(2+1)-dimensional bi-Hamiltonian system obtained from symmetry reduction of (3+1)-dimensional Hirota type equation

Abstract

In this work, we show that the integrable (3+1)-dimensional Hirota type nonlinear differential equation reduces to the (2+1)-dimensional Hirota type equation by symmetry reduction. It is proved that starting from the Dirac constraint analysis and degenerate Lagrangian, obtained (2+1)-dimensional equation is also integrable and admit bi-Hamiltonian structure. Moreover, all the parameters defined in four dimensional system like Lagrangian, Hamiltonian operators, Hamiltonian functions and recursion operator have the same result of the three dimensional system after the reduction.In this work, we show that the integrable (3+1)-dimensional Hirota type nonlinear differential equation reduces to the (2+1)-dimensional Hirota type equation by symmetry reduction. It is proved that starting from the Dirac constraint analysis and degenerate Lagrangian, obtained (2+1)-dimensional equation is also integrable and admit bi-Hamiltonian structure. Moreover, all the parameters defined in four dimensional system like Lagrangian, Hamiltonian operators, Hamiltonian functions and recursion operator have the same result of the three dimensional system after the reduction.