S.V. Kiyashko1, V.O. Afenchenko2, V.V. Chernov3

1–3 Institute of Applied Physics of RAS (Nizhny Novgorod, Russia)
1 kiyashko@ ipfran.ru, 2 afen@ipfran.ru, 3 vcher@ipfran.ru

In nonlinear systems with instability, multistability often occurs. When studying processes occurring in real environments, the problem of finding paths leading to some state of equilibrium and searching for new stable states arises. In two-dimensional systems, they can consist of domains and differ in orientation in space.

The aim of the work is to experimentally determine the features of the generation of multidomain structures in a two-dimensional system in the presence of complex boundaries of a liquid layer that experiences periodic vertical oscillations.

The paper presents the results of an experimental study of the dynamics of roller domains of parametrically excited capillary waves in a square cuvette with a rounded corner and a round insert in the center of the cuvette.In various domains, the rollers were oriented parallel and perpendicular to the boundaries of the rounded corner of the cuvette and the boundaries of the circular insert. It was found that the dynamics of domains is determined by the movement of their fronts, and depending on the initial and boundary conditions, stable two-dimensional roller structures can appear at the edges of a square cuvette with a rounded corner and a circular protrusion in the center of the cuvette. The rollers had different orientations in different domains. In this case, the greatest influence on the dynamics of defects was exerted by roundings with a large radius.

Multistability of the equilibrium states of roller structures was discovered, characterized by the fact that with constant system parameters, various scenarios of domain competition arose, leading to 6 different stable equilibrium states, which differed in the number of domains, their shape and the absence of spatial symmetry.

It was found that during the occurrence of defects and the growth of domain walls, slow liquid flows occur near the boundaries of the cuvette with a rounded corner and the round insert. The rollers begin to move and then a new stable equilibrium state is established in the form of two domains, each containing rounded rollers. It was found that with increasing supercriticality, defects and domain walls occur more frequently, and the equilibrium states begin to slowly change each other. It was experimentally shown that the most stable states of domain equilibrium occur at low supercriticality and symmetrical arrangement of the round protrusion relative to the lateral sides of the cuvette.

The obtained results may be of interest in the study of the processes of establishing stable regimes in active media under strong competition and in the study of the formation of two-dimensional structures from conducting particles capable of scattering electromagnetic waves.

Kiyashko S.V., Afenchenko V.O., Chernov V.V. Multistability of roller structures under parametric excitation of capillary waves in a square cell with a rounded corner and round insert in the center // Nonlinear World. 2026. V. 24. № 3. P. 41–49. DOI: https:// doi.org/10.18127/ j20700970-202603-05

1. Sun H., Ma L., Wang L. Multistability as an indication of chaos in a discharge plasma. Phys. Rev. E 1995. V. 51. № 4. P. 3475.
2. Gelens L., Beri S., Van der Sande, G. et. al. Exploring Multistability in Semiconductor Ring Lasers: Theory and Experiment. Phys. Rev. Lett. 2009. V.102. № 19. P. 193904.
3. Ngonghala C.N., Feudel U., Showalter K. Extreme multistability in a chemical model system. Phys. Rev. E 2011. V.83. № 5. P. 056206.
4. Duncan A., Liao S., Vejchodský T., Erban R., Grima R. Noise-induced multistability in chemical systems: Discrete versus continuum modeling. Phys. Rev. E 2015. V. 91. № 4. P. 042111.
5. Shevtsova V. M., Melnikov D. E., Legros J. C. Multistability of oscillatory thermocapillary convection in a liquid bridge. Phys. Rev. E 2003. V. 68. № 6. P. 066311.
6. Rabinovich M. I., Ezersky A. B., Weidman P. D. The dynamics of patterns. Singapore: World Scientific. 2000. 336 p.
7. Zakharov V.E., L’vov V.S., Musher S.L. Transient behavior of a system of parametrically excited spin waves. Sov. Phys. Solid State 1972. V. 14(3). P. 710–715.
8. Reutov V.P. Tetragonal modulation cells at the parametric excitation of weakly damped capillary waves. European Journal of Mechanics B/Fluids 2011. V. 30. № 3. P. 269–274.
9. Ezersky A.B., Kiyashko S.V., Matusov P.A., Rabinovich M.I. Domain, domain walls and dislocations in capillary ripples. Europhys. Lett. 1994. V. 26. № 3. P. 183–188.
10. Ezersky A.B., Nazarovsky A.V., Kiyashko S.V. Bound states of topological defects in parametrically excited capillary ripples. Physica D 2001. V.152–153. P. 310–324.
11. Afenchenko V.O., Kiyashko S.V., Piskunova L.V. Dvizhenie fronta pri konkurencii rolikovyh domenov parametricheski svyazannyh voln. Izv. RAN. Ser. Fizika. 2004. T. 68. № 12. S.1771–1775 (In Russia).
12. Kiyashko S.V. Dinamika rolikovyh domenov parametricheski vozbuzhdaemyh kapillyarnyh voln. Izv. vuzov. Ser.: Radiofizika. 2008. T. LI. № 4. S. 359–365 (In Russia).
13. Kiyashko S.V., Korzinov L.N., Rabinovich M.I., Tsimring L.S. Rotating spirals in a Faraday experiment. Phys. Rev. E 1996. V. 54. № 5. P. 5037–5040.
14. Edwards W.S., Fauve S. Patterns and quasi-patterns in the Faraday experiment. J. Fluid Mech. 1994. V. 278. P. 123–148.
15. Kiyashko S.V., Afenchenko V.O., Nazarovskij A.V. Mul'tistabil'nost' rolikovyh struktur pri parametricheskom vozbuzhdenii voln v kvadratnoj kyuvete s vnutrennimi granicami. Nelinejnyj mir. 2018. T. 16. № 1. S. 33–40 (In Russia).
16. Kiyashko S.V., Afenchenko V.O., Nazarovskij A.V. Dinamika rolikovyh domenov pri parametricheskom vozbuzhdenii kapillyarnyh voln v kvadratnoj kyuvete s zakruglennym uglom i vnutrennimi granicami. Nelinejnyj mir. 2018. T. 16. № 6. S. 3–10 (In Russia).
17. Kiyashko S.V., Afenchenko V.O. Dinamika rolikovyh domenov pri parametricheskom vozbuzhdenii kapillyarnyh voln v kvadratnoj kyuvete s pryamougol'nym vystupom na granice. Nelinejnyj mir. 2019. T. 17. № 5. S. 5–13 (In Russia).
18. Kiyashko S.V., Afenchenko V.О. Nazarovsky A.V., Dynamics of roll domains in a rounded-corner cell. Phys. Wave Phenomena. 2014. V. 22. P. 132.
19. Kiyashko S.V., Afenchenko V.O., Chernov V.V. Mul'tistabil'nost' rolikovyh struktur pri parametricheskom vozbuzhdenii kapillyarnyh voln v kvadratnoj kyuvete s kruglym vystupom v centre. Nelinejnyj mir. 2024. T. 22. № 1. S. 47–54. DOI: https://doi.org/10.18127/j20700970-202401-06 (In Russia).
20. Kudrolli, Gollub J.P. Localized spatiotemporal chaos in surface waves. Physical Reviev E. 1996. V. 54. № 2. P. R1052–R1055.