High-orders methods and high-precision arithmetics make direct scattering transform for the Korteweg-De Vries equation robust.

Similar to the theory of direct scattering transform for nonlinear wave fields containing solitons within the focusing one-dimensional nonlinear Schrödinger equation [1], we revisit the theory associated with the Korteweg–De Vries equation. We study a crucial fundamental property of the scattering problem for multisoliton potentials demonstrating that in many cases position parameters of solitons cannot be identified with standard machine precision arithmetics making solitons in some sense “uncatchable”. Using the dressing method we find the landscape of soliton scattering coefficients in the plane of the complex spectral parameter for multisoliton wave fields truncated within a finite domain, allowing us to capture the nature of such anomalous numerical errors. They depend on the size of the computational domain L leading to a counterintuitive exponential divergence when increasing L in the presence of a small uncertainty in soliton eigenvalues. Then we demonstrate how one of the scattering coefficients loses its analytical properties due to the lack of the wave-field compact support in case of L→∞. Finally, we show that despite this inherent direct scattering transform feature, the wave fields of arbitrary complexity can be reliably analyzed using high-precision arithmetics and high-order algorithms based on the Magnus expansion [2, 3] providing accurate information about soliton amplitudes, velocities, positions and intensity of the radiation. This procedure is robust even in the presence of noise opening broad perspectives in analyzing experimental data on propagation of surface waves on shallow water.

The work is partially funded by Russian Science Foundation grant No 19-79-30075.

[1] Gelash A., Mullyadzhanov R. Anomalous errors of direct scattering transform // Physical Review E 101 (5), 052206, 2020.

[2] Mullyadzhanov R., Gelash A. Direct scattering transform of large wave packets // Optics Letters 44 (21), 5298-5301, 2019.

[3] Gudko A., Gelash A., Mullyadzhanov R. High-order numerical method for scattering data of the Korteweg—De Vries equation // Journal of Physics: Conference Series 1677 (1), 012011, 2020.