Geometric integration of the paraxial equation

Evolution of solitary waves in photovoltaic-photorefractive crystal satisfy the paraxial equation. The paraxial equation is transformed into the symplectic structure of the infinite dimensional Hamiltonian system. The symplectic structure of the paraxial equation is discretizated by the symplectic method. The corresponding symplectic scheme preserves conservation of discrete energy which reflects conservation of energy of the paraxial equation. The symplectic scheme is applied to simulate the solitary wave behaviors of the paraxial equation. Evolution of the solitary waves with the different applied electric field and the different photovoltaic fields are investigated.