A Fourier-Series Method for Solving Soliton Problems

A Fourier–Galerkin method with an earlier proposed complete orthonormal system of functions in $L^2 ( - \infty ,\infty )$ as the set of trial functions is developed and displayed for the problem of calculating the shape of the one-soliton solution of the Korteweg–de Vries equation. The convergence of the method is investigated through comparison with the analytic solution, which appears to be very good. The truncation and discretization errors are assessed pointwise. The technique developed is also applied to the soliton problem for the so-called Kuramoto–Sivashinsky equation and the obtained soliton shape is compared to the existing difference solution. The quantitative agreement between the Fourier-series-method result and the numerical one is good. In the present paper, however, the soliton solution is obtained for a significantly wider range of phase velocities, which suggests that the spectrum might be continuous. The new technique can also be applied to a variety of other problems involving identification of homoclinic solutions.

[1]  C. Christov.  A Complete Orthonormal System of Functions in $L^2 ( - \infty ,\infty )$ Space , 1982 .

[2]  J. Hyman,et al.  The Kuramoto-Sivashinsky equation: a bridge between PDE's and dynamical systems , 1986 .

[3]  Michel Cosnard,et al.  Algorithm 554: BRENTM, A Fortran Subroutine for the Numerical Solution of Nonlinear Equations [C5] , 1980, TOMS.

[4]  G. Sivashinsky,et al.  On Irregular Wavy Flow of a Liquid Film Down a Vertical Plane , 1980 .

[5]  J. R. Higgins.  Completeness and basis properties of sets of special functions: Bibliography , 1977 .

[6]  Norbert Wiener,et al.  Extrapolation, Interpolation, and Smoothing of Stationary Time Series , 1964 .

[7]  S. V. Alekseenko,et al.  Wave formation on vertical falling liquid films , 1985 .

[8]  Y. Pomeau,et al.  On solitary waves running down an inclined plane , 1983, Journal of Fluid Mechanics.

[9]  J. Boyd.  Spectral methods using rational basis functions on an infinite interval , 1987 .

[10]  S. P. Lin.  Finite amplitude side-band stability of a viscous film , 1974, Journal of Fluid Mechanics.

[11]  P. L. Bhatnagar.  Nonlinear Waves in One-Dimensional Dispersive Systems , 1980 .

[12]  D. J. Benney.  Long Waves on Liquid Films , 1966 .

[13]  J. Boyd.  The Optimization of Convergence for Chebyshev Polynomial Methods in an Unbounded Domain , 1982 .

[14]  John P. Boyd.  The orthogonal rational functions of Higgins and Christov and algebraically mapped Chebyshev polynomials , 1990 .

[15]  T. MacRobert.  Higher Transcendental Functions , 1955, Nature.