Methods of modeling anomalous modes of dynamic processes based on energy estimation

The problem of forecasting methodology for special modes of dynamic processes the nonlinear effect that occurs in the marine environment, called “rogue waves”, is considered. Rogue waves are waves that occur in the ocean, as a rule, suddenly, exist for a short period of time and have a huge destructive potential. There are many directions in the study of this phenomenon based on the application of computer modeling and numerical methods. At the same time, there is a tendency to search for rogue waves not only in hydrodynamics, but also in other subject areas, in which, when constructing models of the phenomena and processes under study, the apparatus for solving the corresponding initial boundary value problems for systems of differential equations is used. As a rule, the authors try to find solutions to differential equations, based on which it is possible to demonstrate the occurrence of abnormally high waves. It should be noted that the search for analytical solutions for some differential equations is an extremely difficult task or even impossible to solve. An alternative approach is proposed that makes it possible to prove the existence of the possibility of an anomaly without the need to solve the corresponding system of differential equations, and a model of a dynamic system is constructed similar to the formalism of Koopman theory which takes into account the asymptotic growth rate of the image of a dynamic operator in the energy space, on the basis of which an ordered hierarchy of classes of dynamic operators arises. The definition of an anomaly in the formalism of the mathematical apparatus under consideration is proposed, while the phenomenon of a rogue wave is interpreted as a special case of the occurrence of an anomalous phenomenon in a hydrodynamic system with a sufficiently high average value of the wave background. Within the framework of the proposed approach, it is possible to formulate the necessary conditions for the occurrence of an abnormal phenomenon and sufficient conditions for the absence of anomalies. A time series processing method is proposed that considers the hypothesis of the frequency of occurrence of anomalous phenomena. The existence of anomalies in magnetohydrodynamic processes is demonstrated, which is proved by constructing a model of magnetic field inversion, and the solution of the corresponding dispersion equation is carried out using a modification of the numerical Ivanisov-Polishchuk method consisting in combining the Ivanisov-Polishchuk algorithm and the Adam optimization method. The results obtained may be in demand for further development of the study of the structure of dynamic systems and for identifying more interdisciplinary connections that allow constructive transfer of some of the results from one subject area to another.

1. Kurkin A.A., Pelinovsky E.N. Rogue Waves: Facts, Theory and Modelling. Moscow; Berlin, 2016, 178 p. (in Russian)

2. Brule S., Enoch S., Guenneau S. On the possibility of seismic rogue waves in very soft soils. arXiv, 2020, arXiv:2004.07037v1. https://doi.org/10.48550/arXiv.2004.07037

3. McAllister M.L., Draycott S., Adcock T.A.A., Taylor P.H., van den Bremer T.S. Laboratory recreation of the Draupner wave and the role of breaking in crossing seas. Journal of Fluid Mechanics, 2019, vol. 860, pp. 767–786. https://doi.org/10.1017/jfm.2018.886

4. Wu X.Y., Tian B., Qu Q.X., Yuan Y.Q., Du X.X. Rogue waves for a (2+1)-dimensional Gross-Pitaevskii equation with time-varying trapping potential in the Bose–Einstein condensate. Computers & Mathematics with Applications, 2020, vol. 79, no. 4, pp. 1023–1030. https://doi.org/10.1016/j.camwa.2019.08.015

5. Yang B., Yang J. Rogue wave patterns in the nonlinear Schrödinger equation. Physica D: Nonlinear Phenomena, 2021, vol. 419, pp. 132850. https://doi.org/10.1016/j.physd.2021.132850

6. Li B.Q. Hybrid breather and rogue wave solution for a (2+1)-dimensional ferromagnetic spin chain system with variable coefficients. International Journal of Computer Mathematics, 2022, vol. 99, no. 3, pp. 506–519. https://doi.org/10.1080/00207160.2021.1922678

7. Liu J.G., Zhu W.H. Multiple rogue wave solutions for (2+1)-dimensional Boussinesq equation. Chinese Journal of Physics, 2020, vol. 67, pp. 492–500. https://doi.org/10.1016/j.cjph.2020.08.008

8. Knuth D.E. Mathematics and computer science: coping with finiteness. Science, 1976, vol. 194, no. 4271, pp. 1235–1242. https://doi.org/10.1126/science.194.4271.1235

9. Kharif C., Pelinovsky E., Slunyaev A. Rogue Waves in the Ocean. Springer, 2009, 216 p.

10. Shamin R.V., Yudin A.V. Simulation of spatiotemporal spread of rogue waves. Doklady Earth Sciences, 2013, vol. 448, no. 2, pp. 240–242. (in Russian). https://doi.org/10.1134/S1028334X13020165

11. Kazankov V.K., Shmeleva A.G., Zaitseva E.V. Unstable plastic flow in structural materials: time series for analysis of experimental data. Materials Physics and Mechanics, 2022, vol. 48, no. 2, pp. 208–216. https://doi.org/10.18149/MPM.4822022_6

12. Kazankov V.K., Peregudin S.I., Kholodova S.E. Mathematical modeling of geophysical processes in a layer of electrically conductive liquid of variable depth. Springer Geology, 2023, pp. 331–341. https://doi.org/10.1007/978-3-031-16575-7_32

13. Ivanisov A.V., Polishchuk V.K. A method of finding the roots of polynomials which converge for any initial approximation. USSR Computational Mathematics and Mathematical Physics, 1985, vol. 25, no. 3, pp. 1–7. https://doi.org/10.1016/0041-5553(85)90066-7

14. Kingma D.P., Ba J. Adam: a method for stochastic optimization. arXiv, 2014, arXiv:1412.6980. https://doi.org/10.48550/arXiv.1412.6980

15. Mikhailov E.A., Khasaeva T.T., Teplyakov I.O. The emergence of contrast structures for galactic magnetic field: theoretical estimates and modeling on GPU. Proceedings of the Institute for System Programming of the RAS (Proceedings of ISP RAS), 2021, vol. 33, no. 6, pp. 253–264. (in Russian). https://doi.org/10.15514/ISPRAS-2021-33(6)-18