Add high-order NSE discretizations: CT4, fast TES4, Padé ES4/6/8, and Chebyshev–Joukowski

[IgorChekhovskoy]

Summary

This pull request adds a collection of high-order discretizations for the Zakharov–Shabat problem:

• slow CT4 reference discretization;
• fast TES4 4A and 4B discretizations;
• fast conservative TES4 Suzuki discretization;
• fast Padé FES4 and FES6 discretizations;
• slow ES6 reference discretization;
• slow ES8 reference discretization;
• fast direct Padé FES8 discretization;
• fast Chebyshev–Joukowski FES8 representation.

The Padé schemes use a common implementation of diagonal [s/s] approximants, supporting degrees up to 7 with scheme-specific minimum degrees.

The complete implementation is submitted as one pull request so that the maintainers can evaluate all available schemes together. I am happy to remove or adjust individual methods if some of them are not considered suitable for FNFT.

References

1. S. Medvedev, I. Vaseva, I. Chekhovskoy, and M. Fedoruk,
   “Numerical algorithm with fourth-order accuracy for the direct Zakharov–Shabat problem,”
   Optics Letters 44, 2264–2267 (2019).
   https://doi.org/10.1364/OL.44.002264

2. S. Medvedev, I. Vaseva, I. Chekhovskoy, and M. Fedoruk,
   “Exponential fourth order schemes for direct Zakharov-Shabat problem,”
   Optics Express 28, 20–39 (2020).
   https://doi.org/10.1364/OE.377140

3. S. Medvedev, I. Chekhovskoy, I. Vaseva, and M. Fedoruk,
   “Conservative multi-exponential scheme for solving the direct Zakharov–Shabat scattering problem,”
   Optics Letters 45, 2082–2085 (2020).
   https://doi.org/10.1364/OL.387436

4. S. Medvedev, I. Chekhovskoy, I. Vaseva, and M. Fedoruk,
   “Fast sixth-order algorithm based on the generalized Cayley transform for the Zakharov-Shabat system associated with nonlinear Schrodinger equation,”
   Journal of Computational Physics 448, 110764 (2022).
   https://doi.org/10.1016/j.jcp.2021.110764

5. S. Medvedev, I. Chekhovskoy, I. Vaseva, and M. Fedoruk,
   “Fast Eighth-Order Padé Schemes Based on Chebyshev Polynomials for the Direct Zakharov-Shabat Problem,”
   arXiv:2608.11892 (2026).
   https://doi.org/10.48550/arXiv.2608.11892

Implementation

• The contribution is integrated as C code following the existing FNFT structure and style.
• Continuous- and discrete-spectrum computations are supported where applicable.
• Tests cover the new discretizations, Padé degrees, Richardson extrapolation, boundary conditions, and 64-bit size-overflow cases.
• Existing public interfaces remain compatible.
• MATLAB MEX option parsing and buffer sizing were updated for the new schemes.

Authorship and licensing

The fast implementations were written by Igor Chekhovskoy. The slow ES8 C reference implementation was written jointly by Igor Chekhovskoy and Irina Vaseva.

No university or institution holds rights to the contributed code.

Testing

Performed locally with MinGW:

• complete build: successful;
• all 14 targeted tests for the contributed schemes: passed;
• complete CTest run: 254/255 passed;
• the only failing test, fnft_kdvp_test_soliton, also fails in the local unmodified baseline and is unrelated to this contribution;
• Padé/Chebyshev boundary and 64-bit overflow tests: passed;
• both modified MEX C sources passed syntax compilation.

MATLAB runtime tests were not performed.