A numerical method for dark solitons has shown a key match with an analytical benchmark: in an inverse-step example, its scattering coefficients and the rate of change of the transmission coefficient accurately matched analytical expressions. The method is designed for the defocusing nonlinear Schrödinger equation, or dNLSE, under continuous-wave boundary conditions. It seeks the discrete eigenvalues and norming constants associated with dark solitons, along with the reflection coefficient for waves in the continuous spectrum.
A fuller map of the wave field
The target is broader than a soliton count alone. Here, “scattering data” means the discrete eigenvalues, the norming constants that complete the dark-soliton data, and the reflection coefficient for continuous-spectrum waves. The authors present the procedure as a generalization of the classical Boffetta–Osborne direct scattering transform to dark solitons in the dNLSE.
To produce those quantities, the scheme numerically solves the auxiliary Zakharov–Shabat scattering problem with continuous-wave boundary conditions. It computes the transfer matrix using the second-order Boffetta–Osborne scheme. The calculation therefore joins a discrete part—the eigenvalues and norming constants—with a continuous-spectrum part represented by the reflection coefficient.
Three tests, three kinds of evidence
The verification used three computational dNLSE wave-field examples containing dark solitons. In the calculation, those inputs were an inverse-step potential, a hyperbolic-tangent potential and an arbitrary-modulated hollow. The inverse-step case was checked against analytical expressions. For the arbitrary-modulated hollow, exact scattering data were unavailable, so the comparison used the most accurate potential discretization as its reference.
In the inverse-step benchmark, the potential contained eight discrete eigenvalues. The numerical scattering coefficients and the rate of change of the transmission coefficient accurately matched the analytical expressions. That benchmark checked both the discrete eigenvalue information and the coefficients produced by the transform.
The hyperbolic-tangent test offered a second benchmark. With amplitude A = 3.4, it contained seven discrete eigenvalues, and symmetric eigenvalues had coinciding norming constants. With grid refinement, the eigenvalues, norming constants and continuous-spectrum results showed second-order convergence, meaning the reported errors fell consistently as the discretization became finer.
One part of that test did not keep improving in the same way. The zero eigenvalue was found with an error of about 10−30 or less even on rough potential discretizations, but further detailization did not produce additional convergence for that eigenvalue. The unusually small initial error and the missing later convergence appeared together in the reported calculation.
The arbitrary-modulated hollow moved the calculation into a case without exact scattering data. The algorithm identified 28 dark solitons, using the most accurate potential discretization as the reference. Its numerical errors also showed second-order convergence. The result supports that convergence pattern for this modeled example, while keeping the soliton count tied to a numerical reference rather than an analytical solution.
A qualified numerical result
Numerical precision became part of the method’s practical setup. The calculation used 100-digit precision for soliton eigenvalues, and anomalous errors were reported in norming-constant calculations.
Taken together, the three examples give a qualified result. The method matched analytical scattering quantities in the eight-eigenvalue inverse-step case, showed second-order convergence in the hyperbolic-tangent and arbitrary-modulated tests, and identified 28 dark solitons in the latter. The evidence supports the proposed procedure on these modeled dNLSE potentials, while keeping two numerical cautions in view: the arbitrary-modulated result relied on a discretized reference, and the zero eigenvalue did not continue to converge after further detailization.
Paper data and sources
Original title: Numerical Direct Scattering Transform for Dark Solitons
Authors: Ilya Mullyadzhanov, Sergey Dremov, Andrey Gelash
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
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