Anomalous errors of direct scattering transform.

Theory of direct scattering transform for nonlinear wave fields containing solitons is revisited to overcome fundamental difficulties hindering its stable numerical implementation. With the focusing one-dimensional nonlinear Schrödinger equation serving as a model, we study a crucial fundamental property of the scattering problem for multisoliton potentials demonstrating that in many cases phase and space 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 analysed using high-precision arithmetics even in the presence of noise opening broad perspectives in nonlinear physics.