Nonlinear systems

The linearity of a system allows researchers to make certain mathematical assumptions and approximations, allowing easier calculation of the results. Since nonlinear systems are not equal to the sum of their parts, they are usually difficult to model, and their behaviors with respect to a given variable (for example, time) are extremely difficult to predict.

Some non-linear systems have exact or integrable solutions, while others have chaotic behavior, therefore they cannot be reduced to a simple form nor can they be solved.

The spectra obtained by applying the inverse scattering technique (TSI) play a crucial role in the physics of non-linear phenomena. They define the long-term evolution of dynamic systems.
We present the TSI spectral portraits for the extensive first-order three-parameter families of doubly periodic solutions of the nonlinear Schrödinger equation that span a wide range of physical phenomena such as modulational instability, shock waves, and many other problems with periodic boundary conditions.
In the article these spectral portraits are related to the parameters of the family. It is shown that there are two qualitatively different types of spectral portraits. The spectra of type A solutions consist of two continuous bands: a band of purely imaginary eigenvalues ​​within the interval [-i, i] and a finite band of complex eigenvalues. In contrast, the spectra of type B solutions have only continuous bands of imaginary eigenvalues, all of them located within the interval [-i, i] and separated by a finite band space. A physical interpretation of these results is given in the article.
Type A solutions can be considered as standing waves of pairs of waves that interact in a nonlinear way and propagate in opposite directions along the transverse axis. This is not the case for type B solutions. The latter are considered incompatible with ABs of a certain frequency range.

The reverse scattering technique is an essential tool in mathematical physics that allows us to solve initial value problems for a number of partial differential equations that model the propagation of non-linear waves in physics. These include the KdV equation, the nonlinear Schrödinger equation, the Sine-Gordon equation, and a multiplicity of equations from other evolutions.
An essential part of this technique is spectral analysis that provides information about the non-linear modes involved in evolution. In particular, this spectrum provides information about the content of solitons in a wave field. The construction of the TSI spectrum for the solution of a non-linear equation is equivalent to Fourier analysis in linear problems. As such, the spectral decomposition of TSI can be used in optical engineering, for example in telecommunications, signal transmission, optical data processing, shock wave analysis and in other problems of non-linear optics.

The TSI spectrum is directly related to the mode spectrum in linear stability analysis of periodic stationary solutions. Therefore, knowledge of the TSI spectrum is important both in theory and in applications. A complete spectral analysis of TSI requires very complex mathematical constructions such as algebraic-geometric approaches, finite gap methods, Riemann surfaces, and theta functions. However, for practical reasons, simple solutions and numerical analysis may be more efficient to visualize so-called "spectral portraits" of NLSE (non-linear Schrödinger equation) solutions of practical interest. Such spectral portraits can be constructed for modulational instability, shock waves, integrable turbulence, super-regular periodic solutions, and other non-linear phenomena common to various branches of physics.

The periodic solutions of the NLSE are an important class of solutions in physics. In principle, they can be written explicitly as a quotient of theta functions. However, this representation has not found many applications since it has an infinite number of free parameters. This general form can be compared to multisoliton solutions with an infinite number of solitons in the solution. When dealing with such solutions, we have to first investigate their fundamental components. A solution of a soliton is well known today. However, the fundamental components of periodic solutions have not yet been fully investigated. These fundamental components are double-period first-order solutions. These are families of three parameters in the form of non-trivial combinations of elliptic Jacobi functions. The free real parameters offer the possibility to adapt the periods and the amplitude of the solution to particular physical conditions. It is important to compare them with the spectral measurements observed in the experiments, but the spectra of the TSI are fundamentally different. These are crucial components of mathematical theory and physical interpretation. The TSI spectra are important for understanding long-term field evolution and for building more complex multi-periodic solutions. Such spectra have never been produced in previous work. We fill this gap in the present manuscript, thus opening avenues for further progress in this area.

The work is a collaboration between the Institute of Optics and the department of theoretical physics of the Research School of Physics of the Australian National University

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