration, preconditioning by high order finite differences, parity symmetry, Aitken and Richardson extrapolation, and the fast Fourier transform are all crucial in reducing our longest runs to about 24 h on a Unix workstation. Although preconditioning by second order differences is the norm, we find that high order preconditioning---as large as 14th order---is more efficient for our one-dimensional problem. We find that the discrepancies are mostly due to differences in convention for (i) definition of the parameter ε and (ii) choice of the far field phase. When these are accounted for, we obtain very good agreement with Grimshaw and Joshi's first order term. However, their second order term, which is predicted to be π2/2∼4.94, is about O(0.0 to -0.2) in our computations. The reason for this difference is still a mystery