NSU Scientists Publish Article on Nonlinear Fourier Transform

NSU graduates and researchers at the NSU Nonlinear Photonics Laboratory, Igor Chekhovskoy, Olga Shtyrina, Mikhail Fedoruk, Sergey Medvedev, and Sergey Turitsyn, published their article, “Nonlinear Fourier Transform for Analysis of Coherent Structures in Dissipative Systems”, in one of the most prestigious physics journals, the “Physical Review Letters”. The Journal’s impact factor is 8.839.

The work is devoted to a new application of the inverse scattering method (IST), also known as nonlinear Fourier transform (NFT). In the 1970’s NSU scientists Vladimir Zakharov and Alexei Shabat demonstrated that you can integrate one of the main models of nonlinear physics, the nonlinear Schrödinger equation (NSE), with the help of IST but you must solve the Zakharov – Shabat spectral problem (ZSh). The new method, analogous to the usual Fourier transform, allows us to simplify the analysis and reduce the complex nonlinear dynamics to a simple evolution in a specific base, the nonlinear signal spectrum. The use of NFT, in contrast to the usual Fourier transform, implies finding the continuous and discrete spectrum of the ZSh operator. Returning to the example of the NSE, the discrete spectrum here will describe the soliton part of the signal, and the continuous represents the dispersive waves signal. 

The difference between the conventional Fourier transform and NFT can be understood using the example of waves at sea. The Fourier transform gives the dependence of the waves amplitude on their length, which is called the spectrum. The NFT makes it possible to determine not only the wave spectrum, but also the presence of “ships at sea”, i.e. solitons. Continuing the analogy, we can say that NFT allows us to determine the relative magnitude of waves. If the wave is small, then the “ships” (solitons) are clearly visible, and if the wave is strong, then the NFT can detect and determine their movement.

The application of NFT to integrable Hamiltonian equations, such as the NSE, is well researched and represents the classical field of mathematical physics. In this paper, the authors investigated the potential of its application to dissipative nonintegrable systems using the Ginzburg-Landau equation (UGL) as an example. Although NFT cannot be used to solve these systems, the authors showed that the evolution of an optical signal obeying an UGL can be described with good accuracy using a finite number of variables using NFT in cases where the discrete component of the spectrum of the ZSh operator for the corresponding NSE is dominant. This corresponds to cases where the ratio of the signal energy associated with the discrete spectrum to the total energy is close to unity.

Thus, it has been shown that NFT can reduce the number of effective degrees of freedom when coherent structures, such as solitons, dominate the dynamics of an optical signal, even when their evolution is unstable. The stationary solutions of the UGL, which are dissipative solitons, are analyzed and the parameters of these solutions are found when the approach to describing the dynamics based on NFT is applicable.


 The figure on the left shows the evolution of the pulse, and on the right the evolution of the continuous and discrete spectrum. It is clear that during propagation, the initial optical pulse passes into asymptotic stability, which can be described with great accuracy by only three points in the discrete spectrum. The energy of the dispersion waves corresponding to the continuous spectrum is small compared to the discrete spectrum.

According to the scientists, this approach can provide new opportunities for the study of complex laser radiation, that consists of a mixture of coherent pulses and dispersive waves.