Conservation laws and symmetry analysis of a generalized Drinfeld-Sokolov system

Tamara M. Garrido; Rafael de la Rosa; Elena Recio; Almudena P. Márquez; Tamara M. Garrido; Rafael de la Rosa; Elena Recio; Almudena P. Márquez

doi:10.3934/math.20231465

AIMS Mathematics

2023, Volume 8, Issue 12: 28628-28645. doi: 10.3934/math.20231465

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Conservation laws and symmetry analysis of a generalized Drinfeld-Sokolov system

Department of Mathematics, University of Cadiz, Puerto Real 11510, Spain

Received: 30 June 2023 Revised: 28 September 2023 Accepted: 12 October 2023 Published: 20 October 2023
MSC : 22E60, 35G50, 35Q53

The generalized Drinfeld-Sokolov system is a widely-used model that describes wave phenomena in various contexts. Many properties of this system, such as Hamiltonian formulations and integrability, have been extensively studied and exact solutions have been derived for specific cases. In this paper we applied the direct method of multipliers to obtain all low-order local conservation laws of the system. These conservation laws correspond to physical quantities that remain constant over time, such as energy and momentum, and we provided a physical interpretation for each of them. Additionally, we investigated the Lie point symmetries and first-order symmetries of the system. Through the point symmetries and constructing the optimal systems of one-dimensional subalgebras, we were able to reduce the system of partial differential equations to ordinary differential systems and obtain new solutions for the system.
- Drinfeld-Sokolov system,
- conservation laws,
- conserved quantities,
- Lie symmetries,
- reductions
Citation: Tamara M. Garrido, Rafael de la Rosa, Elena Recio, Almudena P. Márquez. Conservation laws and symmetry analysis of a generalized Drinfeld-Sokolov system[J]. AIMS Mathematics, 2023, 8(12): 28628-28645. doi: 10.3934/math.20231465

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Abstract

The generalized Drinfeld-Sokolov system is a widely-used model that describes wave phenomena in various contexts. Many properties of this system, such as Hamiltonian formulations and integrability, have been extensively studied and exact solutions have been derived for specific cases. In this paper we applied the direct method of multipliers to obtain all low-order local conservation laws of the system. These conservation laws correspond to physical quantities that remain constant over time, such as energy and momentum, and we provided a physical interpretation for each of them. Additionally, we investigated the Lie point symmetries and first-order symmetries of the system. Through the point symmetries and constructing the optimal systems of one-dimensional subalgebras, we were able to reduce the system of partial differential equations to ordinary differential systems and obtain new solutions for the system.

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