(2+1)-dimensional new bi-hamiltonian integrable system: Symmetries, Noether's theorem and integrals of motion

Abstract

In this work, we investigate a symmetry reduction of the recently discovered (3 + 1)-dimensional equation of the Monge-Amp & egrave;re type. This equation forms a bi-Hamiltonian system using Magri's theorem when expressed in the two-component form. We select a particular linear combination of the Lie point symmetries belonging to this system to conduct symmetry reduction, resulting in a new (2 + 1)-dimensional system in two-component form. Lagrangian and first Hamiltonian densities are then calculated. We employ Dirac's theory of constraints to obtain symplectic and first Hamiltonian operators. Subsequently, we transform the symmetry condition of the reduced system into a skew-factorized form to determine the recursion operator. Applying the recursion operator to the first Hamiltonian operator yields the second Hamiltonian operator. We demonstrate that the reduced system is a bi-Hamiltonian integrable system in the sense of Magri. Lie point symmetries of the reduced system are identified. Finally, we calculate integrals of motion using the inverse Noether theorem and prove that they have the total divergence form.