Numerical algorithms for the direct spectral transform with applications to nonlinear Schroedinger type systems

Journal Article · Fri Nov 20 04:00:00 EST 1998 · Journal of Computational Physics

DOI:https://doi.org/10.1006/jcph.1998.6087· OSTI ID:320967

Burtsev, S; Camassa, R ^[1]; Timofeyev, I ^[2]

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1. Los Alamos National Lab., NM (United States)
2. RPI, Troy, NY (United States). Dept. of Mathematical Sciences

The authors implement two different algorithms for computing numerically the direct Zakharov-Shabat eigenvalue problem on the infinite line. The first algorithm replaces the potential in the eigenvalue problem by a piecewise-constant approximation, which allows one to solve analytically the corresponding ordinary differential equation. The resulting algorithm is of second order in the step size. The second algorithm uses the fourth-order Runge-Kutta method. They test and compare the performance of these two algorithms on three exactly solvable potentials. They find that even though the Runge-Kutta method is of higher order, this extra accuracy can be lost because of the additional dependence of its numerical error on the eigenvalue. this limits the usefulness of the Runge-Kutta algorithm to a region inside the unit circle around the origin in the complex plane of the eigenvalues. For the computation of the continuous spectrum density, this limitation is particularly severe, as revealed by the spectral decomposition of the L{sup 2}-norm of a solution to the nonlinear Schroedinger equation. They show that no such limitations exist for the piecewise-constant algorithm. In particular, this scheme converges uniformly for both continuous and discrete spectrum components.

• Sponsoring Organization: USDOE, Washington, DC (United States); National Science Foundation, Washington, DC (United States)
• DOE Contract Number: W-7405-ENG-36; FG02-93ER25154
• OSTI ID: 320967
• Journal Information: Journal of Computational Physics, Journal Name: Journal of Computational Physics Journal Issue: 1 Vol. 147; ISSN JCTPAH; ISSN 0021-9991
• Country of Publication: United States
• Language: English

Related Subjects

66 PHYSICS
99 GENERAL AND MISCELLANEOUS
ALGORITHMS
COMPARATIVE EVALUATIONS
EIGENVALUES
NONLINEAR OPTICS
NUMERICAL SOLUTION
SCHROEDINGER EQUATION